# The Assouad dimension of Kakeya sets in $\mathbb{R}^{3}$

Hong Wang Courant Institute of Mathematical Sciences, New York University. New
York, NY, USA. Joshua Zahl Department of Mathematics, The University of
British Columbia. Vancouver, BC, Canada.

###### Abstract

This paper studies the structure of Kakeya sets in $\mathbb{R}^{3}$. We show
that for every Kakeya set $K\subset\mathbb{R}^{3}$, there exist well-separated
scales $0<\delta<\rho\leq 1$ so that the $\delta$ neighborhood of $K$ is
almost as large as the $\rho$ neighborhood of $K$. As a consequence, every
Kakeya set in $\mathbb{R}^{3}$ has Assouad dimension 3 and every Ahlfors-David
regular Kakeya set in $\mathbb{R}^{3}$ has Hausdorff dimension 3. We also show
that every Kakeya set in $\mathbb{R}^{3}$ that has “stably equal” Hausdorff
and packing dimension (this is a new notion, which is introduced to avoid
certain obvious obstructions) must have Hausdorff dimension 3.

The above results follow from certain multi-scale structure theorems for
arrangements of tubes and rectangular prisms in three dimensions, and a mild
generalization of the sticky Kakeya theorem previously proved by the authors.

##  1 Introduction

A Kakeya set is a compact subset of $\mathbb{R}^{n}$ that contains a unit line
segment pointing in every direction. The Kakeya set conjecture asserts that
every Kakeya set in $\mathbb{R}^{n}$ has Minkowski and Hausdorff dimension
$n$. This conjecture is proved in the plane [1, 2], and is open in three and
higher dimensions. See [9, 18] for a survey of progress on the Kakeya
conjecture.

In this paper, we study the structure of Kakeya sets in $\mathbb{R}^{3}$. We
show that for every Kakeya set $K\subset\mathbb{R}^{3}$, there exist well-
separated scales $0<\delta<\rho\leq 1$ so that the $\delta$ neighborhood of
$K$ is almost as large as the $\rho$ neighborhood of $K$. The precise
statement is given in Theorem 1.5 below. As a consequence, we prove several
weaker variants of the Kakeya set conjecture: we prove that every Kakeya set
in $\mathbb{R}^{3}$ has Assouad dimension 3, and we prove that every Kakeya
set in $\mathbb{R}^{3}$ with “stably equal” Hausdorff and packing dimension
(see Definition 1.3 below) must have Hausdorff and packing dimension 3. In
order to explain these statements precisely, we require the following
definitions.

######  Definition 1.1.

Let $E\subset\mathbb{R}^{n}$ be non-empty. The _Assouad dimension_ of $E$,
denoted $\mathrm{dim}\,_{\\!\\!A}(E)$, is the infimum of all $\beta\geq 0$ for
which there exist positive constants $C$ and $r_{0}$ so that for all
$0<\rho<r\leq r_{0}$, we have

$\sup_{x\in\mathbb{R}^{n}}\mathcal{E}_{\rho}\big{(}E\cap B(x,r)\big{)}\leq
C(r/\rho)^{\beta}.$

In the above, $\mathcal{E}_{\rho}(X)$ denotes the $\rho$-covering number of
$X$.

The Assouad dimension is always at least as large as the Hausdorff and
Minkowski dimensions, i.e. if $E\subset\mathbb{R}^{n}$ is bounded then we have
$0\leq\mathrm{dim}\,_{H}(E)\leq\underline{\mathrm{dim}\,}_{M}(E)\leq\overline{\mathrm{dim}\,}_{M}(E)\leq\mathrm{dim}\,_{A}(E)\leq
n$. If $E$ is bounded and Ahlfors-David regular, then all of these dimensions
are equal. See [5] for further details on Assouad dimension and its
properties.

Our first result is the following weak version of the Kakeya set conjecture in
$\mathbb{R}^{3}$.

