diff --git "a/4NE3T4oBgHgl3EQfogr8/content/tmp_files/load_file.txt" "b/4NE3T4oBgHgl3EQfogr8/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/4NE3T4oBgHgl3EQfogr8/content/tmp_files/load_file.txt" @@ -0,0 +1,1194 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf,len=1193 +page_content='ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS FEDERICO GLAUDO AND ANDREA CIPRIETTI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Let A be a multiset with elements in an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Let FS(A) be the multiset containing the 2|A| sums of all subsets of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' We study the reconstruction problem “Given FS(A), is it possible to identify A?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=', and we give a satisfactory answer for all abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' We prove that, up to identifying multisets through a natural equivalence relation, the function A �→ FS(A) is injective (and thus the reconstruction problem is solvable) if and only if every order n of a torsion element of the abelian group satisfies a certain number-theoretical property linked to the multiplicative group (Z/nZ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The core of the proof relies on a delicate study of the structure of cyclotomic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Moreover, as a tool, we develop an inversion formula for a novel discrete Radon transform on finite abelian groups that might be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Introduction Let G be an abelian group and let A = {a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' , a|A|} be a finite multiset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=', a set with repeated elements) with elements in G (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='1 for a formal definition of multiset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Its subset sums multiset FS(A), that is, the multiset containing the 2|A| sums over all subsets of A (taking into account multiplicities), is defined as FS(A) := � � i∈I ai : I ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' , |A|} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' We study the following reconstruction question: If one is given FS(A), is it possible to identify A?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' As we will see, this strikingly simple question features a rich structure and its solution spans a wide range of mathematics: from the theory of cyclotomic units, to an inversion formula for a novel discrete Radon transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Before going deeper into the problem, let us give some background on related results in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' If, instead of FS(A), one is given the sums over all the �|A| s � subsets with fixed size equal to s (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=', if s = 2, the sums over all pairs), the reconstruction problem has been studied in the case of a free abelian group G = Zd [SS58;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' GFS62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' For pairs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' s = 2), the reconstruction is possible when the size of A is not a power of 2 [SS58, Theorem 1 and Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' For s-subsets with s > 2, the reconstruction is possible if the size of A does not belong to a finite subset of bad sizes [GFS62, Section 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' See the recent survey [Fom19] for a detailed presentation of the history of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' It might seem that if one is only provided with the sums of s-subsets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=', subsets with size s) then the reconstruction is strictly harder than if one is provided the sums of all subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' This is not true because the information is not ordered and 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='04635v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='NT] 11 Jan 2023 2 FEDERICO GLAUDO AND ANDREA CIPRIETTI thus, even if we have more information, it is also harder to determine which value corresponds to which subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Let us now go back to the reconstruction problem for FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The first important observation is the following one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Given a multiset A and a subset B ⊆ A whose sum equals 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' � b∈B b = 0), if we flip the signs of elements of B then FS does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' So, if A′ := (A \\ B) ∪ (−B), then FS(A) = FS(A′) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' 1 for an explanation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' A B C A′ −(B \\ C) C \\ B Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Proof by picture of A ∼0 A′ =⇒ FS(A) = FS(A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The set C in A (highlighted in gray) and the set (C\\B)∪(−(B\\C)) in A′ (highlighted in gray) have the same sum because the sum of the elements in B is assumed to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Thus, we have a bijection between the subsets of A and A′ which keeps the sum unchanged, hence FS(A) = FS(A���).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Hence, if we only know FS(A), the best we can hope for is to identify the equiv- alence class of A with respect to the following equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Given two multisets A, A′ with elements in G, we say that A ∼0 A′ if and only if A′ can be obtained from A by flipping the signs of the elements of a subset of A with null sum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=', if there exists B ⊆ A, with � b∈B b = 0, such that A′ = (A \\ B) ∪ (−B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' We have already observed that if A ∼0 A′ then FS(A) = FS(A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' If the group is G = Z, this turns out to be an “if and only if” (see Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='3), while if G = Z/2Z it is not (indeed, in Z/2Z one has FS({0, 1}) = {0, 0, 1, 1} = FS({1, 1})).