diff --git "a/d9FIT4oBgHgl3EQfoiuQ/content/tmp_files/load_file.txt" "b/d9FIT4oBgHgl3EQfoiuQ/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/d9FIT4oBgHgl3EQfoiuQ/content/tmp_files/load_file.txt" @@ -0,0 +1,2177 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf,len=2176 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11319v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='CO] 26 Jan 2023 WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY NEIL LYALL ´AKOS MAGYAR Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆ = ∆1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='×∆d ⊆ Rn, where Rn = Rn1 ×· · ·×Rnd with each ∆i ⊆ Rni a non-degenerate simplex of ni points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We prove that any set S ⊆ Rn, with n = n1 + · · · + nd of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of the configuration ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In particular any such set S ⊆ R2d contains a d-dimensional cube of side length λ, for all λ ≥ λ0(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We also prove analogous results with the underlying space being the integer lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof is based on a weak hypergraph regularity lemma and an associated counting lemma developed in the context of Euclidean spaces and the integer lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Existing Results I: Distances and Simplices in Subsets of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that the upper Banach density of a measurable set S ⊆ Rn is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) δ∗(S) = lim N→∞ sup t∈Rn |S ∩ (t + Q(N))| |Q(N)| , where | · | denotes Lebesgue measure on Rn and Q(N) denotes the cube [−N/2, N/2]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A result of Furstenberg, Katznelson, and Weiss [6] states that if S ⊆ R2 has positive upper Banach density, then its distance set {|x − x′| : x, x′ ∈ S} contains all sufficiently large numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that the distance set of any set of positive Lebesgue measure in Rn automatically contains all sufficiently small numbers (by the Lebesgue density theorem) and that it is easy to construct a set of positive upper density which does not contain a fixed distance by placing small balls centered on an appropriate square grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem A (Furstenberg, Katznelson, and Weiss [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If S ⊆ R2 with δ∗(S) > 0, then there exists a λ0 = λ0(S) such that S is guaranteed to contain pairs of points {x1, x2} with |x2 − x1| = λ for all λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This result was later reproved using Fourier analytic techniques by Bourgain in [1] where he established the following more general result for all configurations of n points in Rn whose affine span is n−1 dimensional, namely for all non-degenerate simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem B (Bourgain [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆ ⊆ Rn be a non-degenerate simplex of n points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If S ⊆ Rn with δ∗(S) > 0, then there exists a threshold λ0 = λ0(S, ∆) such that S contains an isometric copy of λ∆ for all λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that a finite point configuration ∆′ is said to be an isometric copy of λ∆ if there exists a bijection φ : ∆ → ∆′ such that |φ(v) − φ(w)| = λ |v − w| for all v, w ∈ ∆, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' if ∆′ is obtained from λ∆ (the dilation of ∆ by a factor λ) via a rotation and translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Bourgain deduced Theorem B as an immediate consequence of the following stronger quantitative result for measurable subsets of the unit cube of positive measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In the proposition below, and throughout this article, we shall refer to a decreasing sequence {λj}J j=1 as lacunary if λj+1 ≤ λj/2 for all 1 ≤ j < J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition B (Bourgain [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆ ⊆ Rn be a non-degenerate simplex of n points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < δ ≤ 1 there exists a constant J = O∆(δ−3n) such that if 1 ≥ λ1 ≥ · · · ≥ λJ is any lacunary sequence and S ⊆ [0, 1]n with |S| ≥ δ, then there exists 1 ≤ j < J such that S contains an isometric copy of λ∆ for all λ ∈ [λj+1, λj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 11B30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The first and second authors were partially supported by grants NSF-DMS 1702411 and NSF-DMS 1600840, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1 2 NEIL LYALL ´AKOS MAGYAR In [12] the authors provided a short direct proof of Theorem B without using Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is based on the observation that uniformly distributed sets S ⊆ Rd contain the expected “number” of isometric copies of dilates λ∆ and that all sets of positive upper density become uniformly distributed at sufficiently large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' However, for the purposes of this paper it will be important to recall Bourgain’s indirect approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' To see that Proposition B implies Theorem B notice that if Theorem B were not to hold for some set S ⊆ Rn of upper Banach density δ∗(S) > δ > 0, then there must exist a lacunary sequence λ1 ≥ · · · ≥ λJ ≥ 1, with J the constant in Proposition B, such that S does not contain an isometric copy of λj∆ for any 1 ≤ j ≤ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Taking a sufficiently large cube Q with side length N ≥ λ1 and |S ∩ Q| ≥ δ|Q| and scaling back Q → [0, 1]n contradicts Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We further note that by taking λj = 2−j in Proposition B we obtain the following “Falconer-type” result for subsets of [0, 1]n of positive Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If ∆ ⊆ Rn is a non-degenerate simplex of n points, then any S ⊆ [0, 1]n with |S| > 0 will necessarily contain an isometric copy of λ∆ for all λ in some interval of length at least exp(−C∆|S|−3n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Bourgain further demonstrated in [1] that no result along the lines of Theorem B can hold for configurations that contain any three points in arithmetic progression along a line, specifically showing that for any n ≥ 1 there are sets of positive upper Banach density in Rn which do not contain an isometric copy of configurations of the form {0, y, 2y} with |y| = λ for all sufficiently large λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This should be contrasted with the following remarkable result of Tamar Ziegler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem C (Ziegler [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let F be any configuration of k points in Rn with n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If S ⊆ Rn has positive upper density, then there exists a threshold λ0 = λ0(S, F) such that Sε contains an isometric copy of λF for all λ ≥ λ0 and any ε > 0, where Sε denotes the ε-neighborhood of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Bourgain’s example was later generalized by Graham [9] to establish that the condition that ε > 0 in Theorem C is necessary and cannot be strengthened to ε = 0 for any given non-spherical configuration F in Rn for any n ≥ 1, that is for any finite configuration of points that cannot be inscribed in some sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We note that the sets constructed by Bourgain and Graham have the property that for any ε > 0 their ε-neighborhoods will contain arbitrarily large cubes and hence trivially satisfy Theorem C with λ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is natural to ask if any spherical configuration F, beyond the known example of simplices, has the property that every positive upper Banach density subset of Rn, for some sufficiently large n, contains an isometric copy of λF for all sufficiently large λ, and even to conjecture that this ought to hold for all spherical configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The first breakthrough in this direction came in [12] when the authors established this for configurations of four points forming a 2-dimensional rectangle in R4 and more generally for any configuration that is the direct product of two non-degenerate simplices in Rn for suitably large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The purpose of this article is to present a strengthening of the results in [12] and to extend them to cover configurations with a higher dimensional product structure in both the Euclidean and discrete settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' New Results I: Rectangles and Products of Simplices in Subsets of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The first main result of this article is the following Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let R be 2d points forming the vertices of a fixed d-dimensional rectangle in R2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (i) If S ⊆ R2d has positive upper Banach density, then there exists a threshold λ0 = λ0(S, R) such that S contains an isometric copy of λR for all λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (ii) For any 0 < δ ≤ 1 there exists a constant c = c(δ, R) > 0 such that any S ⊆ [0, 1]2d with |S| ≥ δ is guaranteed to contain an isometric copy of λR for all λ in some interval of length at least c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover, if R has sidelengths given by t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td, then the isometric copies of λR in both (i) and (ii) above can all be realized in the special form {x11, x12} × · · ·× {xd1, xd2} ⊆ R2 × · · ·× R2 with each |xj2 − xj1| = λtj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The multi-dimensional extension of Szemer´edi’s theorem on arithmetic progressions in sets of positive density due to Furstenberg and Katznelson [5] implies, and is equivalent to the fact, that there are isometric copies of λR in S for arbitrarily large λ, with sides parallel to the coordinate axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 states that there is an isometric copy of λR in S for every sufficiently large λ, but only with sides parallel to given WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 3 2-dimensional coordinate subspaces which provides an extra degree of freedom for each side vector of the rectangle R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A weaker version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, with R2d replaced with R5d, was later established by Durcik and Kovaˇc in [4] using an adaptation of arguments of the second author with Cook and Pramanik in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This approach also makes direct use of the full strength of the multi-dimensional Szemer´edi theorem and as such leads to quantitatively weaker results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our arguments work for more general patterns where d-dimensional rectangles are replaced with direct products of non-degenerate simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆ = ∆1 × · · · × ∆d ⊆ Rn, where Rn = Rn1 × · · · × Rnd and each ∆j ⊆ Rnj is a non-degenerate simplex of nj points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (i) If S ⊆ Rn has positive upper Banach density, then there exists a threshold λ0 = λ0(S, ∆) such that S contains an isometric copy of λ∆ for all λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (ii) For any 0 < δ ≤ 1 there exists a constant c = c(δ, ∆) > 0 such that any S ⊆ [0, 1]n with |S| ≥ δ is guaranteed to contain an isometric copy of λ∆ for all λ in some interval of length at least c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover the isometric copies of λ∆ in both (i) and (ii) above can all be realized in the special form ∆′ 1 × · · × ∆′ d with each ∆′ j ⊆ Rnj an isometric copy of λ∆j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Quantitative Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A careful analysis of our proof reveals that the constant c(δ, ∆) can be taken greater than Wd(C′ ∆δ−3n1···nd)−1 where Wk(m) is a tower of exponentials defined by W1(m) = exp(m) and Wk+1(m) = exp(Wk(m)) for k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Existing Results II: Distances and Simplices in Subsets of Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The problem of counting isomet- ric copies of a given non-degenerate simplex in Zn (with one vertex fixed) has been extensively studied via its equivalent formulation as the number of ways a quadratic form can be represented as a sum of squares of linear forms, see [11] and [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This was exploited by the second author in [16] and [17] to establish analogous results to those described in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 above for subsets of the integer lattice Zn of positive upper density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that the upper Banach density of a set S ⊆ Zn is analogously defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) δ∗(S) = lim N→∞ sup t∈Rn |S ∩ (t + Q(N))| |Q(N)| , where | · | now denotes counting measure on Zn and Q(N) the discrete cube [−N/2, N/2]n ∩ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In light of the fact that any pairs of distinct points {x1, x2} in Zn has the property that the square of the distance between them |x2 − x1|2 is always a positive integer we introduce the convenient notation √ N := {λ : λ > 0 and λ2 ∈ Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem A′ (Magyar [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < δ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If S ⊆ Z5 has upper Banach density at least δ, then there exists an integer q0 = q0(δ) and λ0 = λ0(S) such that S contains pairs of points {x1, x2} with |x2 − x1| = q0λ for all λ ∈ √ N with λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem B′ (Magyar [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < δ ≤ 1 and ∆ ⊆ Z2n+3 be a non-degenerate simplex of n points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (i) If S ⊆ Z2n+3 has upper Banach density at least δ, then there exists an integer q0 = O(exp(C∆δ−13n)) and λ0 = λ0(S, ∆) such that S contains an isometric copy of q0λ∆ for all λ ∈ √ N with λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (ii) If N ≥ exp(2C∆δ−13n), then any S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , N}2n+3 with cardinality |S| ≥ δN 2n+3 will necessarily contain an isometric copy of λ∆ for some λ ∈ √ N with 1 ≤ λ ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that the fact that S ⊆ Zn could fall entirely into a ��xed congruence class of some integer 1 ≤ q ≤ δ−1/n ensures that the q0 that appears in Theorems A′ and B′ above must be divisible by the least common multiple of all integers 1 ≤ q ≤ δ−1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed if S = (qZ)n with 1 ≤ q ≤ δ−1/n then S has upper Banach density at least δ, however the distance between any two points x, y ∈ S is of the form |x − y| = qλ for some λ ∈ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 4 NEIL LYALL ´AKOS MAGYAR However, in both Theorems A′ and Part (i) of Theorem B′, one can take q0 = 1 if the sets S are assumed to be suitably uniformly distributed on congruence classes of small modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This leads via an easy density increment strategy to short new proofs, see [14] for Theorem A′ and Section 8 for Part (i) of Theorem B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The original argument in [17] deduced Theorem B′ from the following discrete analogue of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition B′ (Magyar [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆ ⊆ Z2n+3 be a non-degenerate simplex of n points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < δ ≤ 1 there exist constants J = O∆(δ−3n) and q0 = O(exp(C∆δ−13n)) such that if N ≥ λ1 ≥ · · ≥ λJ ≥ 1 is any lacunary sequence in q0 √ N and S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , N}2n+3 with cardinality |S| ≥ δN 2n+3, then S will necessarily contain an isometric copy of λj∆ for some 1 ≤ j ≤ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' To see that Proposition B′ implies Theorem B′ notice that if Part (i) of Theorem B′ were not to hold for some set S ⊆ Z2n+3 of upper Banach density δ∗(S) > δ > 0 with q0 from Proposition B′, then there must exist a lacunary sequence λ1 ≥ · · · ≥ λJ ≥ 1 in q0 √ N, with J the constant from Proposition B′, such that S does not contain an isometric copy of λj∆ for any 1 ≤ j ≤ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since we can find a sufficiently large cube Q with integer side length N that is divisible by q0 and greater than λ1 such that |S ∩ Q| ≥ δ|Q| , this contradicts Proposition B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Part (ii) of Theorem B′ follows from Proposition B′ by taking λj = 2J−jq0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' New Results II: Rectangles and Products of Simplices in Subsets of Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will also establish the following discrete analogues of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < δ ≤ 1 and R be 2d points forming the vertices of a d-dimensional rectangle in Z5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (i) If S ⊆ Z5d has upper Banach density at least δ, then there exist integers q0 = q0(δ, R) and λ0 = λ0(S, R) such that S contains an isometric copy of q0λR for all λ ∈ √ N with λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (ii) There exists a constant N(δ, R) such that if N ≥ N(δ, R), then any S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , N}5d with cardinality |S| ≥ δN 5d will necessarily contain an isometric copy of λR for some λ ∈ √ N with 1 ≤ λ ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If R has side lengths given by t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td, then each of the isometric copies in (i) and (ii) above can be realized in the form {x11, x12} × · · · × {xd1, xd2} ⊆ Z5 × · · · × Z5 with each |xj2 − xj1| = q0λtj and λtj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our arguments again work for more general patterns where d-dimensional rectangles are replaced with direct products of non-degenerate simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < δ ≤ 1 and ∆ = ∆1 × · · · × ∆d ⊆ Zn, where Zn = Z2n1+3 × · · · × Z2nd+3 and each ∆i ⊆ Z2ni+3 is a non-degenerate simplex of ni points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (i) If S ⊆ Zn has upper Banach density at least δ, then there exist integers q0 = q0(δ, ∆) and λ0 = λ0(S, ∆) such that S contains an isometric copy of q0λ∆ for all λ ∈ √ N with λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (ii) There exists a constant N(δ, ∆) such that if N ≥ N(δ, ∆), then any S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , N}n with cardinality |S| ≥ δN n will necessarily contain an isometric copy of λ∆ for some λ ∈ √ N with 1 ≤ λ ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover, each of the isometric copies in (i) and (ii) above can be realized in the special form ∆′ 1 × · · · × ∆′ d with each ∆′ i ⊆ Z2ni+3 an isometric copy of q0λ∆j and λ∆j, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Quantitative Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A careful analysis of our proof reveals that the constant q0(δ, ∆) (and consequently also N(δ, ∆)) can be taken less than Wd(C′ ∆δ−13n1···nd) where Wk(m) is a tower of exponentials defined by W1(m) = exp(m) and Wk+1(m) = exp(Wk(m)) for k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Notations and Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will consider the parameters d, n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , nd fixed and will not indicate the dependence on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus we will write f = O(g) if |f| ≤ C(n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , nd)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If the implicit constants in our estimates depend on additional parameters ε, δ, K, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' the we will write f = Oε,δ,K,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will use the notation f ≪ g to indicate that |f| ≤ c g for some constant c > 0 sufficiently small for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Given an ε > 0 and a (finite or infinite) sequence L0 ≥ L1 ≥ · · · > 0, we will say that the sequence is ε-admissible if Lj/Lj+1 ∈ N and Lj+1 ≪ ε2Lj for all j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover, if q ∈ N is given and Lj ∈ N for all 1 ≤ j ≤ J, then we will call the sequence L0 ≥ L1 ≥ · · · ≥ LJ (ε, q)-admissible if in addition LJ/q ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Such sequences of scales will often appear in our statements both in the continuous and the discrete case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our proofs are based on a weak hypergraph regularity lemma and an associated counting lemma developed in the context of Euclidean spaces and the integer lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In Section 2 we introduce our approach in the WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 5 model case of finite fields and prove an analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In Section 3 we review Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 for a single simplex and ultimately establish the base case of our general inductive approach to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In