######  Theorem 1.2.

Every Kakeya set in $\mathbb{R}^{3}$ has Assouad dimension 3.

_Remarks_
1\. Theorem 1.2 holds for a slightly more general class of sets, where the
lines satisfy a mild strengthening of the Wolff axioms. See the remarks in
Section 8 for details.

2\. Fraser, Olson and Robinson [4] proved that every half-extended Kakeya set
in $\mathbb{R}^{n}$ (i.e. a subset of $\mathbb{R}^{n}$ containing a half-
infinite line segment in every direction) has Assouad dimension $n$. However
this is a somewhat different question. In brief, Fraser, Olson and Robinson
showed that the Assouad dimension does not increase under a “zooming out”
rescaling, which transforms an extended Kakeya set into one where all lines
pass through the origin (the latter type of set has full Assouad dimension).

Next, we consider Kakeya sets with equal Hausdorff and packing dimension.
Recall that for $E\subset\mathbb{R}^{n}$, the packing dimension is given by

$\mathrm{dim}\,_{P}(E)=\inf\\{\sup\overline{\mathrm{dim}\,}_{M}E_{i}\\},$

where $\overline{\mathrm{dim}\,}_{M}$ denotes the upper Minkowski dimension,
and the infimum is taken over all decompositions $E=\bigcup_{i}E_{i}$ into
countably many sets. We have $\mathrm{dim}\,_{P}(E)\leq\mathrm{dim}\,_{H}(E)$
for every set $E\subset\mathbb{R}^{n}$, and if
$\mathrm{dim}\,_{P}(E)=\mathrm{dim}\,_{H}(E)=\alpha$, then for every
$\varepsilon>0$ there exists a constant $C_{\varepsilon}$ and a measure $\mu$
supported on $E$ so that for $\mu$-a.e. $x\in E$ we have the Frostman-type
estimate

$C_{\varepsilon}^{-1}r^{\alpha+\varepsilon}\leq\mu(B(x,r))\leq
C_{\varepsilon}r^{\alpha-\varepsilon}\quad\textrm{for all}\ 0<r\leq 1.$

We would like to say that every Kakeya set in $\mathbb{R}^{3}$ with equal
Hausdorff and packing dimension must have dimension 3. Unfortunately, we
cannot prove this statement, and indeed this would imply that every Kakeya set
in $\mathbb{R}^{3}$ has packing dimension 3: let $K\subset\mathbb{R}^{3}$ be a
Kakeya set, and let $X\subset\mathbb{R}^{3}$ be a compact set with equal
Hausdorff and packing dimension
$\mathrm{dim}\,_{H}(X)=\mathrm{dim}\,_{P}(X)=\mathrm{dim}\,_{P}(K)$. Then
$K^{\prime}=K\cup X$ is a Kakeya set with equal Hausdorff and packing
dimension, and this common value is $\mathrm{dim}\,_{P}(K)$. In the above
example, the Kakeya set $K^{\prime}$ has equal Hausdorff and packing dimension
for the somewhat trivial reason that it is a union of a Kakeya set with
potentially unequal Hausdorff and packing dimension, and a set of equal
Hausdorff and packing dimension111We could even select $X$ to be a union of
unit line segments pointing in different directions (specifically a
$\mathrm{dim}\,X-1$ dimensional set of directions), and modify $K$ so that
$K^{\prime}$ is a union of unit line segments, with one line segment pointing
in each direction.. In particular, if
$\mathrm{dim}\,_{H}K<\mathrm{dim}\,_{H}K^{\prime}=\alpha$, then an
$\alpha$-dimensional Frostman measure supported on $K^{\prime}$ would be null
on $K$. To exclude this type of situation, we introduce the following
definition.

######  Definition 1.3.

We say a set compact set $K\subset\mathbb{R}^{n}$ is a _Kakeya set with stably
equal Hausdorff and packing dimension_ if for all $\varepsilon>0$, there is a
measure $\mu$ supported on $K$ with the following properties.