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' It is natural to consider the class of abelian groups such that the double implication holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' the fibers of FS coincide with the equivalence classes of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' A group G is FS-regular if, for any two multisets A, A′ with ele- ments in G, it holds FS(A) = FS(A′) if and only if A ∼0 A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' We have already observed that Z/2Z is not FS-regular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' moreover, any group con- taining a subgroup that is not FS-regular cannot be FS-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The next smallest non-FS-regular group is elusive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' in fact, it turns out that Z/nZ is FS-regular for n = 3, 5, 7, 9, 11, 13, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' But Z/17Z is not FS-regular, and then Z/nZ is FS-regular for n = 19, 21, 23, 25, 27, 29 and not FS-regular for m = 31, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' These small exam- ples suggest that the FS-regularity of G may be related to the behavior of powers of two in G (notice that 17, 31, 33 are adjacent to a power of two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Our main result is the characterization of FS-regular groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' In order to state our result, we need to introduce a subset of the natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Let OFS be the set of odd natural numbers n ≥ 1 such that (Z/nZ)∗ is covered by {±2j : j ≥ 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' more precisely, for each x ∈ Z relatively prime with n there exists j ≥ 0 such that either x − 2j or x + 2j is divisible by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS 3 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The first few elements of OFS are OFS = {1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 35, 37, 39, 45, 47, 49, 53, 55, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' }, and the first few missing odd numbers are (2N + 1) \\ OFS = {17, 31, 33, 41, 43, 51, 57, 63, 65, 73, 85, 89, 91, 93, 97, 99, 105, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Let us remark that if n ∈ OFS then also all divisors of n belong to OFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Moreover, one can show that if p, q, r are distinct odd primes, then pqr ̸∈ OFS, and therefore if n ∈ OFS then n has at most two distinct prime factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' We can now state our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='1 (Characterization of FS-regular groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' An abelian group G is FS- regular if and only if ord(g) ∈ OFS for all g ∈ G with finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' As a tool in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='1 (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='1) we define a novel discrete Radon transform for abelian groups and we prove an inversion formula for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' We refer to Section 5 for some motivation on the definition and for an in-depth discussion of the existing related literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Since the invertibility of the Radon transform may have other applications beyond the scope of this paper, we state it here for the interested readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='2 (Invertibility of the discrete Radon transform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Let n, d ≥ 1 be pos- itive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Given a function f : (Z/nZ)d → C, its discrete Radon transform Rf = Rn,df : Hom((Z/nZ)d, Z/nZ) × Z/nZ → C is defined as Rf(ψ, c) = � x: ψ(x)=c f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' This discrete Radon transform is invertible and admits an inversion formula (see Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Sketch of the proof and structure of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Let us briefly describe the strategy that the proof follows, postponing a more detailed presentation to the dedicated sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' For the negative part of the statement, it is sufficient to show that Z/nZ is not FS-regular if n ̸∈ OFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' For this, we construct an explicit counterexample in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Proving that if the orders belong to OFS then the group is FS-regular is more complicated and relies on some nontrivial properties of the units of cyclotomic fields and on the inversion formula for a novel discrete Radon transform on finite abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The proof is divided into three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Step 1: Proof for G = Z/nZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Through the polynomial identity � s∈FS(A) ts ≡ � a∈A (1 + ta) (mod tn − 1), we reduce the FS-regularity of Z/nZ to the study of the kernel of the map Zn ∋ x = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' , xn−1) �→ � n−1 � j=0 (1 + ωj d)xj� d|n, 4 FEDERICO GLAUDO AND ANDREA CIPRIETTI where ωd ∈ C is a d-th primitive root of unity and the codomain of the map consists of tuples indexed by the divisors of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Thanks to a dimensional argument, identifying the kernel of such map is equivalent to identifying its image, which is exactly what we do in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' This is the hardest and most technical proof of the whole paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Up to this point, we have used only that n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The fact that n ∈ OFS is needed in the computation of the rank of the image, which relies heavily on the theory of