Section 4 we address Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 for the direct product of two simplices, this provides a new proof (and strengthening) of the main result of [12] and serves as a gentle preparation for the more complicated general case which we present in the Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4 is outlined in Sections 6 and 7, while a short direct proof of Part (i) of Theorem B′ is presented in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Model case: vector spaces over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In this section we will illustrate our general method by giving a complete proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 in the model setting of Fn q where Fq denotes the finite field of q elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We do this as the notation and arguments are more transparent in this setting yet many of the main ideas are still present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We say that two vectors u, v ∈ Fn q are orthogonal, if x·y = 0, where “·” stands for the usual dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A rectangle in Fn q is then a set R = {x1, y1} × · · ·× {xn, yn} with side vectors yi − xi being pairwise orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The finite field analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 is the following Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < δ ≤ 1 there exists an integer q0 = q0(δ) with the following property: If q ≥ q0 and t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td ∈ F∗ q, then any S ⊆ F2d q with |S| ≥ δ q2d will contain points {x11, x12} × · · · × {xd1, xd2} ⊆ V1 × · · · × Vd with |xj2 − xj1|2 = tj for 1 ≤ j ≤ d where we have written F2d q = V1 × · · · × Vd with Vj ≃ F2 q pairwise orthogonal coordinate subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Overview of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Write F2d q = V1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × Vd with Vj ≃ F2 q pairwise orthogonal coordinate subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any t := (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td) ∈ F∗ q and S ⊆ F2d q we define Nt(1S) := Ex1∈V 2 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xd∈V 2 d � (ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',ℓd)∈{1,2}d 1S(x1ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xdℓd) d � j=1 σtj(xj2 − xj1) where we used the shorthand notation xj := (xj1, xj2) for each 1 ≤ j ≤ d and the averaging notation: Ex∈Af(x) := 1 |A| � x∈A f(x) for a finite set A ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We have also used the notation σt(x) = � q if |x|2 = t 0 otherwise for each t ∈ F∗ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that the function σt may be viewed as the discrete analogue of the normalized surface area measure on the sphere of radius √ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is well-known, see [10], that Ex∈F2q σt(x) = 1 + O(q−1/2) and for all ξ ̸= 0 one has ˆσt(ξ) := Ex∈F2q σt(x) e2πi x·ξ q = O(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if Nt(1S) > 0, then this implies that S contains a rectangle of the form {x11, x12}×· · ·×{xd1, xd2} with xj1, xj2 ∈ Vj and |xj2 − xj1|2 = tj for 1 ≤ j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our approach to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 in fact establishes the following quantitatively stronger result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < ε ≤ 1 there exists an integer q0 = q0(ε) with the following property: If q ≥ q0, then for any S ⊆ F2d q and t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td ∈ F∗ q one has Nt(1S) > � |S| q2d �2d − ε where we have written F2d q = V1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × Vd with Vj ≃ F2 q pairwise orthogonal coordinate subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 6 NEIL LYALL ´AKOS MAGYAR A crucial observation in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 is that the averages Nt(1S) can be compared to ones which can be easily estimated from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We define, for any S ⊆ F2d q , the (unrestricted) count M(1S) := Ex1∈V 2 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xd∈V 2 d � (ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',ℓd)∈{1,2}d 1S(x1ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xdℓd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is easy to see, by carefully applying Cauchy-Schwarz d times to Ex11∈V1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xd1∈Vd1S(x11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd1), that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) M(1S) ≥ � |S| q2d �2d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our approach to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 therefore reduces to establishing that for any ε > 0 one has (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) Nt(1S) = M(1S) + O(ε) + Oε(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The validity of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) will follow immediately from the d = k case of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' However, before we can state this counting lemma we need to introduce some further notation from the theory of hypergraphs, notation that we shall ultimately make use of throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Hypergraph Notation and a Counting Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In order to streamline our notation we will make use the language of hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For J := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', d} and 1 ≤ k ≤ d, we let Hd,k = {e ⊆ J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' |e| = k} denote the full k-regular hypergraph on the vertex set J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For K := {jl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' j ∈ J, l ∈ {1, 2}} we define the projection π : K → J as π(jl) := j and use this in turn to define the hypergraph bundle H2 d,k := {e ⊆ K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' |e| = |π(e)| = k} using the shorthand notation 2 = (2, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 2) to indicate that |π−1(j)| = 2 for all j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Notice when k = d then Hd,d consists of one element, the set e = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', d}, and H2 d,d = { {1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , dld};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ld) ∈ {1, 2}d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let V := F2d q and V = V1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × Vd with Vj ≃ F2 q pairwise orthogonal coordinate subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For a given x = (x11, x12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd1, xd2) ∈ V 2 with xj1, xj2 ∈ Vj and a given edge e = {1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , dld}, we write xe := (x1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xdld).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that the map x → xe defines a projection πe : V 2 → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' With this notation, we can clearly now write Nt(1S) = Ex∈V 2 � e∈H2 d,d 1S(xe) d � j=1 σtj(xj2 − xj1) M(1S) = Ex∈V 2 � e∈H2 d,d 1S(xe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Now for any 1 ≤ k ≤ d and any edge e′ ∈ Hd,k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e′ ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , d}, |e′| = k, we let Ve′ := � j∈e′ Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For every x ∈ V 2 and e ∈ H2 d,k, we define xe := πe(x) where πe : V 2 → Vπ(e) is the natural projection map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our key counting lemma, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 below, which we will establish by induction on 1 ≤ k ≤ d below, is then the statement that given a family of functions fe : Vπ(e) → [−1, 1], e ∈ H2 d,k, the averages (generalizing those discussed above) which are defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3) Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) := Ex∈V 2 � e∈H2 d,k fe(xe) d � j=1 σtj(xj2 − xj1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) M(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) := Ex∈V 2 � e∈H2 d,k fe(xe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' are approximately equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Specifically, one has WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 7 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 (Counting Lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 1 ≤ k ≤ d and 0 < ε ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any collection of functions fe : Vπ(e) → [−1, 1] with e ∈ H2 d,k one has (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = M(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) + O(ε) + Oε(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If we apply this Proposition with d = k and fe = 1S for all e ∈ H2 d,d, then Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 clearly follows given the lower bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will establish Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 by inducting on 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For k = 1 the result follows from the basic observation that if f1, f2 : F2 q → [−1, 1] and let t ∈ F∗ q, then Ex1,x2∈F2q f1(x1)f2(x2) σt(x2 − x1) = � ξ∈F2q ˆf1(ξ) ˆf2(ξ)ˆσt(ξ) = ˆf1(0) ˆf2(0) + O(q−1/2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) = Ex1,x2∈F2q f1(x1)f2(x2) + O(q−1/2) by the properties of the function ˆσ given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' To see how this implies Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 for k = 1 we note that since H2 d,1 = {jl : 1 ≤ j ≤ d, 1 ≤ l ≤ 2} it follows that Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,1) = d � j=1 Exj1,xj2∈F2q fj1(xj1)fj2(xj2) σt(xj2 − xj1) = d � j=1 Exj1,xj2∈F2q fj1(xj1)fj2(xj2) + O(q−1/2) = M(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,1) + O(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The induction step has two main ingredients, the first is an estimate of the type which is often referred to as a generalized von-Neumann inequality, namely Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any collection of functions fe : Vπ(e) → [−1, 1] with e ∈ H2 d,k one has (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) ≤ min e∈H2 d,k ∥fe∥□(Vπ(e)) + O(q−1/2) where for any e ∈ H2 d,k and f : Vπ(e) → [−1, 1] we define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) ∥f∥2k □(Vπ(e)) := Ex∈V 2 π(e) � e∈H2 d,k f(xe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The corresponding inequality for the multilinear expression M(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k), namely the fact that M(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) ≤ � e∈H2 d,k ∥fe∥□(Vπ(e)) ≤ min e∈H2 d,k ∥fe∥□(Vπ(e)) is well-known and is referred to as the Gowers-Cauchy-Schwarz inequality [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The second and main ingredient is an approximate decomposition of a graph to simpler ones, and is essentially the so-called weak (hypergraph) regularity lemma of Frieze and Kannan [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We choose to state this from a somewhat more abstract/probabilistic point of view, a perspective that will be particularly helpful when we consider our general results in the continuous and discrete settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will first introduce this in the case d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A bipartite graph with (finite) vertex sets V1, V2 is a set S ⊆ V1 × V2 and a function f : V1 × V2 → R may be viewed as weighted bipartite graph with weights f(x1, x2) on the edges (x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If P1 and P2 are partitions of V1 and V2 respectively then P = P1 × P2 is a partition V1 × V2 and we let E(f|P) denote the function that is constant and equal to Ex∈Af(x) on each 8 NEIL LYALL ´AKOS MAGYAR atom A = A1 × A2 of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The weak regularity lemma states that for any ε > 0 and for any weighted graph f : V1 × V2 → [−1, 1] there exist partitions Pi of Vi with |Pi| ≤ 2O(ε−2) for i = 1, 2, so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) |Ex1∈V1Ex2∈V2(f − E(f|P))(x1, x2) 1U1(x1)1U2(x2)| ≤ ε for all U1 ⊆ V1 and U2 ⊆ V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Informally this means that the graph f can be approximated with precision ε with the “low complexity” graph E(f, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If we consider the σ-algebras Bi generated by the partitions Pi and the σ-algebra B = B1 ∨ B2 generated by P1 × P2 then we have E(f|B), the so-called conditional expectation function of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover it is easy to see, using Cauchy-Schwarz, that estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) ∥f − E(f|B1 ∨ B2)∥□(V1×V2) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' With this more probabilistic point of view the weak regularity lemma says that the function f can be approximated with precision ε by a low complexity function E(f|B1 � B2), corresponding to σ-algebras Bi on Vi generated by O(ε−2) sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This formulation is also referred to as a Koopman- von Neumann type decomposition, see Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will need a natural extension to k-regular hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' See [22, 8], and also [2] for extension to sparse hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Given an edge e′ ∈ Hd,k of k elements we define its boundary ∂e′ := {f′ ∈ Hd,k−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f′ ⊆ e′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For each f′ = e′\\{j} ∈ ∂e′ let B′ f be a σ-algebra on Vf′ := � j∈f′ Vj and ¯Bf′ := {U × Vj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' U ∈ Bf′} denote its pull-back over the space Ve′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The σ-algebra B = � f′∈∂e′ Bf′ is the smallest σ-algebra on ∂e′ containing ¯Bf′ for all f′ ∈ ∂e′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that the atoms of B are of the form A = � f′∈∂e′ Af′ where Af′ is an atom of ¯Bf′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We say that the complexity of a σ-algebra Bf′ is at most m, and write complex(Bf′) ≤ m, if it is generated by m sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (Weak hypergraph regularity lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 1 ≤ k ≤ d and fe : Vπ(e) → [−1, 1] be a given function for each e ∈ H2 d,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any ε > 0 there exists σ-algebras Bf′ on Vf′ for each f′ ∈ Hd,k−1 such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) complex(Bf′) = O(ε−2k+1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) ∥fe − E(fe| � f′∈∂π(e) Bf′)∥□(Vπ(e)) ≤ ε for all e ∈ H2 d,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof of Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 are presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We close this subsection by demon- strating how these lemmas can be combined to establish Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ε > 0, 2 ≤ k ≤ d and assume that the lemma holds for k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 that there exists σ-algebras Bf′ of complexity O(ε−2k+1) on Vf′ for each f′ ∈ Hd,k−1 for which (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) holds for all e ∈ H2 d,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For each e ∈ H2 d,k we let ¯fe := E(fe| � f′∈∂π(e) Bf′) and write fe = ¯fe + he.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and multi-linearity we have that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13) Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = Nt( ¯fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) + O(ε) + O(q−1/2) and also by the Gowers-Cauchy-Schwarz inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14) M(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = M( ¯fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) + O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The conditional expectation functions ¯fe are linear combinations of the indicator functions 1Ae of the atoms Ae of the σ-algebras Be := � f′∈∂π(e) Bf′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the number of terms in this linear combination is at most 2Cε−2k+1 , with coefficients at most 1 in modulus, plugging these into the multi-linear expressions Nt( ¯fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) and M( ¯fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) one obtains a linear combination of expressions of the form Nt(1Ae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) and M(1Ae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) respectively with each Ae being an atoms of Be for all e ∈ H2 d,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The key observation is that these expressions are at level k − 1 instead of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed, 1Ae = � f′∈∂π(e) 1Aef′ where Aef′ = A′ ef′ × Vj, with A′ ef′ being an atom of Bf′ when f′ = π(e)\\{j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If e = (j1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , jl, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , jklk), WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 9 let pf′(e) := (j1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , jklk) ∈ H2 d,k−1, obtained from e by removing the jl-entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then we have 1Aef′ (xe) = 1A′ ef′(xp′ f(e)) since xjl ∈ Vj, and hence 1Ae(xe) = � f′∈∂π(e) 1A′ ef′ (xp′ f(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It therefore follows that Nt(1Ae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = Ex∈V 2 � e∈H2 d,k � f′∈∂π(e) 1A′ ef′(xpf′ (e)) d � j=1 σtj(xj2 − xj1) = Ex∈V 2 � f∈H2 d,k−1 � e∈H2 d,k, f′∈∂π(e) pf′ (e)=f 1A′ ef′(xpf′ (e)) � �� � =:gf d � j=1 σtj(xj2 − xj1) = Nt(gf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ H2 d,k−1) and similarly that M(1Ae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = M(gf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ H2 d,k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It then follows from the induction hypotheses that Nt(1Ae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = M(1Ae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) + O(ε1) + Oε1(q−1/2) for any ε1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If we choose ε1 := 2−C1 ε−2k+1 , with C1 ≫ 1 sufficiently large, then ε1 2Cε−2k+1 = O(ε) and it follows that Nt( ¯fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = M( ¯fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) + O(ε) + Oε(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This, together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14), establishes that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) hold for d = k as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We start by observing the following consequence of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6), namely that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) ���Ex1,x2∈F2qf1(x1)f2(x2)σt(x2 − x1) ��� 2 ≤ Ex1,x2∈F2qf1(x1)f1(x2) + O(q−1/2) for any f1, f2 : F2 q → [−1, 1] and t ∈ F∗ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Now, fix an edge, say e0 = (11, 21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', k1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Partition the edges e ∈ H2 d,k into three groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' the first group consisting of edges e for which 1 /∈ π(e), the second where 11 ∈ e and write e = (11, e′) with e′ ∈ H2 d−1,k−1 and the third when 12 ∈ e, using the notation H2 d−1,k−1 := {(j2l2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , jklk)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Accordingly we can write (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = Ex∈V 2 � 1/∈π(e) fe(xe) � e′∈H2 d−1,k−1 f(11,e′)(x11, xe′) � e′∈H2 d−1,k−1 f(12,e′)(x12, xe′) d � j=1 σtj(xj2−xj1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If for given x ∈ V1 and x′ = (x21, x22, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd1, xd2) ∈ V 2 2 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × V 2 d we define g1(x, x′) := � e′∈H2 d−1,k−1 f(11,e′)(x, xe′) and g2(x, x′) := � e′∈H2 d−1,k−1 f(12,e′)(x, xe′) then we can write Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) = Ex21,x22,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xd1,xd2 � 1/∈π(e) fe(xe) d � j=2 σtj(xj2 − xj1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) × Ex11,x12 g1(x11, x′)g2(x12, x′) σt1(x12 − x11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) we can estimate the inner sum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) by the square root of Ex11,x12 g1(x11, x′)g1(x12, x′) + O(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 10 NEIL LYALL ´AKOS MAGYAR Thus by Cauchy-Schwarz, and the fact that fe : Vπ(e) → [−1, 1] for all e ∈ H2 d,k, we can conclude that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='18) Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k)2 ≤ Ex11,x12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xd1,xd2 � e′∈H2 d−1,k−1 f(11,e′)(x11, xe′)f(11,e′)(x12, xe′) d � j=2 σtj(xj2 − xj2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The expression on the right hand side of the inequality above is similar to that