  * •

There is a constant $C_{\varepsilon}$ so that for all
$x\in\operatorname{supp}(\mu),$ we have

$C_{\varepsilon}^{-1}r^{\alpha+\varepsilon}\leq\mu(B(x,r))\leq
C_{\varepsilon}r^{\alpha-\varepsilon},\quad\textrm{for all}\ 0\leq r\leq 1,$

where $\alpha=\mathrm{dim}\,_{H}(K)$.

  * •

There is a positive (two-dimensional Lebesgue) measure set of directions
$\Omega\subset S^{2},$ so that for each direction $e\in\Omega$, there is a
line $\ell_{e}$ pointing in direction $e$ for which
$\ell_{e}\cap\operatorname{supp}(\mu)$ has positive one-dimensional Lebesgue
measure.

In particular, if $K$ is a Kakeya set with unequal Hausdorff and packing
dimension, then the set $K^{\prime}=K\cup X$ described above will not have
stably equal Hausdorff and packing dimension222Unless the set $X$ is itself a
Kakeya set with stably equal Hausdorff and packing dimension..

######  Theorem 1.4.

Every Kakeya set in $\mathbb{R}^{3}$ with stably equal Hausdorff and packing
dimension has Hausdorff and packing dimension 3.

Theorems 1.2 and 1.4 will follow from a common discretized Kakeya estimate,
which is the main technical contribution of this paper. In the statement
below, $N_{\rho}(X)$ denotes the $\rho$-neighborhood of the set $X$, a
$\delta$-tube is the $\delta$-neighborhood of a unit line segment, and say two
$\delta$-tubes are essentially distinct if neither tube is contained in the
2-fold dilate of the other; see Section 2 for precise definitions.

######  Theorem 1.5.

For all $\varepsilon>0$, there exists $\eta,\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let $\mathbb{T}$ be a set
of essentially distinct $\delta$-tubes in $B(0,1)\subset\mathbb{R}^{3}$, and
suppose that each convex set of volume $V$ contains at most
$\delta^{-\eta}V(\\#\mathbb{T})$ tubes from $\mathbb{T}$. For each
$T\in\mathbb{T}$, let $Y(T)\subset T$ be a measurable set, and suppose that
$\sum_{\mathbb{T}}|Y(T)|\geq\delta^{\eta}\sum_{\mathbb{T}}|T|$.

Then there exists $\rho,r\in[\delta,1]$ with $\rho\leq\delta^{\eta}r$ and a
ball $B$ of radius $r$ so that

$\Big{|}B\cap
N_{\rho}\Big{(}\bigcup_{T\in\mathbb{T}}Y(T)\Big{)}\Big{|}\geq(\rho/r)^{\varepsilon}|B|.$
(1.1)

######  Remark 1.6.

In the above theorem, we have required that the tubes be essentially distinct
and that every convex set $U\subset\mathbb{R}^{3}$ contain at most
$\delta^{-\eta}|U|(\\#\mathbb{T})$ tubes from $\mathbb{T}$. This condition is
satisfied, for example, if $\\#\mathbb{T}=\delta^{-2}$ and the tubes in
$\mathbb{T}$ point in $\delta$-separated directions. Indeed; for the latter
condition we need only consider convex sets $U\subset
B(0,1)\subset\mathbb{R}^{3}$ of diameter roughly 1, and every set of this form
is comparable to an ellipsoid whose axes have lengths $s\leq t\leq w$ with
$w\sim 1$. If $T$ is a $\delta$-tube contained in $U$, then the direction of
$T$ lies in a rectangular sector of $S^{2}$ of dimensions roughly $s\times t$;
if the tubes in $\mathbb{T}$ point in $\delta$-separated directions, we
conclude that $\\#\\{T\in\mathbb{T}\colon T\subset U\\}\lesssim
st\delta^{-2}\lesssim|U|(\\#\mathbb{T})$. The anti-clustering condition
imposed in Theorem 1.5 and will be discussed further in Section 2.2.