cyclotomic units (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' This step is carried out in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Step 2: Z/nZ is FS-regular =⇒ (Z/nZ)d is FS-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Take A, A′ multisets with elements in (Z/nZ)d such that FS(A) = FS(A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Given a homomorphism ψ : (Z/nZ)d → Z/nZ, by linearity, it holds FS(ψ(A)) = FS(ψ(A′)), and since Z/nZ is FS-regular this implies that ψ(A) ∼0 ψ(A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' So, we know that ψ(A) ∼0 ψ(A′) for all homomorphisms ψ : (Z/nZ)d → Z/nZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' In order to deduce that A ∼0 A′, we introduce a novel discrete Radon transform (see Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='1) and we prove an inversion formula (see Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='2) which may be of indepen- dent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' This allows us to reconstruct a multiset B ∈ M((Z/nZ)d) from its projections {φ(B) : φ ∈ Hom((Z/nZ)d, Z/nZ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' This step is performed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Step 3: G is FS-regular =⇒ G ⊕ Z is FS-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' In this step, we exploit crucially that Z is totally ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The argument is short and purely combinatorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' This is done in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Once these three steps are established, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='1 follows naturally, as shown in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Let us remark here that our proof is not constructive, hence it does not provide an efficient algorithm to find the ∼0-equivalence class of A if FS(A) is known1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' To make the paper accessible to a broad audience, in Section 2 we recall basic facts about multisets, abelian groups, and cyclotomic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The authors are thankful to Fabio Ferri for providing valu- able suggestions and references about the theory of cyclotomic units, and also to Michele D’Adderio and Elia Bru`e for their comments and feedback on an early ver- sion of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The second author is supported by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' DMS-1926686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Notation and Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Multisets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' A multiset with elements in a set X is an unordered collection of elements of X which may contain a certain element more than once [Bli89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' For example, {1, 1, 2, 2, 3} is a multiset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Rigorously, a multiset A is encoded by a function µA : X → Z≥0 (Z≥0 denotes the set of nonnegative integers) such that µA(x) represents the multiplicity of the element x in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' For example, if A = {1, 1, 2, 2, 3} then µA(1) = 2, µA(2) = 2, µA(3) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' 1The nonconstructive part of the proof is contained Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' In fact, we show that a certain map is injective by proving its surjectivity and then applying a standard dimension argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' This kind of reasoning does not produce an efficient way to invert the map we have proven to be injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS 5 A multiset A is finite if � x∈X µA(x) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The cardinality of a finite multiset A ∈ M(X) is given by |A| := � x∈X µA(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Given a set X, let us denote with M(X) the family of finite multisets with elements in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Let us define the usual set operations on multisets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Notice that all of them are the natural generalization of the standard version when one takes into account the multiplicity of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Fix two multisets A, B ∈ M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Membership: We say that x ∈ X is an element of A, denoted by x ∈ A, if µA(x) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Inclusion: We say that A is a subset of B, denoted by A ⊆ B, if µA(x) ≤ µB(x) for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Union: The union A∪B ∈ M(X) is defined as µA∪B(x) := µA(x)+µB(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Hence, {1} ∪ {1, 2} = {1, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Cartesian product: The Cartesian product A × B ∈ M(X × X) is defined as µA×B((x1, x2)) = µA(x1)µB(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Difference: If A ⊆ B, the difference B \\A is defined as µB\\A(x) := µB(x)−µA(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Pushforward: Given a function f : X → Y , the pushforward f(A) ∈ M(Y ) of the multiset A (denoted also by {f(a) : a ∈ A}) is defined as µf(A)(y) = � x∈f −1(y) µA(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Power set: The power set of A (the family of subsets of A), denoted by P(A) ∈ M(M(X)), is a multiset defined recursively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' For the empty mul- tiset, we have P(∅) := {∅};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' otherwise let a ∈ A be an element of A and define P(A) := P(A \\ {a}) ∪ � A′ ∪ {a} : A′ ∈ P(A \\ {a}) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Notice that |P(A)| = 2|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Whenever we iterate over the subsets of A (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=', {f(A′) : A′ ⊆ A} or � A′⊆A f(A′)), the iteration has to be understood over P(A) (hence the subsets are counted with multiplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Taking the complement