in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) except for the following changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The functions fe for 1 /∈ e are eliminated i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' replaced by 1, as well as the factor σt1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The functions f(12,e′), are replaced by f(11,e′) for all e′ ∈ H2 d−1,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Repeating the same procedure for j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , k one eliminates all the factors σtj for 1 ≤ j ≤ k, moreover all the functions fe for edges e such that j /∈ π(e) for some 1 ≤ j ≤ k, which leaves only the edges e so that π(e) = (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , k), moreover for such edges the functions fe are eventually replaced by fe0 = f11,21,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The factors σtj(xj2 −xj1) are not changed for j > k however as the function fe0 does not depend on the variables xjl for j > k, averaging over these variables gives rise to a factor of 1 + O(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus one obtains the following final estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='19) Nt(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k)2k ≤ Ex11,x12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xk1,xk2 � π(e)=(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',k) fe0(xe) + O(q−1/2) = ∥fe0∥2k □(Vπ(e0)) + O(q−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This proves the lemma, as it is clear that the above procedure can be applied to any edge in place of e0 = (11, 21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', k1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For a function fe : Vπ(e) → [−1, 1] and a σ-algebra Bπ(e) on Vπ(e) define the energy of fe with respect to Bπ(e) as E(fe, Bπ(e)) := ∥E(fe|Bπ(e))∥2 2 = Ex∈Vπ(e) |E(fe|Bπ(e))(x)|2, and for a family of functions fe and σ-algebras Bπ(e), e ∈ H2 d,k its total energy as E(fe, Bπ(e);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ H2 d,k) := � e∈H2 d,k E(fe, Bπ(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will show that if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) does not hold for a family of σ-algebras Bπ(e) = � f′∈∂π(e) Bf′ , then the σ-algebras Bf′ can be refined so that the total energy of the system increases by a quantity depending only on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the functions fe are bounded the total energy of the system is O(1), the energy increment process must stop in Oε(1) steps, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The idea of this procedure appears already in the proof of Szemer´edi’s regularity lemma [20], and have been used since in various places [7, 22, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Initially set Bf′ := {∅, Vf′} and hence Bπ(e) = {∅, Vπ(e)} to be the trivial σ-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Assume that in general (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) does not hold for a family of σ-algebras Bf′, with f′ ∈ Hd,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then there exists an edge e ∈ H2 d,k so that ∥ge∥□(Vπ(e)) ≥ ε, with ge := fe − E(fe|Bπ(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let e = (11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , k1) for simplicity of notation, hence π(e) = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then, with notation x′ = (x12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xk2), one has ε2k ≤ ∥ge∥2k □(Vπ(e)) = Ex11,x12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xk1,xk2 � l1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',lk=1,2 ge(x1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xklk) ≤ Ex12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xk2 ���Ex11,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xk1ge(x11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xk1) k � j=1 hj,x′(x11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xj−1 1, xj+1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xk1) ��� for some functions hj,x′ that are bounded by 1 in magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed if and edge e ̸= (11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , k1) then xe does not depend at least one of the variables xj1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus there must be an x′ for which the inner sum in the above expression is at least ε2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Fix such an x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Decomposing the functions hj,x′ into their positive and negative parts and then writing them as an average of indicator functions, one obtains that there sets Bj ⊆ Vπ(e)\\{j} such that ���Ex11,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xk1ge(x11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xk1) k � j=1 1Bj(x11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xj−1 1, xj+1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xk1) ��� ≥ 2−k ε2k WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 11 which can be written more succinctly, using the inner product notation, as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='20) ���⟨fe − E(fe|Bπ(e)), k � j=1 1Bj⟩ ��� ≥ 2−k ε2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For f′ = ∂π(e)\\{j} let B′ f′ be the σ-algebra generated by Bf′ and the set Bj and let B′ π(e) := � f′∈∂π(e) B′ f′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the functions 1Bj are measurable with respect to the σ-algebra B′ π(e) for all 1 ≤ j ≤ k, we have that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='21) ⟨fe − E(fe|B′ π(e)), k � j=1 1Bj⟩ = 0 and hence, by Cauchy-Schwarz, that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='22) ∥E(fe|B′ π(e)) − E(fe|Bπ(e))∥2 2 = ∥E(fe|B′ π(e))∥2 2 − ∥E(fe|Bπ(e))∥2 2 ≥ 2−2k ε2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that the first equality above follows from the fact that conditional expectation function E(f|B) is the orthogonal projection of f to the subspace of B-measurable functions in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This also implies that energy of a function is always increasing when the underlying σ-algebra is refined, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='22) tells us that the energy of fe is increased by at least ck ε2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For f′ /∈ ∂π(e) we set B′ f′ := Bf′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then the total energy of the family fe with respect to the system B′ π(e) = � f′∈∂π(e) B′ f′, e ∈ H2 d,k is also increased by at least ck ε2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is clear that the complexity of the σ-algebras Bf′ are increased by at most 1, hence, as explained above, the lemma follows by applying this energy increment process at most O(ε−2k+1) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The base case of an inductive strategy to establish Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 In this section we will ultimately establish the base case of our more general inductive argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We however start by giving a quick review of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 when d = 1 (which contains both Theorem B and Corollary B as stated in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1), namely the case of a single simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This was originally addressed in [1] and revisited in [12] and [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A Single Simplex in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q ⊆ Rn be a fixed cube and let l(Q) denotes its side length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆0 = {v1 = 0, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , vn} ⊆ Rn be a fixed non-degenerate simplex and define tkl := vk · vl for 2 ≤ k, l ≤ n where “ · ” is the dot product on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Given λ > 0, a simplex ∆ = {x1 = 0, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xn} ⊆ Rn is isometric to λ∆0 if and only if xk · xl = λ2tkl for all 2 ≤ k, l ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus the configuration space Sλ∆0 of isometric copies of λ∆0 is a non-singular real variety given by the above equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let σλ∆0 be natural normalized surface area measure on Sλ∆0, described in [1], [12], and [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is clear that the variable x1 can be replaced by any of the variables xi by redefining the constants tkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any family of functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn : Q → [−1, 1] and 0 < λ ≪ l(Q) we define the multi-linear expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) N 1 λ∆0,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn) := x1∈Q ˆ x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xn f1(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' fn(xn) dσλ∆0(x2 − x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xn − x1) dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We note that all of our functions are 1-bounded and both integrals, in fact all integrals in this paper, are normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that we are using the normalized integral notation ffl A f := 1 |A| ´ A f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the normalized measure σλ∆0 is supported on Sλ∆0 we will not indicate the support of the variables (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xn) explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note also that if S ⊆ Q is a measurable set and N 1 λ∆0,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) > 0 then S must contain an isometric copy of λ∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The following proposition (with Q = [0, 1]n) is a quantitatively stronger version of Proposition B that appeared in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and hence immediately establishes Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 for d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < ε ≤ 1 there exists an integer J = O(ε−2 log ε−1) with the following property: Given any lacunary sequence l(Q) ≥ λ1 ≥ · · · ≥ λJ and S ⊆ Q, there is some 1 ≤ j < J such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) N 1 λ∆0,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) > � |S| |Q| �n − ε for all λ ∈ [λj+1, λj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 12 NEIL LYALL ´AKOS MAGYAR Our approach to establishing Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 is to compare the above expressions to simpler ones for which it is easy to obtain lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Given a scale 0 < λ ≪ l(Q) we define the multi-linear expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3) M1 λ,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn) := t∈Q x1,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xn∈t+Q(λ) f1(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' fn(xn) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dxn dt where Q(λ) = [− λ 2 , λ 2 ]n and t + Q(λ) is the shift of the cube Q(λ) by the vector t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if S ⊆ Q is a set of measure |S| ≥ δ|Q| for some δ > 0, then for a given ε > 0, H¨older implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) M1 λ,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) = t∈Q � x∈t+Q(λ) 1S(x) dx �n dt ≥ � t∈Q x∈t+Q(λ) 1S(x) dx dt �n ≥ δn − O(ε), for all scales 0 < λ ≪ ε l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that for any ε > 0 we call a sequence L1 ≥ · · · ≥ LJ ε-admissible if Lj/Lj+1 ∈ N and Lj+1 ≪ ε2Lj for all 1 ≤ j < J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that given any lacunary sequence l(Q) ≥ λ1 ≥ · · · ≥ λJ′ with J′ ≫ (log ε−1) J, one can always finds an ε-admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJ such that for each 1 ≤ j < J the interval [Lj+1, Lj] contains at least two consecutive elements from the original lacunary sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In light of this observation, and the one above regarding a lower bound for M1 λ,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S), our proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 reduces to establishing the following “counting lemma”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer J1 = O(ε−2) such that for any ε-admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJ1 and S ⊆ Q there is some 1 ≤ j < J1 such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) N 1 λ∆0,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) = M1 λ,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) + O(ε) for all λ ∈ [Lj+1, Lj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There are two main ingredients in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, this will be typical to all of our arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The first ingredient is a result which establishes that the our multi-linear forms N 1 λ∆0,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn) are controlled by an appropriate norm which measures the uniformity of distribution of functions f : Q → [−1, 1] with respect to particular scales L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This is analogous to estimates in additive combinatorics [8] which are often referred to as generalized von-Neumann inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The result below was proved in [12] for Q = [0, 1]n, however a simple scaling of the variables xi transfers the result to an arbitrary cube Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 (A Generalized von-Neumann inequality [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ε > 0, 0 < λ ≪ l(Q), and 0 < L ≪ ε6λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any collections of functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn : Q → [−1, 1] we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) |N 1 λ∆0,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn)| ≤ min i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',n ∥fi∥U1 L(Q) + O(ε) where for any f : Q → [−1, 1] we define (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) ∥f∥2 U1 L(Q) := t∈Q ��� x∈t+Q(L) f(x) dx ��� 2 dt with t + Q(L) denoting the shift of the cube Q(L) = [− L 2 , L 2 ]n by the vector t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The corresponding inequality for the multilinear expression M1 λ,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn), namely the fact that M1 λ,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn) ≤ min i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',n ∥fi∥U1 L(Q) + O(ε) whenever 0 < L ≪ ε6λ follows easily from Cauchy-Schwarz together with the simple observation that ∥f∥U1 L(Q) ≤ ∥f∥U1 L′(Q) + O(ε) whenever L′ ≪ εL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The second key ingredient, proved in [13] and generalized in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 below, is a Koopman-von Neumann type decomposition of functions where the underlying σ-algebras are generated by cubes of a fixed length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' To recall it, let Q ⊆ Rn be a cube, L > 0 be scale that divides l(Q), Q(L) = [− L 2 , L 2 ]n, and GL,Q denote the collection of cubes t + Q(L) partitioning the cube Q and ΓL,Q denote the grids corresponding to the centers of the cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By a slightly abuse of notation we also write GL,Q for the σ-algebra generated by the WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 13 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that the conditional expectation function E(f|GL,Q) is constant and equal to ffl A f on each cube A ∈ GL,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (A Koopman-von Neumann type decomposition [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≤ 1 and Q ⊆ Rn be a cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer ¯J1 = O(ε−2) such that for any ε-admissible sequence l(Q) ≥ L1 ≥ · · · ≥ L ¯ J1 and function f : Q → [−1, 1] there is some 1 ≤ j < ¯J1 such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) ∥f − E(f|GLj,Q)∥U1 Lj+1 (Q) ≤ ε Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let GLj,Q be the grid obtained from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 for the functions f = 1S for some fixed ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ¯f := E(f|GLj,Q), then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) and multi-linearity, we have N 1 λ∆0,Q(f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f) = N 1 λ∆0,Q( ¯f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f) + O(ε), and also M1 λ,Q(f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f) = M1 λ,Q( ¯f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f) + O(ε) provided for ε−6Lj+1 ≪ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus in showing (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) one can replace the functions f with ¯f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If we make the additional assumption that λ ≪ εLj then it is easy to see, using the fact that the function ¯f is constant on the cubes Qt(Lj) ∈ GLj,Q, that N 1 λ∆0,Q( ¯f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f) = M1 λ,Q( ¯f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f) + O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the condition ε−6Lj+1 ≪ λ ≪ εLj can be replaced with Lj+1 ≪ λ ≪ Lj if one passes to a subsequence of scales, for example L′ j = L5j, this completes the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The base case of a general inductive strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In this section, as preparation to handle the case of products of simplices, we prove a parametric version of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, namely Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 below, which will serve as the base case for later inductive arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q = Q1 × · · · × Qd with Qi ⊆ Rni be cubes of equal side length l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let L be a scale dividing l(Q) and for each t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td) ∈ ΓL,Q let Qt(L) = t + Q(L) and Qti(L) = ti + Qi(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that Qt(L) = Qt1(L) × · · · × Qtd(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Here Q(L) = [− L 2 , L 2 ]n and Qi(L) = [− L 2 , L 2 ]ni for each 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆0 i = {vi 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , vi ni} ⊆ Rni be a non-degenerate simplex for each 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 (Parametric Counting Lemma on Rn for Simplices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≤ 1 and R ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer J1 = J1(ε, R) = O(R ε−4) such that for any ε-admissible sequence of scales L0 ≥ L1 ≥ · · · ≥ LJ1 with the property that L0 divides l(Q) and collection of functions f i,r k,t : Qti(L0) → [−1, 1] with 1 ≤ i ≤ d, 1 ≤ k ≤ ni, 1 ≤ r ≤ R and t ∈ ΓL0,Q there exists 1 ≤ j < J1 and a set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε|ΓL0,Q| such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) N 1 λ∆0 i ,Qti (L0)(f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f i,r ni,t) = M1 λ,Qti (L0)(f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f i,r ni,t) + O(ε) for all λ ∈ [Lj+1, Lj] and t /∈ Tε uniformly in 1 ≤ i ≤ d and 1 ≤ r ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 will follow from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and the following generalization of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 in which we simultaneously consider a family of functions supported on the subcubes in a partition of an original cube Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 (A simultaneous Koopman-von Neumann type decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≤ 1, m ≥ 1, and Q ⊆ Rn be a cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer ¯J1 = O(mε−3) such that for any ε-admissible sequence L0 ≥ L1 ≥ · · · ≥ L ¯ J1 with the property that L0 divides l(Q), and collection of functions f1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fm,t : Qt(L0) → [−1, 1] defined for each t ∈ ΓL0,Q, there is some 1 ≤ j < ¯J1 and a set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε|ΓL0,Q| such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) ∥fi,t − E(fi,t|GLj,Qt(L0))∥U1 Lj+1(Qt(L0)) ≤ ε for all 1 ≤ i ≤ m and t /∈ Tε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 14 NEIL LYALL ´AKOS MAGYAR Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Fix 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For 1 ≤ k ≤ ni and t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td) ∈ ΓL0,Q , we will abuse notation and write f i,r k,t(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd) := f i,r k,t(xi) for (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd) ∈ Qt(L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If we apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 to the family of functions f i,r k,t on Qt(L0) for 1 ≤ i ≤ d, 1 ≤ k ≤ ni, and 1 ≤ r ≤ R, with m = (n1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' + nd)R, then this produces a grid GLj,Q for some 1 ≤ j ≤ ¯J1 = O(ε−3R), and a set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε|ΓL0,Q|, such that ∥f i,r k,t − E(f i,r k,t|GLj,Q)∥U1 Lj+1 (Qt(L0)) ≤ ε uniformly for 1 ≤ i ≤ d, 1 ≤ k ≤ ni and 1 ≤ r ≤ R for t /∈ Tε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since f i,r k,t(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd) = f i,r k,t(xi) for (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd) ∈ Qt(L0) it is easy to see that ∥f i,r k,t − E(f i,r k,t|GLj,Q)∥U1 Lj+1 (Qt(L0)) = ∥f i,r k,t − E(f i,r k,t|GLj,Qi)∥U1 Lj+1(Qti (L0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ¯f i,r k,t := E(f i,r k,t|GLj,Qi) , then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, one has N 1 λ∆0 i ,Qti(L0)(f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f i,r ni,t) = N 1 λ∆0 i ,Qti (L0)( ¯f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f i,r ni,t) + O(ε), and M1 λ,Qti (L0)(f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f i,r ni,t) = M1 λ,Qti (L0)( ¯f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f i,r ni,t) + O(ε) for all t /∈ Tε provided ε−6Lj+1 ≪ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Finally, if we also have λ ≪ εLj then it is easy to see that N 1 λ∆0 i ,Qti (L0)( ¯f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f i,r ni,t) = M1 λ,Qti (L0)( ¯f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ¯f i,r ni,t) + O(ε) as the functions ¯f i,r k,t are constant on cubes Qti(Lj) of GLj,Qi, which are of size Lj ≪ εL0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Passing first to a subsequence of scales, for example L�� j = L5j, the condition ε−6Lj+1 ≪ λ ≪ εLj can be replaced with Lj+1 ≪ λ ≪ Lj so this completes the proof of the Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ We conclude this section with a sketch of the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' These arguments are standard, see for example the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 given in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' First we make an observation about the U 1 L(Q)-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Suppose 0 < L′ ≪ ε2L with L′ dividing L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If s ∈ ΓL′,Q and t ∈ Qs(L′) then |t − s| = O(L′) and hence x∈Qt(L) g(x) dx = x∈Qs(L) g(x) dx + O(L′/L) for any function g : Q → [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover, since the cube Qs(L) is partitioned into the smaller cubes Qt(L′), we have by Cauchy-Schwarz ��� x∈Qs(L) g(x) dx ��� 2 ≤ Et∈ΓL′,Qs(L) ��� x∈Qt(L′) g(x) dx ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' From these observations it is easy to see that ∥g∥2 U1 L(Q) = t∈Q ��� x∈Qt(L) g(x) dx ��� 2 dt ≤ Et∈ΓL′,Q ��� x∈Qt(L′) g(x) dx ��� 2 + O(L′/L) and we note that the right side of the above expression is ∥E(g|GL′,Q)∥2 L2(Q) since the conditional expectation function E(g|GL′,Q) is constant and equal to ffl x∈Qt(L′) g(x) dx on the cubes Qt(L′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Suppose that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) does not hold for some 1 ≤ i ≤ m for every t in some set Tε ⊆ ΓL0,Q of size |Tε| > ε |ΓL0,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If we apply the above observation to g := fi,t − E(fi,t|GLj,Qt(L0)), for every t ∈ Tε, we obtain by orthogonality that m � i=1 ∥E(fi,t|GLj+2,Qt(L0))∥2 L2(Qt(L0)) ≥ m � i=1 ∥E(fi,t|GLj,Qt(L0))∥2 L2(Qt(L0)) + cε2 for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 15 If we now define fi : Q → [−1, 1] such that fi|(Qt(L0)) = fi,t, for 1 ≤ i ≤ m, average over t ∈ ΓL0,Q, and use the fact ∥fi∥2 L2(Q) = Et∈ΓL0,Q∥fi,t∥2 L2(Qt(L0)), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) m � i=1 ∥E(fi|GLj+2,Q)∥2 L2(Q) ≥ m � i=1 ∥E(fi|GLj,Q)∥2 L2(Q) + cε3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is clear that the sums in the above expressions are bounded by m for all j ≥ 1, thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) cannot hold for some 1 ≤ j ≤ ¯J1 for ¯J1 := C m ε−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This implies that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) must hold for some 1 ≤ j ≤ ¯J1, for all 1 ≤ i ≤ m and all t /∈ Tε for a set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε |ΓL0,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Product of two simplices in Rn Although not strictly necessary, we discuss in this section the special case d = 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This already gives an improvement of the main results of [12], but more importantly serves as a gentle preparation for the more complicated general case, presented in the Section 5, which involve both a plethora of different scales and the hypergraph bundle notation introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 with d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q = Q1 × Q2 with Q1 ⊆ Rn1 and Q2 ⊆ Rn2 be cubes of equal side length l(Q) and ∆0 = ∆0 1 × ∆0 2 with ∆0 1 = {v11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , v1n1} ⊆ Rn1 and ∆0 2 = {v11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , v2n2} ⊆ Rn2 two non-degenerate simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In order to “count” configurations of the form ∆ = ∆1 × ∆2 ⊆ Rn1+n2 with ∆1 and ∆2 isometric copies of λ∆0 1 and λ∆0 2 respectively for some 0 < λ ≪ l(Q) in a set S ⊆ Q we introduce the multi-linear expression N 2 λ∆0,Q({fkl}) := x11∈Q1 x21∈Q2 ˆ x12,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',x1n1 ˆ x22,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',x2n2 n1 � k=1 n2 � l=1 fkl(x1k, x2l) dσλ∆0 1(x12 − x11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , x1n1 − x11) dσλ∆0 2(x22 − x21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , x2n2 − x21) dx21 dx11 for any family of functions fkl : Q1 × Q2 → [−1, 1] with 1 ≤ k ≤ n1 and 1 ≤ l ≤ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed, if fkl = 1S for all 1 ≤ k ≤ n1 and 1 ≤ l ≤ n2 then the above expression is 0 unless there exists a configuration ∆ ⊆ S of the form ∆1 × ∆2 with ∆1 and ∆2 isometric copies of λ∆0 1 and λ∆0 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The short argument presented in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 demonstrating how both Theorem B and Corollary B follow from Proposition B, and hence from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, applies equally well to each of our main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This reduces our main theorems to analogous quantitative results involving an arbitrary lacunary sequence of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In the case d = 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 this stronger quantitative result takes the following form: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < ε ≪ 1 there exists an integer J = O(exp(Cε−13)) with the following property: Given any lacunary sequence l(Q) ≥ λ1 ≥ · · · ≥ λJ and S ⊆ Q, there is some 1 ≤ j < J such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) N 2 λ∆0,Q({1S}) > � |S| |Q| �n1n2 − ε for all λ ∈ [λj+1, λj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our approach to establishing Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 is again to compare the above expressions to simpler ones for which it is easy to obtain lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < λ ≪ l(Q) and family of functions fkl : Q1×Q2 → [−1, 1] with 1 ≤ k ≤ n1 and 1 ≤ l ≤ n2 we consider M2 λ,Q({fkl}) := t∈Q x1∈(t1+Q1(λ))n1 x2∈(t2+Q2(λ))n2 n1 � k=1 n2 � l=2 fkl(x1k, x2l) dx2 dx1 dt where t = (t1, t2) ∈ Q1 × Q2, xi = (xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xini) and Qi(λ) = [− λ 2 , λ 2 ]ni for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if S ⊆ Q is a set of measure |S| ≥ δ|Q| for some δ > 0, then careful applications of H¨older’s inequality give M2 λ,Q({1S}) ≥ t∈Q � (x1,x2)∈t+Q(λ) 1S(x1, x2) dx1dx2 �n1n2 dt ≥ δn1n2 − O(ε) for all scales 0 < λ ≪ ε l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 16 NEIL LYALL ´AKOS MAGYAR In light of the observation above, and the discussion preceding Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, we see that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, and hence Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 when d = 2, will follows as a consequence of the following Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer J2 = O(exp(Cε−12)) such that for any ε-admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJ2 and S ⊆ Q there is some 1 ≤ j < J2 such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) N 2 λ∆0,Q({1S}) = M2 λ,Q({1S}) + O(ε) for all λ ∈ [Lj+1, Lj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There are again two main ingredients in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The first establishes that the our multi-linear forms N 2 λ∆0,Q({fkl}) are controlled by an appropriate box-type norm attached to a scale L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q = Q1 × Q2 be a cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any scale 0 < L ≪ l(Q) and function f : Q → R we define its local box norm at scale L to be (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3) ∥f∥4 □L(Q1×Q2) := t∈Q ∥f∥4 □(t+Q(L)) dt where Q(L) = [− L 2 , L 2 ]n1+n2 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) ∥f∥4 □( � Q) := x11,x12∈ � Q1 x21,x22∈ � Q2 f(x11, x21)f(x12, x21)f(x11, x22)f(x12, x22) dx11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dx22 for any cube �Q ⊆ Q of the form �Q = �Q1 × �Q2 with �Qj ⊆ Qj for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 (A Generalized von-Neumann inequality [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ε > 0, 0 < λ ≪ l(Q), and 0 < L ≪ ε24λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any collections of functions fkl : Q1 × Q2 → [−1, 1] with 1 ≤ k ≤ n1 and 1 ≤ l ≤ n2 we have both (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) |N 2 λ∆0,Q({fkl})| ≤ min 1≤k≤n1, 1≤l≤n2 ∥fkl∥□L(Q1×Q2) + O(ε) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) |M2 λ,Q({fkl})| ≤ min 1≤k≤n1, 1≤l≤n2 ∥fkl∥□L(Q1×Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The result above was essentially proved in [12] for the multi-linear forms N 2 λ∆0,Q when Q = [0, 1]n1+n2, however a simple scaling argument transfers the result to an arbitrary cube Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For completeness we include its short proof in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The second and main ingredient is an analogue of a weak form of Szemer´edi’s regularity lemma due to Frieze and Kannan [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The more probabilistic formulation, we will use below, can be found for example in [21], [22], and [23], and is also sometimes referred to as a Koopman-von Neumann type decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any cube Q ⊆ Rn and scale L > 0 that divides l(Q) we will let Q(L) = [− L 2 , L 2 ]n and GL,Q denote the collection of cubes Qt(L) = t + Q(L) partitioning the cube Q and let ΓL,Q denote grid corresponding to the centers of these cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will say that a finite σ-algebra B on Q is of scale L if it contains GL,Q and for simplicity of notation will write Bt for B|Qt(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that if we have two σ-algebras B1 on a cube Q1 and B2 on Q2 then by B1 ∨ B2 we mean the σ-algebra on Q = Q1 × Q2 generated by the sets B1 × B2 with B1 ∈ B1 and B2 ∈ B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall also that we say the complexity of a σ-algebra B is at most m, and write complex(B) ≤ m, if it is generated by m sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (Weak regularity lemma in Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1 and Q = Q1 × Q2 with Q1 ⊆ Rn1 and Q2 ⊆ Rn2 be cubes of equal side length l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer ¯J2 = O(ε−12) such that for any ε4-admissible sequence l(Q) ≥ L1 ≥ · · · ≥ L ¯ J2 and function f : Q → [−1, 1] there is some 1 ≤ j ≤ ¯J2 and a σ-algebra B of scale Lj on Q such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) ∥f − E(f|B)∥□Lj+1(Q1×Q2) ≤ ε which has the additional local structure that for each t = (t1, t2) ∈ ΓLj,Q there exist σ-algebras B1,t on Qt1(Lj) and B2,t on Qt2(Lj) with complex(Bi,t) = O(j) for i = 1, 2 such that Bt = B1,t ∨ B2,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Comparing the above statement to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 for d = 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e to the weak regularity lemma, note that the σ-algebra B of scale Lj has a direct product structure only locally, inside each cube Qt(Lj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover this product structure varies with t ∈ ΓLj,Q, however the “local complexity” remains uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 17 Assuming for now the validity of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 we prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will make crucial use of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3, namely our parametric counting lemma on Rn for simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1, ε1 := exp(−C1ε−12) for some C1 ≫ 1, and {Lj}j≥1 be an ε1- admissible sequence of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Set R = ε ε−1 1 and J1(ε1, R) be the parameter appearing in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3, noting that J1(ε1, R) = O(ε−5 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For L ∈ {Lj}j≥1 write index(L) = j if L = Lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We now choose a subsequence {L′ j} ⊆ {Lj} so that L′ 1 = L1 and index(L′ j+1) ≥ index(L′ j) + J1(ε1, R) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, with fkl = f := 1S for all 1 ≤ k ≤ n1 and 1 ≤ l ≤ n2, guarantees the existence of a σ-algebra B of scale L′ j on Q such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) ∥f − E(f|B)∥□L′ j+1(Q1×Q2) ≤ ε for some 1 ≤ j ≤ Cε−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover, we know that B has the additional local structure that for each t = (t1, t2) ∈ ΓL′ j,Q there exist σ-algebras B1,t on Qt1(L′ j) and B2,t on Qt2(L′ j) with complex(Bi,t) = O(ε−12) for i = 1, 2 such that Bt = B1,t ∨ B2,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus, if we let R1,t and R2,t denote the number of atoms in B1,t and B2,t respectively, then we can assume, by formally adding the empty set to these collections of atoms if necessary, that R1,t = R2,t = R′ := exp(Cε−12) for all t ∈ ΓL′ j,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If we let ¯f := E(f|B1 ∨ B2), then by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and multi-linearity we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) N 2 λ∆0,Q({f}) = N 2 λ∆0,Q({ ¯f}) + O(ε) and M2 λ,Q({f}) = M2 λ,Q({ ¯f}) + O(ε) provided for ε−24L′ j+1 ≪ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For a given t ∈ ΓQ,L′ j write ¯ft for the restriction of ¯f to the cube Qt(L′ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By localization, one then has (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) N 2 λ∆0,Q({ ¯f}) = Et∈ΓL′ j ,Q N 2 λ∆0,Qt(L′ j)({ ¯ft}) + O(ε), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) M2 λ,Q({ ¯f}) = Et∈ΓL′ j,Q M2 λ,Qt(L′ j)({ ¯ft}) + O(ε) provided one also insists that λ ≪ ε L′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For given t ∈ ΓL′ j,Q, the functions ¯ft(x1, x2) are linear combinations of functions of the form 1Ar1 1,t(x1)1Ar2 2,t(x2), where {Ar1 1,t}1≤r1≤R′ and {Ar2 2,t}1≤r2≤R′ are the collections of the atoms of the σ-algebras B1,t and B2,t defined on the cubes Qt1(L′ j) and Qt2(L′ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus for each t ∈ ΓL′ j,Q one has ¯ft = R′ � r1=1 R′ � r2=1 αr,t1Ar1 1,t × 1Ar2 2,t where r = (r1, r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Plugging these linear expansions into the multi-linear expressions in above one obtains N 2 λ∆0,Qt(L′ j)({ ¯ft}) = � r={rkl}kl αr,t N 2 λ∆0,Qt(L′ j)({1A r1,kl 1,t × 1A r2,kl 2,t }) using the notations rkl = (r1,kl, r2,kl), αr,t = � kl αrkl,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Notice that the product n1 � k=1 n2 � l=1 1A r1,kl 1,t (x1k)1A r2,kl 2,t (x2l) is nonzero only if Ar1,kl 1,t = Ar1,k 1,t , that is if r1,kl = r1,k for all 1 ≤ l ≤ n2, as the atoms Ar 1,t are all disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Similarly, one has that r2,kl = r2,l for all 1 ≤ k ≤ n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus, in fact (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) N 2 λ∆0,Qt(L′ j)({ ¯ft}) = � r={rkl}kl αr,t N 2 λ∆0,Qt(L′ j)({1A r1,k 1,t × 1A r2,l 2,t }) and similarly (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13) M2 λ,Qt(L′ j)({ ¯ft}) = � r={rkl}kl αr,t M2 λ,Qt(L′ j)({1A r1,k 1,t × 1A r2,l 2,t }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note, that indices r are running through the index set [1, R′]n1 × [1, R′]n2 of size at most R if C1 ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 18 NEIL LYALL ´AKOS MAGYAR The key observation is that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14) N 2 λ∆0,Qt(L′ j)(1A r1,k 1,t × 1A r2,l 2,t ) = N 1 λ∆0 1,Qt1 (L′ j)(1A r1,1 1,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1A r1,n1 1,t ) N 1 λ∆0 2,Qt2 (L′ j)(1A r2,1 2,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1A r2,n2 2,t ) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) M2 λ,Qt(L′ j)(1A r1,k 1,t × 1A r2,l 2,t ) = M1 λ,Qt1 (L′ j)(1A r1,1 1,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1A r1,n1 1,t ) M1 λ,Qt2 (L′ j)(1A r2,1 2,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1A r2,n2 2,t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let r = {(r1,k, r2,l)}kl and g1,r k,t := 1A r1,k 1,t , g2,r l,t := 1A r2,l 2,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Writing j′ := index(L′ j) and J′ := index(L′ j+1), one may apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 for the families of functions g1,r k,t, g2,r l,t , where 1 ≤ k ≤ n1, 1 ≤ l ≤ n2 and r = (r1,k, r2,l)kl ∈ [1, R′]n1 × [1, R′]n2, with respect to the ε1-admissible sequence of scales Lj′+1 ≥ Lj′+2 ≥ · · · ≥ LJ′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This is possible as J′ − j′ = J1(ε1, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then there is a scale Lj with j′ ≤ j < J′ so that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) N 1 λ∆0 1,Qt1 (L′ j)(g1,r 1,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , g1,r n1t) = M1 λ,Qt1 (L′ j)(g1,r 1,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , g1,r n1,t) + O(ε1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) N 1 λ∆0 2,Qt2 (L′ j)(g2,r 1,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , g2,r n2,t) = M1 λ,Qt2 (L′ j)(g2,r 1,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , g2,r n2,t) + O(ε1), for all λ ∈ [Lj+1, Lj] uniformly in r = {(r1,k, r2,l)}kl and t /∈ Tε1 ⊆ ΓL′ j,Q, for a set of size |Tε1| ≤ ε1|ΓL′ j,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13), we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='18) N 2 λ∆0,Qt(L′ j)({ ¯ft}) = M2 λ,Qt(L′ j) ({ ¯ft}) + O(ε) for t /∈ Tε1, as |αr,t| ≤ 1 and Rε1 ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Finally, since |Tε1| ≤ ε1|ΓL′ j,Q|, by averaging in t ∈ ΓL′ j,Q, one has N 2 λ∆0,Q({ ¯f}) = M2 λ,Q ({ ¯f}) + O(ε) using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) and the Proposition follows by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) with an index 1 ≤ j < J2 = O(ε−12ε−5 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' First we note that if χL := L−n1[−L/2,L/2]n and ψL := χL ∗ χL, then ψL(x2 − x1) = ˆ t χL(x1 − t)χL(x2 − t) dt and hence for any function f : Q → [−1, 1], with Q ⊆ Rn being a cube of side length l(Q), one has ∥f∥2 U1 L(Q) = x1∈Q ˆ x2 f(x1)f(x2)ψL(x2 − x1) dx1dx2 + O(L/l(Q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Write x′ := (x21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , x2n2) and let gk,x′(x) := �n2 l=1 fkl(x, x2l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then one may write N 2 λ∆0,Q({fkl}) = x21∈Q2 ˆ x22,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',x2n2 N 1 λ∆0 1,Q1(g1,x′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , gn1,x′) dσλ∆0 2(x22 − x21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , x2n2 − x21) dx21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Using estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6), the above observation, and Cauchy-Schwarz one has |N 2 λ∆0,Q({fkl})|2 ≤ x11∈Q1 ˆ x12 ψL(x12 − x11) N 1 λ∆0 2,Q2(h1,x11,x12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , hn2,x11,x12) dx11dx12 + O(ε4) provided 0 < λ ≪ l(Q) and 0 < L ≪ ε24λ where hl,x11,x12(x) = f1l(x11, x)f1l(x12, x) for 1 ≤ l ≤ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Applying the same procedure again ultimately gives |N 2 λ∆0,Q({fkl})|4 ≤ ∥f11∥4 □L(Q1×Q2) + O(ε4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The same estimate can of course be given for any function fkl in place of f11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This establishes (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) is established similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 19 Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For each t = (t1, t2) ∈ ΓL1,Q we will let B1,t(L1) := {∅, Qt1(L1)} and B2,t(L1) := {∅, Qt2(L1)}, in other words the trivial σ-algebras on Qt1(L1) and Qt2(L1) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) holds with B(L1) = GL1,Q, noting that Bt(L1) := B1,t(L1) ∨ B2,t(L1) in this case, then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We now assume that we have developed, for each t = (t1, t2) ∈ ΓLj,Q, σ-algebras B1,t(Lj) on Qt1(Lj) and B2,t(Lj) on Qt2(Lj) with complex(Bi,t(Lj)) ≤ j for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let B(Lj) be the σ-algebra such that Bt(Lj) = B1,t(Lj) ∨ B2,t(Lj) for all t ∈ ΓLj,Q and assume that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) does not hold, namely that ∥g∥□Lj+1(Q) ≥ ε where g := f − E(f|B(Lj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By the definition of the local box norm this means that t∈Q ∥g∥4 □(t+Q(Lj+1)) dt ≥ ε4 and hence, as Lj+2 ≪ ε4Lj+1, it is easy to see that Es∈ΓLj+2,Q ∥g∥4 □(s+Q(Lj+2)) ≥ ε4/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This implies that there is a set S ⊆ ΓLj+2,Q of size |S| ≥ (ε4/4)|ΓLj+2,Q| such that for all s = (s1, s2) ∈ S, one has that ∥g∥4 □(Qs(Lj+2)) ≥ ε4/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It therefore follows, as is well-known see for example [12] or [23], that there exist sets B1,s ⊆ Qs1(Lj+2) and B2,s ⊆ Qs2(Lj+2) such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='19) x1∈Qs1 (Lj+2) x2∈Qs2 (Lj+2) g(x1, x2) 1B1,s(x1)1B2,s(x2) dx1 dx2 ≥ ε4/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For a given s ∈ ΓLj+2,Q there is a unique t = t(s) such that Qs(Lj+2) ⊆ Qt(Lj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let B′ 1,s(Lj+2) := B1,t(Lj)|Qs1 (Lj+2) and B′ 2,s(Lj+2) := B2,t(Lj)|Qs2 (Lj+2) noting that complex(B′ i,s(Lj+2)) ≤ j for i = 1, 2, as the complexity of a σ-algebra does not increase when restricted to a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If, for i = 1, 2, we let Bi,s(Lj+2) denote the σ-algebra generated by B′ i,s(Lj+2) and the set Bi,s if s ∈ S and let Bi,s(Lj+2) := B′ i,s(Lj+2) otherwise, then clearly complex(Bi,s(Lj+2)) is at most j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We now define B(Lj+2) to the the sigma algebra of scale Lj+2 with the property that Bs(Lj+2) = B1,s(Lj+2) ∨ B2,s(Lj+2) for all s ∈ ΓLj+2,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Using the inner product notation ⟨f, g⟩Q = ffl Q f(x)g(x) dx we can rewrite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='19) as ⟨f − E(f|B(Lj)) , 1B1,s × 1B2,s ⟩Qs(Lj+2) ≥ ε4/16 for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the function 1B1,s × 1B2,s is measurable with respect to B(Lj+2) one clearly has ⟨f − E(f|B(Lj+2)) , 1B1,s × 1B2,s⟩Qs(Lj+2) = 0 and hence ⟨E(f|B(Lj+2)) − E(f|B(Lj)) , 1B1,s × 1B2,s⟩Qt(Lj+2) ≥ ε4/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It then follows from Cauchy-Schwarz and orthogonality that ∥E(f|B(Lj+2))∥2 L2(Qs(Lj+2)) − ∥E(f|B1(Lj))∥2 L2(Qs(Lj+2)) ≥ ε8/256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since |S| ≥ (ε4/4)|ΓLj+2,Q| averaging over all s ∈ ΓLj+2,Q gives ∥E(f|B(Lj+2))∥2 L2(Q) ≥ ∥E(f|B(Lj))∥2 L2(Q) + ε12/210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Trivially both sides are at most 1 thus the process must stop at a step j = O(ε−12) where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) holds for a σ-algebra of “local complexity” at most j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This proves the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2: The general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' After these preparations we will now consider the general case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q = Q1×· · ·×Qd ⊆ Rn with Qi ⊆ Rni cubes of equal side length l(Q) and ∆0 = ∆0 1 ×· · ·×∆0 d with each ∆i ⊆ Rni a non-degenerate simplex of ni points for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will use a generalized version of the hypergraph terminology introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In particular, for a vertex set I = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', d} and set K = {il;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1 ≤ i ≤ d, 1 ≤ l ≤ ni} we will let π : K → I denote the 20 NEIL LYALL ´AKOS MAGYAR projection defined by π(il) := i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' As before we will let Hd,k := {e ⊆ I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' |e| = k} denote the complete k-regular hypergraph with vertex set I, and for the multi-index n = (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , nd) define the hypergraph bundle Hn d,k := {e ⊆ K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' |e| = |π(e)| = k} noting that |π−1(i)| = ni for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In order to parameterize the vertices of direct products of simplices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' sets of the form ∆ = ∆1×· · ·×∆d with ∆i ⊆ Qi, we consider points x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd) with xi = (xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xini) ∈ Qni i for each i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Now for any 1 ≤ k ≤ d and any edge e′ ∈ Hd,k we will write Qe′ := � i∈e′ Qi, and for every x ∈ Qn1 1 × · · · × Qnd d and e ∈ Hn d,k we define xe := πe(x), where πe : Qn1 1 × · · · × Qnd d → Qπ(e) is the natural projection map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Writing ∆i = {xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xini} we have that ∆1 × · · · × ∆d = {xe : e ∈ Hn d,d} since every edge xe is of the form (x1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xdld).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We can therefore identify points x with configurations of the form ∆1 × · · · × ∆d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < λ ≪ l(Q) the measures dσλ∆0 i , introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, are supported on points (y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , yni) for which the simplex ∆i = {0, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , yni} is isometric to λ∆0 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For simplicity of notation we will write ˆ xi f(xi) dσλ i (xi) := xi1∈Qi ˆ xi2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xini f(xi) dσλ∆0 i (xi2 − xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xini − xi1) dxi1 Note that the support of the measure dσλ i is the set of points xi so that the simplex ∆i := {xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xini} is isometric to λ∆0 i and xi1 ∈ Qi, moreover the measure is normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus if S ⊆ Q is a set then the density of configurations ∆ in S of the form ∆ = ∆1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × ∆d with each ∆i ⊆ Qi an isometric copy of λ∆0 i is given by the expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) N d λ∆0,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) := ˆ x1 · · ˆ xd � e∈Hn d,d 1S(xe) dσλ 1 (x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dσλ d (xd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 reduces to establishing the following stronger quantitative result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < ε ≪ 1 there exists an integer Jd = Jd(ε) with the following property: Given any lacunary sequence l(Q) ≥ λ1 ≥ · · · ≥ λJd and S ⊆ Q, there is some 1 ≤ j < Jd such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) N d λ∆0,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) > � |S| |Q| �n1··· nd − ε for all λ ∈ [λj+1, λj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Quantitative Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A careful analysis of our proof reveals that there is a choice of Jd(ε) which is less than Wd(log(C∆ε−3)), where Wk(m) is again the tower-exponential function defined by W1(m) = exp(m) and Wk+1(m) = exp(Wk(m)) for k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 0 < λ ≪ l(Q) and set S ⊆ Q we define the expression: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3) Md λ,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) := t∈Q Md t+Q(λ)(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) dt where Q(λ) = [− λ 2 , λ 2 ]n and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) Md � Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) := x1∈ � Qn1 1 · · xd∈ � Q nd d � e∈Hn d,d 1S(xe) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dxd for any cube �Q ⊆ Q of the form �Q = �Q1 × · · · × �Qd with �Qi ⊆ Qi for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if S ⊆ Q is a set of measure |S| ≥ δ|Q| for some δ > 0, then careful applications of H¨older’s inequality give Md λ,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) ≥ t∈Q � (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xd)∈t+Q(λ) 1S(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dxd �n1··· nd dt ≥ δn1··· nd − O(ε) for all scales 0 < λ ≪ ε l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In light of the discussion above, and that preceding Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, we see that Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, and hence Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 in general, will follows as a consequence of the following WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 21 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer Jd = Jd(ε) such that for any ε-admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJd and S ⊆ Q there is some 1 ≤ j < Jd such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) N d λ∆0,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) = Md λ,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) + O(ε) for all λ ∈ [Lj+1, Lj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The validity of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 will follow immediately from the d = k case of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Reduction of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 to a more general “local” counting lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any given 1 ≤ k ≤ d and collection of functions fe : Qπ(e) → [−1, 1] with e ∈ Hn d,k we define the following multi-linear expressions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) N d λ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) := ˆ x1 · · ˆ xd � e∈Hn d,k fe(xe) dσλ 1 (x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='dσλ d (xd) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) Md λ,Q(fe ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) := t∈Q Md t+Q(λ)(fe ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) dt where Q(λ) = [− λ 2 , λ 2 ]n and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) Md � Q(fe ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) := x1∈ � Qn1 1 · · xd∈ � Q nd d � e∈Hn d,k fe(xe) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dxd for any cube �Q ⊆ Q of the form �Q = �Q1 × · · · × �Qd with �Qi ⊆ Qi for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our strategy to proving Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 is the same as illustrated in the finite field settings, that is we would like to compare averages Nλ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) to those of Md λ,Q(fe ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k), at certain scales λ ∈ [Lj+1, Lj], inductively for 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' However in the Euclidean case, an extra complication emerges due to the fact the (hypergraph) regularity lemma, the analogue of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, does not produce σ-algebras Bf, for f ∈ Hn d,k−1, on the cubes Qf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In a similar manner to the case for d = 2 discussed in the previous section, we will only obtain σ-algebras “local” on cubes Qtf(L0) at some scale L0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This will have the effect that the functions fe will be replaced by a family of functions fe,t, where t runs through a grid ΓL0,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' To be more precise, let L > 0 be a scale dividing the side-length l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For t ∈ ΓL,Q and e′ ∈ Hd,k we will use te′ to denote the projection of t onto Qe′ and Qte′ (L) := te′ + Qe′(L) to denote the projection of the cube Qt(L) centered at t onto Qe′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is then easy to see that for any ε > 0 we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) N d λ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Et∈ΓL,Q N d λ∆0,Qt(L)(fe,t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) Md λ,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Et∈ΓL,Q Md λ,Qt(L)(fe,t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) provided 0 < λ ≪ εL where fe,t denotes the restriction of a function fe to the cube Qt(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' At this point the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 reduces to showing that the expressions in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) only differ by O(ε) at some scales λ ∈ [Lj+1, Lj], given an ε-admissible sequence L0 ≥ L1 ≥ · · · ≥ LJ, for any collection of bounded functions fe,t, e ∈ Hn d,k, t ∈ ΓL0,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed, our crucial result will be the following Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 (Local Counting Lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1 and M ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer Jk = Jk(ε, M) such that for any ε-admissible sequence of scales L0 ≥ L1 ≥ · · · ≥ LJk with the property that L0 divides l(Q), and collection of functions f m e,t : Qtπ(e)(L0) :→ [−1, 1] with e ∈ Hn d,k, 1 ≤ m ≤ M and t ∈ ΓL0,Q there exists 1 ≤ j < Jk and a set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε|ΓL0,Q| such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) N d λ∆0,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Mλ,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) for all λ ∈ [Lj+1, Lj] and t /∈ Tε uniformly in e ∈ Hn d,k and 1 ≤ m ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 22 NEIL LYALL ´AKOS MAGYAR 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will prove Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 by induction on 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For k = 1 this is basically Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed, in this case for a given t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td) ∈ ΓL0,Q and edge e ∈ Hn d,1 = {il : 1 ≤ i ≤ d, 1 ≤ l ≤ ni} we have that f m e,t(xe) = f m il,t(xil) with xil ∈ Qti(L0) and hence both N d λ∆0,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,1) = d � i=1 N 1 λ∆0 i ,Qti (L0)(f m i1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f m ini,t) Md λ,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,1) = d � i=1 M1 λ,Qti (L0)(f m i1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f m ini,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 there exists an 1 ≤ j < J1 = O(Mε−4) and an exceptional set Tε ⊆ ΓL0,Q of size |Tε| ≤ ε|ΓL0,Q|, such that uniformly for t /∈ Tε and for 1 ≤ i ≤ d, one has N 1 λ∆0 i ,Qti(L0)(f m i1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f m ini,t) = M1 λ,Qti (L0)(f m i1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f m ini,t) + O(ε) hence N d λ∆0,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,1) = Md λ,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,1) + O(ε) as the all factors are trivially bounded by 1 in magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This implies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For the induction step we again need two main ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The first establishes that the our multi-linear forms N d λ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) are controlled by an appropriate box-type norm attached to a scale L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q = Q1 × · · · × Qd and 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any scale 0 < L ≪ l(Q) and function f : Qe′ → [−1, 1] with e′ ∈ Hd,k we define its local box norm at scale L by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) ∥f∥2k □L(Qe′ ) := s∈Qe′ ∥f∥2k □(s+Q(L)) ds where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13) ∥f∥2k □( � Q) := x11,x12∈ � Q1 · · xk1,xk2∈ � Qk � (ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',ℓk)∈{1,2}k f(x1ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xkℓk) dx11 dx12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dxk1 dxk2 for any cube �Q of the form �Q = �Q1 × · · · × �Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 (Generalized von-Neumann inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ε > 0, 0 < λ ≪ l(Q) and 0 < L ≪ (ε2k)6λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any 1 ≤ k ≤ d and collection of functions fe : Qπ(e) → [−1, 1] with e ∈ Hn d,k we have both (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14) |N d λ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k)| ≤ min e∈Hn d,k ∥fe∥□L(Qπ(e)) + O(ε) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) |Md λ,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k)| ≤ min e∈Hn d,k ∥fe∥□L(Qπ(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The crucial ingredient is the following analogue of the weak hypergraph regularity lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (Parametric weak hypergraph regularity lemma for Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1, M ≥ 1, and 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists ¯Jk = O(Mε−2k+3) such that for any ε2k-admissible sequence L0 ≥ L1 ≥ · · · ≥ L ¯ Jk with the property that L0 divides l(Q) and collection of functions f m e,t : Qtπ(e)(L0) → [−1, 1] with e ∈ Hn d,k, 1 ≤ m ≤ M, and t ∈ ΓL0,Q there is some 1 ≤ j < ¯Jk and σ-algebras Be′,t of scale Lj on Qte′ (L0) for each t ∈ ΓL0,Q and e′ ∈ Hd,k such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) ∥f m e,t − E(f m e,t|Bπ(e),t)∥□Lj+1(Qtπ(e)(L0)) ≤ ε uniformly for all t /∈ Tε, e ∈ Hn d,k, and 1 ≤ m ≤ M, where Tε ⊆ ΓL0,Q with |Tε| ≤ ε|ΓL0,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 23 Moreover, the σ-algebras Be′,t have the additional local structure that the exist σ-algebras Be′,f′,s on Qsf′ (Lj) with complex(Be′,f′,s) = O(j) for each s ∈ ΓLj,Q, e′ ∈ Hd,k, and f′ ∈ ∂e′ such that if s ∈ Qt(L0), then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) Be′,t �� Qse′ (Lj) = � f′∈∂e′ Be′,f′,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 is the parametric and simultaneous version of the extension of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7 to the product of d simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The difference is that in the general case one has to deal with a parametric family of functions f m e,t as t is running through a grid ΓL0,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The essential new content of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 is that one can develop σ-algebras Be′,t on the cubes Qt(L0) with respect to the family of functions f m e,t such that the local structure described above and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) hold simultaneously for almost all t ∈ ΓL0,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Assume the Proposition holds for k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ε > 0, ε1 := exp (−C1ε−2k+3) for some large constant C1 = C1(n, k, d) ≫ 1, and {Lj}j≥1 be an ε1-admissible sequence of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Set F(ε) := Jk−1(ε1, M) with M = ε ε−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For L ∈ {Lj}j≥1 we again write index(L) = j if L = Lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We now choose a subsequence {L′ j} ⊆ {Lj} so that L′ 0 = L0 and index(L′ j+1) ≥ index(L′ j) + F(ε) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 then guarantees the existence of σ-algebras Be′,t of scale L′ j on Qte′ (L0) for each t ∈ ΓL0,Q and e′ ∈ Hd,k, with the local structure described above, such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='18) ∥f m e,t − E(f m e,t|Bπ(e),t)∥□L′ j+1(Qtπ(e)(L0)) ≤ ε uniformly for all t /∈ T ′ ε, e ∈ Hn d,k, and 1 ≤ m ≤ M, for some 1 ≤ j < ¯Jk(ε, M) = O(Mε−2k+3), where T ′ ε ⊆ ΓL0,Q with |T ′ ε| ≤ ε|ΓL0,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ¯f m e,t := E(f m e,t|Bπ(e),t) for t ∈ ΓL0,Q and e ∈ Hn d,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If t /∈ T ′ ε, then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) we have both (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='19) N d λ∆0,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = N d λ∆0,Qt(L0)( ¯f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='20) Md λ,Qt(L0)(f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Md λ,Qt(L0)( ¯f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' provided (ε−2k)6L′ j+1 ≪ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For given s ∈ ΓL′ j,Qt(L0) one may write ¯f m e,s for the restriction of ¯f m e,t on the cube Qs(L′ j) ⊆ Qt(L0), as s uniquely determines t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By localization, provided λ ≪ εL′ j, we then have both (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='21) N d λ∆0,Qt(L0)( ¯f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Es∈ΓL′ j,Qt(L0)N d λ∆0,Qs(L′ j)( ¯f m e,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='22) Md λ,Qt(L0)( ¯f m e,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Es∈ΓL′ j,Qt(L0)Md λ,Qs(L′ j)( ¯f m e,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For a fixed cube Qs(L′ j) we have that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='23) ¯f m e,s = Re,s � re=1 αs,re,m 1Are π(e),s where {Are π(e),s}1≤r≤Re,s is the family of atoms of the σ-algebra Bπ(e),t restricted to the cube Qs(L′ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that |αs,re| ≤ 1 and |Re,s| = O(exp (Cε−2k+3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By adding the empty set to the collection of atoms one may assume |Re,s| = R := exp (Cε−2k+3) for all e ∈ Hn d,k and s ∈ ΓL′ j,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then, by multi-linearity, using the notations r = (re)e∈Hn d,k and αr,s = � e αs,re, one has both (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='24) N d λ∆0,Qs(L′ j)( ¯f m s,e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = � r αs,r,m N d λ∆0,Qs(L′ j)(1Are π(e),s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='25) Md λ,Qs(L′ j)( ¯f m s,e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = � r αs,r,m Md λ,Qs(L′ j)(1Are π(e),s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The key observation is that these expressions in the sum above are all at level k − 1 instead of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' To see this let e = (i1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , imlm, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , iklk) so e′ = π(e) = (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , im, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , ik).