Theorem 1.5 can be amplified to obtain the following variant, which says that
if $\mathbb{T}$ is a collection of $\delta$-tubes satisfying the anti-
clustering condition described above, then there are two well-separated scales
$\rho<r$ so that the $\rho$-neighborhood of $\bigcup_{\mathbb{T}}T$ is almost
as large as its $r$-neighborhood.

######  Corollary 1.7.

For all $\varepsilon>0$, there exists $\eta,\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let $\mathbb{T}$ be a set
of essentially distinct $\delta$-tubes in $B(0,1)\subset\mathbb{R}^{3}$, and
suppose that each convex set of volume $V$ contains at most
$\delta^{-\eta}V(\\#\mathbb{T})$ tubes from $\mathbb{T}$. For each
$T\in\mathbb{T}$, let $Y(T)\subset T$ be a measurable set, and suppose that
$\sum_{\mathbb{T}}|Y(T)|\geq\delta^{\eta}\sum_{\mathbb{T}}|T|$.

Then there exist sets $Y^{\prime}(T)\subset Y(T),$ $T\in\mathbb{T}$, with
$\sum_{\mathbb{T}}|Y^{\prime}(T)|\gtrsim|\log\delta|^{-2}\sum_{\mathbb{T}}|Y(T)|$,
and there exist scales $\rho,r\in[\delta,1]$ with $\rho\leq\delta^{\eta}r$ so
that

$\Big{|}N_{\rho}\Big{(}\bigcup_{T\in\mathbb{T}}Y^{\prime}(T)\Big{)}\Big{|}\geq(\rho/r)^{\varepsilon}\Big{|}N_{r}\Big{(}\bigcup_{T\in\mathbb{T}}Y^{\prime}(T)\Big{)}\Big{|}.$
(1.2)

###  1.1 Main ideas, and a sketch of the proof

Recall that every Kakeya set contains a two-dimensional family of unit line
segments pointing in different directions. In the discretized setting, this
corresponds to a set of $\delta$-tubes pointing in $\delta$-separated
directions. The requirement that the line segments (or $\delta$-tubes) point
in different directions can be thought of as a type of anti-clustering
condition—without this condition, for example, we could construct a two-
dimensional family of coplanar line segments in $\mathbb{R}^{3}$, and the
union of these line segments would have dimension at most 2.

In [17], Wolff imposed an anti-clustering condition on his collection of
$\delta$-tubes in $\mathbb{R}^{3}$ that is slightly weaker than requiring the
tubes to point in $\delta$-separated directions. First, Wolff required that
for $\delta\leq\rho\leq 1$, each $\rho$-tube must contain at most
$(\rho/\delta)^{2}$ $\delta$-tubes; if we think of $\delta$-tubes as points in
the Grassmannian of (affine) lines in $\mathbb{R}^{3}$, then Wolff’s first
requirement corresponds to a 2-dimensional Frostman non-concentration
condition on the set of tubes. Second, Wolff required that for each
$\delta\leq\rho\leq 1$, at most $\rho/\delta$ $\delta$-tubes can be contained
in any rectangular prism of dimensions $\delta\times\rho\times 1$. This second
condition prevents the tubes from clustering into the $\delta$-neighborhood of
a plane. These two conditions are now referred to as the _Wolff axioms_.

In our arguments below, we will consider several anti-clustering conditions
that are related to the Wolff axioms. The first one already appeared in the
statement of Theorem 1.5; it is a mild generalization of the Wolff axioms,
which we call the _Convex Wolff Axioms_. Next, we say a set of $\delta$-tubes
$\mathbb{T}$ satisfies the _Convex Wolff Axioms at every scale_ if for every
$\delta\leq\rho\leq 1$, the tubes in $\mathbb{T}$ can be covered by
$\rho$-tubes, so that for each $\rho$-tube $T_{\rho}$ in this cover, the
portion of $\mathbb{T}$ contained inside $T_{\rho}$ can be re-scaled to create
a collection of $(\delta/\rho)$-tubes that satisfy the convex Wolff axioms.
Finally, we say a set of tubes $\mathbb{T}$ satisfies a _Frostman Condition at
exponent $\alpha$_ if every $\rho$-tube contains at most
$\rho^{\alpha}(\\#\mathbb{T})$ tubes from $\mathbb{T}$. Precise versions of
these definitions are given in Section 2.2.