is an involution of the power set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=', P(A) = {A \\ A′ : A′ ∈ P(A)}, and we have the following identity for the power set of a union P(A ∪ B) = {A′ ∪ B′ : (A′, B′) ∈ P(A) × P(B)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Sum (and product): If the set X is an additive abelian group, we can define the sum � A ∈ X of the elements of A as � A := � x∈X µA(x)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Analogously, if X is a multiplicative abelian group, one can define the prod- uct � A of the elements of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Abelian Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Let us recall some basic facts about abelian groups that we will use extensively later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' 6 FEDERICO GLAUDO AND ANDREA CIPRIETTI Any finitely generated abelian group is isomorphic to a finite product of cyclic groups [Lan02, Chapter I, Section 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' We denote with Z/nZ the cyclic group with n elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Given some elements g1, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' , gk ∈ G of an abelian group, we denote with ⟨g1, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' , gk⟩ the subgroup generated by such elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Given an element g ∈ G, its order (which may be equal to ∞) is denoted by ord(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' For an abelian group G, its rank rk(G) is the cardinality of a maximal set of Z-independent2 elements of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Let us list some useful properties of the rank (see [Lan02, Chapter I and XVI]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Any finitely generated abelian group G is isomorphic to Zrk(G) ⊕ G′ where G′ is a finite abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Given two abelian groups G, H, it holds rk(G ⊕ H) = rk(G) + rk(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' For a homomorphism φ : G → H of abelian groups, it holds rk(G) = rk(ker φ) + rk(Im φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' An abelian group has null rank if and only if all elements have finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Let G1, G2, G3 be three abelian groups and φ1 : G1 → G2, φ2 : G2 → G3 be two homomorphisms with full rank, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' rk(Im φ1) = rk(G2) and rk(Im φ2) = rk(G3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Then φ2 ◦ φ1 : G1 → G3 has full rank as well, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' rk(Im φ2 ◦ φ1) = rk(G3) Given an abelian group G, let us denote with G ⊗ Q its tensor product (as a Z-module) with Q (see [Lan02, Chapter XVI]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The dimension of G ⊗ Q as vector space over Q coincides with rk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' For a homomorphism φ : G → H of abelian groups, let φ⊗Q : G⊗Q → H⊗Q be its tensorization with Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' It holds rk(Im φ) = dimQ(Im (φ ⊗ Q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Units of cyclotomic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Given n ≥ 1, let ωn := exp(2πi/n) be the prim- itive n-th root of unity with minimum positive argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The algebraic number field Q(ωn) is called cyclotomic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' It is well-known that the ring of integers of Q(ωn) coincides with Z[ωn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Our main focus is the group of units of Q(ωn), that consists of the invertible elements of its ring of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' For 0 < r < n and s ≥ 1 coprime with n, the element ξ := 1−ωrs n 1−ωrn is a unit of Q(ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Indeed ξ = 1+ωr n +· · ·+ω(s−1)r n ∈ Z[ωn] and, if u ∈ N is such that n divides us − 1, then ξ−1 = 1 − ωrus n 1 − ωrs n = 1 + ωrs n + · · · + ω(u−1)rs n ∈ Z[ωn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' It turns out that these units are sufficient to generate a subgroup of finite index of the units of Q(ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The following statement follows from [Was97, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='3 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' For any odd n ≥ 3, the multiplicative group Cn ⊆ C generated by �1 − ωrs n 1 − ωrn : 0 < r < n, s ≥ 1 coprime with n � is a subgroup of finite index of the units of Q(ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' 2Some elements g1, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' , gk ∈ G are Z-independent if, whenever � i aigi = 0 for some a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' , ak ∈ Z, it holds a1 = a2 = · · · = ak = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS 7 Thus, applying Dirichlet’s unit Theorem (see [Mar77, Theorem 38]), we are able to compute the rank of Cn (since it coincides with the rank of the group of units of Q(ωn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' For any odd n ≥ 3, we have rk(Cn) = ϕ(n) 2 − 1, where ϕ is Euler’s totient function (and Cn is defined in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The units of Q(ωn) satisfy a family of nontrivial relations known as distribution relations (see [Was97, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' 151]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' We recall here the relations in the form we will need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Notice that 1+ωj n is a unit for 1 ≤ j < n because of the identity 1+ωj n = 1−ω2j n 1−ωj n ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content='3 (Distribution relations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Let n ≥ 1 be an odd integer and let p be one of its prime divisors3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' For any 0 ≤ j < n p , the identity p−1 � k=0 (1 + ωj+kn/p n ) = 1 + ωjp n holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfogr8/content/2301.04635v1.pdf'} +page_content=' The numbers {1 + ωj+kn/p n }0≤k