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If f′ = e′\\{im} then recall that the 24 NEIL LYALL ´AKOS MAGYAR edge pf′(e) = (i1l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , iklk) ∈ Hn d,k−1 is obtained from e by removing the imlm-entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus, for any atom Ae′,s of Bs,e′(L′ j) we have by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17), that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='26) 1Ae′,s(xe) = � f′∈∂e′ 1Ae′,f′,s,(xpf′ (e)) where Ae′,f′,s is an atom of the σ-algebra Be′,f′,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='27) � e∈Hn d,k 1Are π(e),s(xe) = � f∈Hn d,k−1 � e∈Hn d,k,f′∈∂π(e) pf′ (e)=f 1Are π(e),f′,s(xf) = � f∈Hn d,k−1 gr f,s (xf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It follows that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='28) N d λ∆0,Qs(L′ j)(1Are π(e),s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = N d λ∆0,Qs(L′ j) (gr f,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) and hence that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='29) N d λ∆0,Qs(L′ j)( ¯f m e,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = � r αs,r,m N d λ∆0,Qs(L′ j) (gr f,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) and similarly (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='30) Md λ,Qs(L′ j)( ¯f m e,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = � r αr,s,m Md λ,Qs(L′ j) (gr f,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that number of index vectors r = (re)e∈Hn d,k is RD with D := |Hn d,k| and hence RD ≤ M if C1 ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Writing j′ := index(L′ j) and J′ := index(L′ j+1) it then follows from our inductive hypothesis functions, applied with respect to the ε1-admissible sequence of scales Lj′+1 ≥ Lj′+2 ≥ · · · ≥ LJ′−1 which is possible as J′ − j′ ≫ Jk−1(ε1, RD), that there is a scale Lj with j′ ≤ j < J′ so that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='31) Nλ∆0,Qs(L′ j) (gr s,f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) = Mλ,Qs(L′ j) (gr s,f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) + O(ε1) for all λ ∈ [Lj+1, Lj] uniformly in r for s /∈ Sε1, where Sε1 ⊆ ΓL′ j,Q is a set of size |Sε1| ≤ ε1|ΓL′ j,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the cubes Qt(L0) form a partition of Q as t runs through the grid ΓL0,Q the relative density of the set Sε1 can substantially increase only of a few cubes Qt(L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed, it is easy to see that |T ′′ ε1| ≤ ε1/2 1 |ΓL0,Q| for the set T ′′ ε1 := {t ∈ ΓL0,Q : |Sε1 ∩ Qt(L0)| ≥ ε1/2 1 |ΓL′ j,Q ∩ Qt(L0)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We claim that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) holds for λ ∈ [Lj+1, Lj] uniformly in t /∈ Tε := T ′ ε ∪ T ′′ ε1, e ∈ Hn d,k, and 1 ≤ m ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed, from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='18), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='31) and the fact that |αs,r| ≤ 1, it follows N d λ∆0,Qs(L′ j) ( ¯fe,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Md λ,Qs(L′ j) ( ¯fe,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) for s /∈ Sε1 ∩ Qt(L0) since RDε1 ≪ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Finally, the fact that t /∈ T ′′ ε1 together with localization, namely (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='21) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='22), ensures that averaging over ΓL′ j,Qt(L0) gives N d λ∆0,Qt(L0) ( ¯fe,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Md λ,Qt(L0) ( ¯fe,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) + O(ε1/2 1 ) which in light of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='19), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='20), and the fact that ε1 ≪ ε2 complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The argument is similar to that of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Fix an edge, say e0 = (11, 12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', 1k), and partition the edges e ∈ Hn d,k in to as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let H0 be the set of those edges e for which 1 /∈ π(e), and for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , n1 let Hl denote the collection of edges of the form e = (1l, j2l2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , jklk), in other words e ∈ Hl if e = (1l, e′) for some edge e′ = (j2l2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , jklk) ∈ Hn d−1,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Accordingly write � e∈Hn d,k fe(xe) = � e∈H0 fe(xe) n1 � l=1 � e′∈Hn d−1,k−1 f1l,e′(x1l, xe′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 25 For x ∈ Q1 and x′ = (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xd) with xi ∈ Qni i , define (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='32) gl(x, x′) := � e′∈Hn d−1,k−1 f1l,e′(x1l, xe′) Then one may write (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='33) N d λ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' xd � e∈H0 fe(xe) � x1 n1 � l=1 gl(x1l, x′) dσλ 1 (x1) � dσλ d (xd) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dσλ 2 (x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For the inner integrals we have, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6), the estimate � x1 n1 � l=1 gl(x1l, x′) dσλ 1 �2 ≤ ∥g1∥2 U1 L(Q) + O(ε2k) = y11 ˆ y12 g1(y11)g1(y12)ψ1 L(y12 − y11) dy11 dy12 + O(ε2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' provided 0 < L ≪ (ε2k)6λ, where as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 we use the notation ψi L(y2 − y1) = ˆ t χi L(y1 − t)χi L(y2 − t) dt with χi L := L−ni1[−L/2,L/2]ni for 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By Cauchy-Schwarz we then have ���N d λ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k ��� 2 ≤ ˆ y1 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' xd � e′∈Hn d−1,k−1 f11,e′(x11, xe′)f11,e′(x12, xe′) dσλ d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dσλ 2 dω1 L(y1) + O(ε2k) where dωi L(yi) = |Qi|−1ψi L(yi2 − yi1) dyi1 dyi2 with yi = (yi1, yi2) ∈ Q2 i for 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The expression we have obtained above is similar to the one in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) except for the following changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The variable x1 ∈ Qn1 1 is replaced by y1 ∈ Q2 1 and the measure dσλ 1 by dω1 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The functions f1l,e′ are replaced by f11,e′, for 1 ≤ l ≤ n1, while the functions fe for all e ∈ Hn d,k such that 1 /∈ π(e) are eliminated, that is replaced by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Repeating the same procedure for i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , k replaces all variables xi with variables yi as well as the measures dσλ i with dωi L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The procedure eliminates all functions fe when e is an edge such that i /∈ π(e) for some 1 ≤ i ≤ k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' for the remaining edges, when π(e) = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , k), it replaces the functions fe with fe0 = f11,21,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For k < i the variables xi and the measures dσλ i are not changed, however integrating in these variables will have no contribution as the measures are normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus one obtains the following final estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='34) ���Nλ∆0,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k ��� 2k ≤ 1 |Q1| ˆ y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1 |Qk| ˆ yk � e∈H2 k,k fe0(ye) k � i=1 ψi L(yi2 − yi1) dyi1 dyi2 + O(ε2k) noting that these integrals are not normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus, one may write the expression in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='34), using a change of variables yi1 := yi1 − ti, yi2 := yi2 − ti, as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='35) 1 |Q1| ˆ t1 y1∈t1+Q1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1 |Qk| ˆ tk yk∈tk+Qk � e∈H2 k,k fe0(ye) dy1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' dyk dt = ∥fe0∥2k □L(Qπ(e0)) + O(ε2k) where the last equality follows from the facts that the function fe0 is supported on the cube Qπ(e0) and hence the integration in t is restricted to the cube Q + Q(L), giving rise an error of O(L/l(Q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14) follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='34) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='35) noting that the above procedure can be applied to any e ∈ Hn d,k in place of e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) is established similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For j = 0 we set Be′,t(L0) := {Qt(L0), ∅} and Be′,f′,s(L0) := {Qsf′ (L0), ∅} for e′ ∈ Hd,k, f′ ∈ ∂e′, and t, s ∈ ΓL0,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will develop σ-algebras Be′,t(Lj) of scale Lj such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) holds with complex(Be′,f′,s(Lj)) ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We define the total energy of a family of functions f m e,t with respect to a family of σ-algebras Be′,t(Lj) as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='36) E(f m e,t|Be′,t(Lj)) := Et∈ΓL0,Q M � m=1 � e∈Hn d,k ∥E(f m e,t|Bπ(e),t(Lj))∥2 L2(Qtπ(e)(L0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 26 NEIL LYALL ´AKOS MAGYAR Since |f m e,t| ≤ 1 for all e, m, and t it follows that the total energy is bounded by M · |Hn d,k| = O(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Our strategy will be to show that if (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) does not hold then there exist a family of σ-algebras Be′,t(Lj+2) such that the total energy of the family of functions f m e,t is increased by at least ckε2k+3 with respect to this new family of σ-algebras, and at the same time ensuring that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) remains valid with complex(Be′,f′,s(Lj+2)) ≤ j + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This iterative process must stop at some j = O(M ε−2k+3) proving the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Assume that we have developed σ-algebras Be′,t(Lj) and Be′,f′,s(Lj) of scale Lj such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) holds with complex(Be′,f′,s(Lj)) ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) does not hold then |Tε| ≥ ε|ΓL0,Q| for the set Tε := {t ∈ ΓL0,Q : ∥f m e,t − E(f m e,t|Bπ(e),t(Lj))∥□Lj+1 (Qtπ(e) (L0)) ≥ ε for some e ∈ Hn d,k and 1 ≤ m ≤ M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Fix t ∈ Tε and let e ∈ Hn d,k and 1 ≤ m ≤ M be such that ∥f m e,t − E(f m e,t|Bπ(e),t(Lj))∥□Lj+1(Qtπ(e)(L0)) ≥ ε and write e′ := π(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Consider the partition of the cube Qte′ (L0) into small cubes Qse′ (Lj+2) where se′ ∈ ΓLj+2,Qe′ ∩Qte′ (L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By the localization properties of the □Lj+1(Q)-norm, and the fact that Lj+2 ≪ ε2kLj+1 we have that ∥f∥2k □Lj+1(Qte′ (L0)) ≤ Ese′ ∈ΓLj+2,Qte′ (L0) ∥f∥2k □(Qse′ (Lj+2)) + ε2k 2 for any function f : Qte′(L0) → [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus there exists a set Sε,e,t ⊆ ΓLj+2,Qte′ (L0) of size |Sε,e,t| ≥ ε2k 4 |ΓLj+2,Qte′ (L0)| such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='37) ∥f m e,t − E(f m e,t|Be′,t(Lj))∥2k □(Qse′ (Lj+2) ≥ ε2k 4 for all se′ ∈ Sε,e,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For a given cube Q and functions f, g : Q → R, define the normalized inner product of f and g as ⟨f, g⟩Q := Q f(x)g(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then by the well-known property of the □-norm, see for example [23] or the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='37) that there exits sets Bf′,se′ ,t ⊆ Qsf′ (Lj+2) for f′ ∈ ∂e′ such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='38) � f m e,t − E(f m e,t|Be′,t(Lj)) , � f′∈∂e′ 1Bf′,se′ ,t � Qse′ (Lj+2) ≥ ε2k 2k+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If s ∈ ΓLj+2,Q then there is a unique t = t(s) ∈ ΓL0,Q such that s ∈ Qt(L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If t ∈ Tε and se′ ∈ Sε,e,t then we define the σ-algebras Bf′,e′,s(Lj+2) on Qsf′ (Lj+2) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Write Bf′,e′,s = Bf′,se′ ,t where t = t(s) and let Bf′,e′,s(Lj+2) be the σ-algebra generated by the set Bf′,e′,s and the σ-algebra Bf′,e′,s′(Lj) restricted to Qsf′ (Lj+2) where s′ ∈ ΓLj,Q is the unique element so that s ∈ Qs′(Lj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that that the complexity of the σ-algebra Bf′,e′,s(Lj+2) is at most one larger then the complexity of the σ-algebra Bf′,e′,s′(Lj) as restricting a σ-algebra to a set does not increase its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If t = t(s) /∈ Tε or se′ /∈ Sε,e,t then let Bf′,e′,s(Lj+2) be simply the restriction of Bf′,e′,s′(Lj) to the cube Qsf′ (Lj+2), or equivalently define the sets Bf′,e′,s := Qsf′ (Lj+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Finally, let (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='39) Be′,s(Lj+2) := � f′∈∂e′ Bf′,e′,s(Lj+2) be the corresponding σ-algebra on the cube Qse′ (Lj+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the cubes Qse′ (Lj+2) partition the cube Qte′ (L0) as se′ runs through the grid ΓLj+2,Qe′ ∩ Qte′ (L0), these σ-algebras define a σ-algebra Be′,t(Lj+2) on Qte′(L0), such that its restriction to the cubes Qse′ (Lj+2) is equal to the σ-algebras Be′,s(Lj+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 27 Since the function � f′∈∂e′ 1Bf′,e′,s is measurable with respect to the σ-algebra Be′,t(Lj+2) restricted to the cube Qse′ (Lj+2) one clearly has (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='40) ⟨ f m e,t − E(f m e,t|Be′,t(Lj+2)), � f′∈∂e′ 1Bf′,e′,s ⟩Qse′ (Lj+2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' and hence, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='38), that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='41) ⟨ E(f m e,t|Be′,t(Lj+2)) − E(f m e,t|Be′,t(Lj)), � f′∈∂e′ 1Bf′,e′,s ⟩Qse′ (Lj+2) ≥ ε2k 2k+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It then follows from Cauchy-Schwarz and orthogonality, using the fact that the σ-algebra Be′,t(Lj+2)) is a refinement of Be′,t(Lj+2), that ∥E(f m e,t|Be′,t(Lj+2))−E(f m e,t|Be′,t(Lj))∥2 L2(Qse′ (Lj+2)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='42) = ∥E(f m e,t|Be′,t(Lj+2))∥2 L2(Qse′ (Lj+2)) − ∥E(f m e,t|Be′,t(Lj))∥2 L2(Qse′ (Lj+2)) ≥ � ε2k 2k+2 �2 for se′ ∈ Sε,e,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since |Sε,e,t| ≥ ε2k 4 |ΓLj+2,Qte′ (L0)| averaging over se′ ∈ ΓLj+2,Qte′ (L0) implies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='43) ∥E(f m e,t|Be′,t(Lj+2))∥2 L2(Qte′ (L0)) ≥ ∥E(f m e,t|Be′,t(Lj))∥2 L2(Qte′ (L0)) + ε2k+2 22k+6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' At this point we have shown that if t ∈ Tε then there exists an edge e ∈ Hn d,k, 1 ≤ m ≤ M, and σ-algebras Be′,t(Lj+2)) of scale Lj+2 on Qte′ (L0), with e′ = π(e), such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='43) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For all e′′ ∈ Hd,k with e′′ ̸= e′ let Bf′,e′′,s(Lj+2) be the restriction of the σ-algebra Bf′,e′′,s′(Lj) to the cube Qsf′ (Lj+2), where s′ is such that s ∈ Qs′(Lj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='39) this implies that Be′′,s(Lj+2) is also the restriction of Be′′,s′(Lj) to the cube Qse′′ (Lj+2), and hence the σ-algebra Be′′,t(Lj+2) is generated by the grid GLj+2,Qte′′ (L0) and the σ-algebra Be′′,t(Lj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We have therefore defined a family of the σ-algebras Be′,t(Lj+2) for e′ ∈ Hd,k, satisfying M � m=1 � e∈Hn d,k ∥E(f m e,t|Bπ(e),t(Lj+2))∥2 L2(Qtπ(e) (L0)) ≥ M � m=1 � e′∈Hn d,k ∥E(f m e,t|Bπ(e),t(Lj))∥2 L2(Qtπ(e)(L0)) + ε2k+2 22k+6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Using the fact that |Tε| ≥ ε|ΓL0,Q| and averaging over t ∈ ΓL0,Q it follows using the notations of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='36) that E(f m e,t|Be′,t(Lj+2)) ≥ E(f m e,t|Be′,t(Lj)) + ε2k+3 22k+6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' As the total energy E(f m e,t|Be′,t(Lj)) is bounded by O(M), the process must stop at a step j = O(M ε−2k+3) where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) holds for a σ-algebra of “local complexity” at most j, completing the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The base case of an inductive strategy to establish Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4 In this section we will ultimately establish the base case of our more general inductive argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will however start by giving a (new) proof of Theorem B′, namely the case d = 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A Single Simplex in Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆0 = {v1 = 0, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , vn1} be a fixed non-degenerate simplex of n1 points in Zn with n = 2n1 + 3 and define tkl := vk · vl for 2 ≤ k, l ≤ n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall, see [17], that a simplex ∆ = {m1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1} ⊆ Zn is isometric to λ∆0 if and only if mk · ml = λ2tkl for all 2 ≤ k, l ≤ n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any positive integer q and λ ∈ q √ N we define Sλ∆0,q(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1) : Zn(n1−1) → {0, 1} be the function whose value is 1 if mk · ml = λ2tkl with both mk and ml in (qZ)n for all 2 ≤ k, l ≤ n1 and is equal to 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is a well-known fact in number theory, see [11] or [17], that for n ≥ 2n1 + 1 we have that � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1 Sλ∆0,q(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1) = ρ(∆0) (λ/q)(n−n1)(n1−1)(1 + O(λ−τ)) 28 NEIL LYALL ´AKOS MAGYAR for some absolute constant τ > 0 and some constant ρ(∆0) > 0, the so-called singular series, which can be interpreted as the product of the densities of the solutions of the above system of equations among the p-adics and among the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus if we define σλ∆0,q := ρ(∆0)−1(λ/q)−(n−n1)(n1−1)Sλ∆0,q then σλ∆0,q is normalized in so much that � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1 σλ∆0,q(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1) = 1 + O(λ−τ) for some absolute constant τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q ⊆ Zn be a fixed cube and let l(Q) denotes its side length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any family of functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1 : Q → [−1, 1] and 0 < λ ≪ l(Q) we define the following two multi-linear expressions (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) N 1 λ∆0,q,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1) := Em1∈Q � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1 f1(m1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' fn1(mn1) σλ∆0,q(m2 − m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1 − m1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) M1 λ,q,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1) := Et∈Q Em1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1∈t+Q(q,λ) f1(m1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' fn1(mn1) where Q(q, λ) := [− λ 2 , λ 2 ]n ∩ (qZ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if S ⊆ Q and N 1 λ∆0,q,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) > 0 then S must contain an isometric copy of λ∆0, while if |S| ≥ δ|Q| for some δ > 0 then as before H¨older implies that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3) M1 λ,q,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) ≥ δn − O(ε) for all scales λ ∈ q √ N with 0 < λ ≪ ε l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that for any 0 < ε ≪ 1 and positive integer q we call a sequence L1 ≥ · · · ≥ LJ (ε, q)-admissible if Lj/Lj+1 ∈ N and Lj+1 ≪ ε2Lj for all 1 ≤ j < J and LJ/q ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if λ1 ≥ · · · ≥ λJ′ ≥ 1 is any lacunary sequence in q √ N with J′ ≫ (log ε−1) J + log q, one can always finds an (ε, q)-admissible sequence of scales L1 ≥ · · · ≥ LJ with the property that for each 1 ≤ j < J the interval [Lj+1, Lj] contains at least two consecutive elements from the original lacunary sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In light of these observations we see that the following “counting lemma” ultimately establishes a quanti- tatively stronger version of Proposition B′ that appeared in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 and hence immediately establishes Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4 for d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1 and qj := q1(ε)j for j ≥ 1 with q1(ε) := lcm{1 ≤ q ≤ Cε−10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists J1 = O(ε−2) such that for any (ε, qJ1)-admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJ1 and S ⊆ Q there is some 1 ≤ j < J1 such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) N 1 λ∆0,qj,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) = M1 λ,qj,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) + O(ε) for all λ ∈ qj √ N with Lj+1 ≤ λ ≤ Lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' As in the continuous setting the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 has two main ingredients, namely Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In these lemmas, and for the remainder Sections 6 and 7, we will continue to use the notation q1(ε) := lcm{1 ≤ q ≤ Cε−10} for any given ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 (A Generalized von Neumann inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1, q, q′ ∈ N with qq1(ε)|q′, and λ ∈ q √ N with λ ≪ l(Q) and 1 ≪ L ≪ ε10λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any collection of functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1 : Q → [−1, 1] we have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) |N 1 λ∆0,q,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1)| ≤ min 1≤i≤n1 ∥fi∥U1 q′,L(Q) + O(ε) where for any function f : Q → [−1, 1] we define (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) ∥f∥U1 q,L(Q) := � 1 |Q| � t∈Q |f ∗ χq,L(t)|2�1/2 WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 29 with χq,L denoting the normalized characteristic function of the cubes Q(q, L) := [− L 2 , L 2 ]n ∩ (qZ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any cube Q ⊆ Zn of side length l(Q) and q, L ∈ N satisfying q ≪ L with L dividing l(Q), we shall now partition Q into cubic grids Qt(q, L) = t + ((qZ)n ∩ Q(L)), with Q(L) = [− L 2 , L 2 ]n as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' These grids form the atoms of a σ-algebra Gq,L,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if q|q′ and L′|L then Gq,L,Q ⊆ Gq′,L′,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (A Koopman-von Neumann type decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1 and qj := q1(ε)j for all j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer ¯J1 = O(ε−2) such that any (ε, q ¯ J1)- admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ L ¯ J1 and function f : Q → [−1, 1] there is some 1 ≤ j < ¯J1 such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) ∥f − E(f|Gqj,Lj,Q)∥U1 qj+1,Lj+1(Q) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The reduction of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 to these two lemmas is essentially identical to the analogous argument in the continuous setting as presented at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, we choose to omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will rely on some prior exponential sum estimates, specifically Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4 in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' First we deal with the case n1 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By the change of variables m1 := m1, mi := mi − m1 for 2 ≤ i ≤ n1, one may write N 1 λ∆0,q,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1) := Em1∈QN � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1 f1(m1)f2(m1 + m2) · · · fn1(m1 + mn1) σλ∆0,q(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We now write σλ∆0,q(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1) = σλ∆0′,q(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1−1) σ m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1−1 λ,q (mn1) where ∆0′ = {v1 = 0, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , vn1−1} and for each m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1−1 ∈ (qZ)n we are using σ m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1−1 λ,q (m) denote the (essentially) normalized indicator function of the subset of (qZ)n that contains m if and only if m · mk = λ2tkn1 for all 2 ≤ k ≤ n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Using the fact that |fi| ≤ 1, together with Cauchy-Schwarz and Plancherel, one can then easily see that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) |N 1 λ∆0,q,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1)|2 ≤ |Q|−1 ˆ ξ∈Tn | �fn1(ξ)|2Hλ,q(ξ) dξ with Hλ,q(ξ) = � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1 σλ∆0′,q(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mn1−1) | � σ m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mn1−1 λ,q (ξ)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It then follows by Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4 in [17], with δ = ε4 and after rescaling by q, that in addition to being non-negative and uniformly bounded in ξ we in fact have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) Hλ,q(ξ) = O(ε) whenever ����qξ − l q1(ε) ���� ≥ q ε4λ, for all l ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We note that the expression Hλ,q(ξ) may be interpreted as the Fourier transform of the indicator function of the set of integer points on a certain variety, and estimate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) indicates that this concentrates near rational points of small denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is this crucial fact from number theory which makes results like Theorem B′ possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since �χq,L(ξ) = qn Ln � m∈[− L 2 , L 2 )n, q|m e−2πim·ξ it is easy to see that �χq,L(l/q) = 1 for all l ∈ Zn and that there exists some absolute constant C > 0 such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) 0 ≤ 1 − �χq,L(ξ)2 ≤ C L |ξ − l/q| for all ξ ∈ Tn and l ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is then easy to see using our assumption that qq1(ε)|q′ that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) 0 ≤ Hλ,q(ξ)(1 − �χq′,L(ξ)2) ≤ Cε 30 NEIL LYALL ´AKOS MAGYAR for some constant C > 0 uniformly in ξ ∈ Tn provided L ≪ ε5λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Substituting inequality (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8), we obtain |N 1 λ∆0,q,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fn1)|2 ≤ |Q|−1 �ˆ | ˆfn1(ξ)|2Hλ(ξ)�χq′,L(ξ)2 dξ + ˆ | ˆfn1(ξ)|2Hλ(ξ)(1 − �χq′,L(ξ)2) dξ � ≤ ∥fn1∥2 U1 q′,L(Q) + O(ε) provided L ≪ ε5λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This proves Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 for k ≥ 3, as it is clear that by re-indexing the above estimate holds for any of the functions fi in place of fn1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For n1 = 2 an easy modification of arguments in [14], specifically the proof of Lemma 3 therein, establishes that |N 1 λ∆0,q,Q(f1, f2)|2 ≤ ∥fi∥2 U1 q′,L(Q) + O(ε) for i = 1, 2 provided L ≪ ε5λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let q, L ∈ N such that L|N, q|L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The “modulo q” grids Qt(q, L) = t+Q(q, L) partition the cube Q with t running through the set Γq,L,Q = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', q}n + ΓL,Q, where ΓL,Q denote the centers of the “integer” grids t + Q(L) in an initial partition of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let q′, L′ be positive integers so that q|q′, L′|L and L′ ≪ ε2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If s ∈ Γq′,L′,Q and t ∈ Qs(q′, L′) then |t − s| = O(L′) and hence Ex∈Qt(q,L)g(x) = Ex∈Qs(q,L)g(x) + O(L′/L) for any function g : Q → [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover, since the cube Qs(q, L) is partitioned into the smaller cubes Qt(q′, L′), we have by Cauchy-Schwarz |Ex∈Qs(q,L) g(x)|2 ≤ Et∈Γq′,L′,Qs(q,L)|Ex∈Qt(q′,L′)g(x)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' From this it is easy to see that ∥g∥2 U1 q,L(Q) = Et∈Q|Ex∈Qt(q,L)g(x)|2 ≤ Et∈Γq′,L′,Q |Ex∈Qt(q′,L′)g(x)|2 + O(L′/L) and we note that the right side of the above expression is ∥E(g|Gq′,L′,Q)∥2 L2(Q) since the conditional expecta- tion function E(g|Gq′,L′,Q) is constant and equal to Ex∈Qt(q′,L′)g(x) on the cubes Qt(q′, L′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Now suppose (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) does not hold for some j ≥ 1, that is ∥f − E(f|Gqj,Lj,Q)∥2 U1 qj+1,Lj+1 (Q) ≥ ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since Lj+2 ≪ ε2Lj+1, Lj+2|Lj, and qj+1|qj+2 we can apply the above observations to g := f − E(f|Gqj,Lj,Q) and obtain, by orthogonality, that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) ∥E(f|Gqj+2,Lj+2,Q)∥2 L2(Q) ≥ ∥E(f|Gqj,Lj,Q)∥2 L2(Q) + cε2 for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the above expressions are clearly bounded by 1, the above procedure must stop in O(ε−2) steps at which (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) must hold for some 1 ≤ j ≤ ¯J1(ε) with ¯J1(ε) = O(ε−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The base case of our general inductive strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q = Q1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × Qd with Qi ⊆ Z2ni+3 be cubes of equal side length l(Q) and ∆0 i ⊆ Z2ni+3 be a non-degenerate simplex of ni points for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We note that for any q0 ∈ N and scale L0 dividing l(Q) if t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , td) ∈ Γq0,L0,Q, then the corresponding grids Qt(q0, L0) in the partition of Q take the form Qt(q0, L0) = Qt1(q0, L0) × · · · × Qtd(q0, L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' As in the continuous setting we will ultimately need a parametric version of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, namely Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (Parametric Counting Lemma on Zn for Simplices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≤ 1 and R ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer J1 = J1(ε, R) = O(R ε−4) such that for any (ε, qJ1)-admissible sequence of scales L0 ≥ L1 ≥ · · · ≥ LJ1 with L0 dividing l(Q) and qj := q0q1(ε)j for 0 ≤ j ≤ J1 with q0 ∈ N, and collection of functions f i,r k,t : Qti(q0, L0) → [−1, 1] with 1 ≤ i ≤ d, 1 ≤ k ≤ ni, 1 ≤ r ≤ R and t ∈ Γq0,L0,Q there exists 1 ≤ j < J1 and a set Tε ⊆ Γq0,L0,Q of size |Tε| ≤ ε|Γq0,L0,Q| such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13) N 1 λ∆0 i ,qj,Qti (q0,L0)(f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f i,r ni,t) = M1 λ,qj,Qti (q0,L0)(f i,r 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , f i,r ni,t) + O(ε) WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 31 for all λ ∈ qj √ N with Lj+1 ≤ λ ≤ Lj and t /∈ Tε uniformly in 1 ≤ i ≤ d and 1 ≤ r ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This proposition follows, as the analogous result did in the continuous setting, from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and the follow parametric version of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 (A simultaneous Koopman-von Neumann type decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1, m ≥ 1, and Q ⊆ Zn be a cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists an integer ¯J1 = O(mε−3) such that for any (ε, q ¯ J1)-admissible sequence L0 ≥ L1 ≥ · · · ≥ L ¯ J1 with L0 dividing l(Q) and qj := q0q1(ε)j for 0 ≤ j ≤ ¯J1 with q0 ∈ N, and collection of functions f1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' fm,t : Qt(q0, L0) → [−1, 1] defined for each t ∈ Γq0,L0,Q, there is some 1 ≤ j < ¯J1 and a set Tε ⊆ Γq0,L0,Q of size |Tε| ≤ ε|Γq0,L0,Q| such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14) ∥fi,t − E(fi,t|Gqj,Lj,Qt(q0,L0)∥U1 qj+1,Lj+1 (Qt(q0,L0)) ≤ ε for all 1 ≤ i ≤ m and t /∈ Tε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 above is of course the discrete analogue of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Since the proofs of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 are almost identical to the arguments presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 we choose to omit these details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4: The general case After the preparations in Section 6 we can proceed very similarly as in Section 5 to prove our main result in the discrete case, namely Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The main difference will be that given 0 < ε ≪ 1 and 1 ≤ k ≤ d, we construct a positive integer qk(ε) and assume that all our sequences of scales will be (ε, qk(ε))-admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The cubes Qt(L) will be naturally now be replaced by the grids Qt(q, L) of the form that already appear in Section 6 where we always assume q|L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆0 = ∆0 1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × ∆0 d with each ∆0 i ⊆ Z2ni+3 a non-degenerate simplex of ni points for 1 ≤ i ≤ d and Q = Q1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × Qd ⊆ Zn with Qi ⊆ Z2ni+3 cubes of equal side length l(Q) (taken much larger than the diameter of ∆0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will use the same parameterizations in terms of hypergraph bundles Hn d,k and corresponding notations as in Section 5 to count the configurations ∆ = ∆1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × ∆d ⊆ Q with each ∆i ⊆ Qi an isometric copy of λ∆0 i for some λ ∈ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Given any positive integer q and λ ∈ q √ N we will make use of the notation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) � xi f(xi) σi λ,q(xi) := Exi1∈Qi � xi2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',xini f(xi) σλ∆0 i ,q(xi2 − xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xini − xi1) dxi1 with σλ∆0 i ,q as defined in the previous section and xi = (xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xini) ∈ Qni i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if S ⊆ Q then the density of configurations ∆ in S, of the form ∆ = ∆1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × ∆d with each ∆i ⊆ Qi an isometric copy of λ∆0 i for some λ ∈ q √ N is given by the expression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) N d λ∆0,q,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) := � x1 · · � xd � e∈Hn d,d 1S(xe) σ1 λ,q(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' σd λ,q(xd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' More generally, for any given 1 ≤ k ≤ d and a family of functions fe : Qπ(e) → [−1, 1] with e ∈ Hn d,k we define the multi-linear expression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3) N d λ∆0,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) := � x1 · · � xd � e∈Hn d,k fe(xe) σ1 λ,q(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='σd λ,q(xd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' as well as (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) Md λ,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) := Et∈Q Md t+Q(q,L) (fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) where Q(q, L) = Q1(q, L) × · · · × Qd(q, L) with each Qi(q, L) = (qZ ∩ [− L 2 , L 2 ])2ni+3 and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) Md � Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) := Ex1∈ � Qn1 1 · · · Exd∈ � Q nd d � e∈Hn d,k fe(xe) for any cube �Q ⊆ Q of the form �Q = �Q1 × · · · × �Qd with �Qi ⊆ Qi for 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 32 NEIL LYALL ´AKOS MAGYAR We note that it is easy to show, as in the continuous, that if S ⊆ Q with |S| ≥ δ|Q| for some δ > 0 then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6) Md λ,q,Q(1S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) ≥ δn1··· nd − O(ε) for all scales λ ∈ q √ N with 0 < λ ≪ ε l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In light of this observation and the discussion preceding Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4 reduces, as it did in the continuous setting, to the following Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exist positive integers Jd = Jd(ε) and qd(ε) such that for any (ε, qd(ε)Jd)-admissible sequence of scales l(Q) ≥ L1 ≥ · · · ≥ LJ1 and S ⊆ Q there is some 1 ≤ j < Jd such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7) N d λ∆0,qj,Q(1S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) = Md λ,qj,Q(1S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,d) + O(ε), for all λ ∈ qj √ N with Lj+1 ≤ λ ≤ Lj with qj := qd(ε)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Quantitative Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' A careful analysis of our proof reveals that there exist choices of Jd(ε) and qd(ε) which are less than Wd(log(C∆ε−3)) and Wd(C∆ε−13) respectively where Wk(m) is again the tower- exponential function defined by W1(m) = exp(m) and Wk+1(m) = exp(Wk(m)) for k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 follows along the same lines as the analogous result in the continuous setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' As before we will compare the averages N d λ∆0,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) to those of Md λ,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k), at certain scales q and λ ∈ q √ N with with Lj+1 ≤ λ ≤ Lj, inductively for 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' As the arguments closely follow those given in Section 5 we will be brief and emphasize mainly just the additional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Reduction of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 to a more general “local” counting lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any given 1 ≤ k ≤ d and a family of functions fe : Qπ(e) → [−1, 1] with e ∈ Hn d,k it is easy to see that for any ε > 0, scale L0 > 0 dividing the side-length l(Q), and q0|q we have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) N d λ∆0,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Et∈Γq0,L0,Q N d λ∆0,q,Qt(q0,L0)(fe,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) Md λ,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Et∈ΓL,Q Md λ,q,Qt(q0,L0)(fe,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) provided 0 < λ ≪ εL0 where fe,t denotes the restriction of a function fe to the cube Qt(q0, L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 reduces to showing that the expressions in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='8) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='9) only differ by O(ε) for all scales λ ∈ q √ N with Lj+1 ≤ λ ≤ Lj, given an (ε, q)-admissible sequence L0 ≥ L1 ≥ · · · ≥ LJ, for any collection of bounded functions fe,t, e ∈ Hn d,k, t ∈ Γq0,L0,Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Indeed, our crucial result will be the following Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (Local Counting Lemma in Zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1 and q0, M ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exist positive integers Jk = Jk(ε, M) and qk(ε) such that for any (ε, qJd)-admissible sequence of scales L0 ≥ L1 ≥ · · · ≥ LJ1 with L0 dividing l(Q) and qj := q0 qk(ε)j for j ≥ 1, and collection of functions f m e,t : Qtπ(e)(q0, L0) :→ [−1, 1] with e ∈ Hn d,k, 1 ≤ m ≤ M and t ∈ Γq0,L0,Q there exists 1 ≤ j < Jk and a set Tε ⊆ Γq0,L0,Q of size |Tε| ≤ ε|Γq0,L0,Q| such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='10) N d λ∆0,qj,Qt(q0,L0)(fe,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = Md λ,qj,Qt(q0,L0)(fe,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) + O(ε) for all λ ∈ qj √ N with Lj+1 ≤ λ ≤ Lj and t /∈ Tε uniformly in e ∈ Hn d,k and 1 ≤ m ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if k = d, L0 = l(Q), q0 = M = 1, then |Γq0,L0,Q| = 1, and moreover if fe,t = 1S for all e ∈ Hn d,k for a set S ⊆ Q, then Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 reduces to precisely Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' In fact, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 is a parametric, multi-linear and simultaneous extension of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 which we need in the induction step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' when going from level k − 1 to level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 33 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will prove Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 by induction on 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For k = 1 this is basically Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, exactly as it was in the base case of the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For the induction step we will again need two main ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The first establishes that the our multi- linear forms N d λ∆0,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) are controlled by a box-type norm attached to scales q′ and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q = Q1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' × Qd with Qi ⊆ Z2ni+3 be cubes of equal side length l(Q) and 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any scale 0 < L ≪ l(Q) and function f : Qe′ → [−1, 1] with e′ ∈ Hd,k we define its local box norm at scales q′ and L by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) ∥f∥2k □q′,L(Qe′ ) := Es∈Qe′ ∥f∥2k □(Qs(q′,L)) where (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) ∥f∥2k □( � Q) := Ex11,x12∈ � Q1 · · · Exk1,xk2∈ � Qk � (ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',ℓk)∈{1,2}k f(x1ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , xkℓk) for any cube �Q of the form �Q = �Q1 × · · · × �Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We note that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) are special cases of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='11) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='12) with k = d, n = (2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 2), and fe = f for all e ∈ Hn d,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 (A Generalized von-Neumann inequality on Zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 1 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1, q, q′ ∈ N with qq1(ε)|q′, and λ ∈ q √ N with λ ≪ l(Q) and 1 ≪ L ≪ (ε2k)10λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any collection of functions fe : Qπ(e) → [−1, 1] with e ∈ Hn d,k we have both (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13) |N d λ∆0,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k)| ≤ min e∈Hn d,k ∥fe∥□q′,L′ (Qπ(e)) + O(ε) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14) |Md λ,q,Q(fe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k)| ≤ min e∈Hn d,k ∥fe∥□q′,L′ (Qπ(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof of inequalities (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='13) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='14) follow exactly as in the continuous case, see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1, using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 in place of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The crucial ingredient is again a parametric weak hypergraph regularity lemma, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 adapted to the discrete settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The proof is essentially the same as in the continuous case, with exception that the □Lj-norms are replaced by □qj,Lj-norms where qj = q0qj is a given sequence of positive integers and L0 ≥ L1 ≥ · · · ≥ LJ is an (ε, qJ)-admissible sequence of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' To state it we say that a σ-algebra B on a cube Q is of scale (q, L) if it is refinement of the grid Gq,L,Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' if its atoms partition each cube Qt(q, L) of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will always assume that q|L and L|l(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall also that we say the complexity of a σ-algebra B is at most m, and write complex(B) ≤ m, if it is generated by m sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (Parametric weak hypergraph regularity lemma for Zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1, 1 ≤ k ≤ d, q0, q, L0, M ∈ N, and let qj := q0qj for j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' There exists ¯Jk = O(Mε−2k+3) such that for any (ε2k, q ¯ Jk)-admissible sequence L0 ≥ L1 ≥ · · · ≥ L ¯ Jk with the property that L0 divides l(Q) and collection of functions f m e,t : Qtπ(e)(q0, L0) → [−1, 1] with e ∈ Hn d,k, 1 ≤ m ≤ M, and t ∈ Γq0,L0,Q there is some 1 ≤ j < ¯Jk and σ-algebras Be′,t of scale (qj, Lj) on Qte′ (q0, L0) for each t ∈ Γq0,L0,Q and e′ ∈ Hd,k such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) ∥f m e,t − E(f m e,t|Bπ(e),t)∥□qj+1,Lj+1 (Qtπ(e) (L0)) ≤ ε uniformly for all t /∈ Tε, e ∈ Hn d,k, and 1 ≤ m ≤ M, where Tε ⊆ Γq0,L0,Q with |Tε| ≤ ε|Γq0,L0,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Moreover, the σ-algebras Be′,t have the additional local structure that the exist σ-algebras Be′,f′,s on Qsf′ (qj, Lj) with complex(Be′,f′,s) = O(j) for each s ∈ Γqj,Lj,Q, e′ ∈ Hd,k, and f′ ∈ ∂e′ such that if s ∈ Qt(q0, L0), then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='16) Be′,t �� Qse′ (qj,Lj) = � f′∈∂e′ Be′,f′,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 34 NEIL LYALL ��AKOS MAGYAR The proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 follows exactly as the corresponding proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 in the continuous setting, so we will omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We will however provide some details of how one deduces Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, from Lemmas 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The arguments are again very similar to those in the continuous setting, however one needs to make a careful choice of the integers qk(ε), appearing in the statement of the Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 2 ≤ k ≤ d and assume that the lemma holds for k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1 and ε1 := exp (−C1ε−2k+3) for some large constant C1 = C1(n, k, d) ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We then define qk(ε) := qk−1(ε1) recalling that q1(ε) := lcm{1 ≤ q ≤ Cε−10} and note that it is easy to see by induction that qk(ε)|qk(ε′) for 0 < ε′ ≤ ε and qk−1(ε)|qk(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We further define the function F(ε) := Jk−1(ε1, M) with M = ε ε−1 1 and recall that qj := q0 qk(ε)j for j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We now proceed exactly as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 but with {Lj}j≥1 being a (ε1, q � J)-admissible sequence of scales, with �J ≫ F(ε) ¯Jk(ε, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We again choose a subsequence {L′ j} ⊆ {Lj} so that L′ 0 = L0 and index(L′ j+1) ≥ index(L′ j) + F(ε) + 2, but also now set q′ j = qj′, where j′ := index(L′ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 then guarantees the existence of σ-algebras Be′,t of scale (q′ j, L′ j) on Qte′ (q0, L0) for each t ∈ Γq0,L0,Q and e′ ∈ Hd,k, with the local structure described above, such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='15) holds uniformly for all t /∈ T ′ ε, e ∈ Hn d,k, and 1 ≤ m ≤ M, for some 1 ≤ j < ¯Jk(ε, M) = O(Mε−2k+3), where T ′ ε ⊆ Γq0,L0,Q with |T ′ ε| ≤ ε|Γq0,L0,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Arguing as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 we can conclude from this that for each j′ ≤ l < J′ we have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='17) N d λ∆0,ql,Qs(q′ j,L′ j)(f m e,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = � r αs,r,m N d λ∆0,ql,Qs(q′ j,L′ j) (gr f,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) + O(ε) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='18) Md λ,ql,Qs(q′ j,L′ j)(f m e,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' e ∈ Hn d,k) = � r αr,s,m Md λ,ql,Qs(q′ j,L′ j) (gr f,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) + O(ε) provided (ε−2k)10L′ j+1 ≪ λ with λ ∈ ql √ N, where each |αs,re| ≤ 1 and number of index vectors r = (re)e∈Hn d,k is RD with D := |Hn d,k| and hence RD ≤ M if C1 ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' By induction, we apply Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 to the sequence of scales L′ j = Lj′ ≥ Lj′+1 ≥ · · · ≥ LJ′ = L′ j+1 with ε1 > 0 and for ql := q′ j qk(ε)l−j′ = qj′ qk−1(ε1)l−j′ where j′ ≤ l ≤ J′ with respect to the family of functions gr s,f : Qsf(q′ j, L′ j) → [−1, 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This is possible as J′ − j′ ≫ Jk−1(ε1, RD) and our sequence of scales is (ε1, qJ′)-admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus there exists an index j′ ≤ l < J′ such that for all λ ∈ ql √ N with Ll+1 ≤ λ ≤ Ll we have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='19) N d λ∆0,ql,Qs(q′ j,L′ j) (gr f,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) = Md λ,ql,Qs(q′ j,L′ j) (gr f,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' f ∈ Hn d,k−1) + O(ε1) uniformly in r for s /∈ Sε1, where Sε1 ⊆ Γq′ j,L′ j,Q is a set of size |Sε1| ≤ ε1|Γq′ j,L′ j,Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The remainder of the proof follows as just as it did for Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Appendix: A short direct proof of Part (i) of Theorem B′ We conclude by providing a short direct proof of Part (i) of Theorem B′, namely the following Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 (Magyar [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < δ ≤ 1 and ∆ ⊆ Z2k+3 be a non-degenerate simplex of k points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If S ⊆ Z2k+3 has upper Banach density at least δ, then there exists an integer q0 = q0(δ) and λ0 = λ0(S, ∆) such that S contains an isometric copy of q0λ∆ for all λ ∈ √ N with λ ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any ε > 0 we define qε := lcm{1 ≤ q ≤ Cε−10} with C > 0 a (sufficiently) large absolute constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Following [14] we further define S ⊆ Zn to be ε-uniformly distributed (modulo qε) if its relative upper Banach density on any residue class modulo qε never exceeds (1 + ε2) times its density on Zn, namely if δ∗(S | s + (qεZ)d) ≤ (1 + ε2) δ∗(S) WEAK HYPERGRAPH REGULARITY AND APPLICATIONS TO GEOMETRIC RAMSEY THEORY 35 for all s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , qε}d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It turns out that this notion is closely related to the U 1 q,L(Q)-norm introduced in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall that for any cube Q ⊆ Zn and function f : Q → [−1, 1] we define (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1) ∥f∥U1 q,L(Q) := � 1 |Q| � t∈Q |f ∗ χq,L(t)|2�1/2 with χq,L denoting the normalized characteristic function of the cubes Q(q, L) := [− L 2 , L 2 ]n ∩ (qZ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that the U 1 q,L(Q)-norm measures the mean square oscillation of a function with respect to cubic grids of size L and gap q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' The following observation from [14] (specifically Lemmas 1 and 2) is key to our short proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If S ⊆ Zn be ε-uniformly distributed with δ := δ∗(S) > 0, then there exists an integer L = L(S, ε) > 0 and cubes Q of arbitrarily large side length l(Q) with l(Q) ≫ ε−4L such that ∥1S − δ1Q∥U1 qε,L(Q) = O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let ∆0 = {v1 = 0, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , vk} be a fixed non-degenerate simplex of k points in Zn with n = 2k + 3 and define tij := vi · vj for 2 ≤ i, j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We now define a function which counts isometric copies of λ∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recall, see [17], that a simplex ∆ = {m1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mk} ⊆ Zn is isometric to λ∆0 if and only if mi · mj = λ2tij for all 2 ≤ i, j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any λ ∈ √ N we define Sλ∆0(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mk) : Zn(k−1) → {0, 1} be the function whose value is 1 if mi · mj = λ2tij for all 2 ≤ i, j ≤ k and is equal to 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is a well-known fact in number theory, see [11] or [17], that for n ≥ 2k + 1 we have that � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mk Sλ∆0(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mk) = ρ(∆0) λ(n−k)(k−1)(1 + O(λ−τ)) for some absolute constant τ > 0 and constant ρ(∆0) > 0, the so-called singular series, which can be interpreted as the product of the densities of the solutions of the above system of equations among the p-adics and among the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus if we define σλ∆0 := ρ(∆0)−1λ−(n−k)(k−1)Sλ∆0 then σλ∆0 is normalized in so much that � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mk σλ∆0(m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mk) = 1 + O(λ−τ) for some absolute constant τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let Q ⊆ Zn be a fixed cube and let l(Q) denotes its side length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For any family of functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fk : Q → [−1, 1] and 0 < λ ≪ l(Q) we define (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2) N 1 λ∆0,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fk) := Em1∈Q � m2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=',mk f1(m1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' fk(mk) σλ∆0(m2 − m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , mk − m1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' It is clear that if f1 = · · · = fk = 1S restricted to Q, then the above expression is a normalized count of the isometric copies of λ∆0 in S ∩ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 will follow from Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 and the following special case (with q = 1) of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 (A Generalized von Neumann inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If λ ∈ √ N with λ ≪ l(Q) and 1 ≪ L ≪ ε10λ then for any collection of functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fk : Q → [−1, 1] we have (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3) |N 1 λ∆0,Q(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , fk)| ≤ min 1≤j≤k ∥fj∥U1 qε,L(Q) + O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' This compares with the purely number theoretic fact that the number of simplices ∆ = {v1 = 0, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , vk} ⊆ Zn isometric to λ∆0 is asymptotic to ρ(∆0) λ(n−k)(k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Thus, under the same conditions as in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2, we have (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4) N 1 λ∆0,Q(1Q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1Q) = 1 + O(λ−τ) + O(ε) 36 NEIL LYALL ´AKOS MAGYAR provided one also has λ ≪ εl(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Let 0 < ε ≪ δk and S ⊆ Zn be a set of upper Banach density δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' We assume first that S is ε-uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Select a scale L = L(ε, S) and a sufficiently large cube Q so that the conclusion of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' For a given λ ∈ √ N with λ ≪ εl(Q) and L ≪ ε10λ write 1S = δ1Q + g and substitute this decomposition into the multi-linear expression N 1 λ∆0,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Then by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='2 and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3)-(8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='4), we have that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='5) N 1 λ∆0,Q(1S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' , 1S) ≥ δk − O(ε) and we can conclude that S must contain an isometric copy of λ∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' If S is not ε-uniformly distributed, then its upper Banach density is increased to at least δ1 := (1 + ε2)δ when restricted to a residue class s+(qεZ)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Identify s+(qεZ)n with Zn and simultaneously the set S|s+(qεZ)n with a set S1 ⊆ Zn, via the map y → q−1 ε (y − s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Note that if S1 is ε-uniformly distributed then it contains an isometric copy of λ∆0 for all sufficiently large λ ∈ √ N and hence S contains an isometric copy of qελ∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Repeating the above procedure one arrives to a set Sj = q−j ε (S − sj) ⊆ Zn for some sj ∈ Zn in j = O(log ε−1) steps which contains an isometric copy of λ∆0 for all sufficiently large λ ∈ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' □ References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Bourgain, A Szemer´edi type theorem for sets of positive density in Rk, Israel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 54 (1986), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 3, 307–316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Conlon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Fox, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Zhao, A relative Szemeredi theorem, Geometric and Functional Analysis 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='3 (2015): 733-762.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Cook, ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Magyar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Pramanik, A Roth-type theorem for dense subsets of Rd, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 49 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 4, 676-689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Durcik, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Kovaˇc, Boxes, extended boxes, and sets of positive upper density in the Euclidean space, arXiv 1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='08692 [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Furstenberg, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Katznelson, An ergodic Szemer´edi theorem for commuting trnasformations, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Analyse Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 31 (1978), 275-291 [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Furstenberg, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Katznelson and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Weiss, Ergodic theory and configurations in sets of positive density, Mathematics of Ramsey theory, 184–198, Algorithms Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', 5, Springer, Berlin, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Frieze, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Kannan, The regularity lemma and approximation schemes for dense problems, In Foundations of Computer Science (1996) Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 37th Annual Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', 12-20 [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Gowers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Hypergraph regularity and the multidimensional Szemer´edi theorem, Annals of Mathematics (2007), 897- 946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Graham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Recent trends in Euclidean Ramsey theory, Discrete Mathematics 136, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1-3 (1994), 119-127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Iosevich and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Rudnev, Erd˝os distance problem in vector spaces over finite fields, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 359, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 12 (2007), 6127-6142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' [11] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='Kitaoka, Siegel modular forms and representation by quadratic forms Lectures on Mathe- matics and Physics, Tata Institute of Fundamental Research, Springer-Verlag, (1986) [12] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lyall and ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Magyar, Product of simplices and sets of positive upper density in Rd, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' of the Cambridge Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (2018), 25-51 [13] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lyall and ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Magyar, Distance Graphs and sets of positive upper density in Rd, to appear in Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' PDE [14] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lyall and ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Magyar, Distances and trees in dense subsets of Zd, arXiv 1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='09298 [15] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lyall, ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Magyar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Parshall, Spherical configurations over finite fields, to appear in Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' [16] ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Magyar, Distance sets of large sets of integer points, Israel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', v (2008) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' [17] ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Magyar, k-point configurations in sets of positive density of Zn, Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', v 146/1, (2009) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 1-34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' [18] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Parshall, Simplices over finite fields, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='6 (2017), 2323-2334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Siegel, On the theory of indefinite quadratic forms, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' (2) 45 (1944), 577-622 [20] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', Szemer´edi, On sets of integers containing no k elements in arithmetic progression, Acta Arith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 27 (1975), 199-245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' [21] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Tao, Szemer´edi’s regularity lemma revisited, arXiv preprint math/0504472 (2005) [22] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Tao, A variant of the hypergraph removal lemma, Journal of Combinatorial Theory, Series A 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='7 (2006): 1257-1280 [23] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Tao, The ergodic and combinatorial approaches to Szemer´edi’s theorem, Additive combinatorics, 145–193, CRM Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Lecture Notes, 43, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=', Providence, RI, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' [24] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Tao and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Vu, Additive combinatorics, Cambridge Studies in Advanced Mathematics, 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' xviii+512 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' [25] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Ziegler, Nilfactors of Rm-actions and configurations in sets of positive upper density in Rm, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' 99 (2006), 249-266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content=' Department of Mathematics, The University of Georgia, Athens, GA 30602, USA Email address: lyall@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='edu Email address: magyar@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FIT4oBgHgl3EQfoiuQ/content/2301.11319v1.pdf'}