In Section 4, we show that if a set of tubes $\mathbb{T}$ satisfies the Convex
Wolff Axioms and also satisfies a Frostman condition, then either it must
satisfy the Convex Wolff Axioms at every scale, or else the conclusion of
Theorem 1.5 must hold for $\mathbb{T}$ (or both). In broad strokes, the
argument is as follows. Suppose that $\mathbb{T}$ satisfies a Frostman
condition but fails to satisfy the Convex Wolff Axioms at every scale. Then
the tubes in $\mathbb{T}$ must cluster into rectangular prisms of dimensions
$s\times t\times 1$ for some $\delta\leq s<\\!\\!<t\leq 1.$ Each such prism is
essentially the intersection of the $s$-neighborhood of a plane with the
$t$-neighborhood of a line. First, we consider the case that when two such
prisms intersect, the corresponding planes make angle $\theta>\\!\\!>s/t$.
Then the union of the $s$-neighborhood of these planes will contain a ball of
radius $r=\theta t,$ which is much larger than $s$, and hence (1.1) holds.
Second, we consider the case that when two such prisms intersect, the
corresponding planes make angle $\leq s/t$. This forces the $s\times t\times
1$ prisms cluster into rectangular prisms of dimensions $s^{\prime}\times
t^{\prime}\times 1$, where $s^{\prime}\ll t^{\prime}$ are substantially larger
than $s$ and $t$, respectively. We can then repeat this argument at a larger
scale. After finitely many iterations of this argument, we must find ourselves
in the First Case. This concludes the argument.

The argument described above is a key geometric input for the proof of Theorem
1.5. With this geometric input, the proof of Theorem 1.5 proceeds by
contradiction: suppose there exists a counter-example to Theorem 1.5, i.e. a
set of tubes satisfying the Convex Wolff Axioms, for which (1.1) fails for
every pair of scales $\rho,r$ and every ball $B$ of radius $r$. The first step
is to find a “worst possible” such counter-example to Theorem 1.5, i.e. a set
of $\delta$-tubes $\mathbb{T}$ of largest possible cardinality satisfying the
Convex Wolff Axioms, for which the reverse of Inequality (1.1) holds for every
pair of scales $\rho,r$ and every ball $B$ of radius $r$, with the term
$(\rho/r)^{\varepsilon}$ replaced by $(\rho/r)^{\omega},$ for the largest
possible exponent $\omega$.

Since the conclusion of Theorem 1.5 is agnostic to location and scale, if we
select $s\in[\delta,1]$; consider only those tubes from $\mathbb{T}$ that are
contained in a $s$-tube; and re-scale, then the reverse of Inequality (1.1)
must continue to hold, with the same exponent $\omega$. But this means that
the newly obtained collection of tubes (obtained by restricting $\mathbb{T}$
to a $s$-tube and rescaling) cannot be larger (relative to the new scale
$\delta/s$) than the original set $\mathbb{T}$, since $\mathbb{T}$ was a set
of largest possible cardinality for which the reverse of Inequality (1.1)
holds at every location and scale with exponent $\omega$. The above argument
shows that $\mathbb{T}$ must satisfy a Frostman condition. We can now apply
the geometric input discussed above (and proved in Section 4): $\mathbb{T}$
satisfies the Convex Wolff Axioms (by assumption), and we just established
that $\mathbb{T}$ satisfies a Frostman condition. On the other hand, we are
assuming that $\mathbb{T}$ does not satisfy the conclusion of Theorem 1.5. We
conclude that $\mathbb{T}$ satisfies the Convex Wolff Axioms at every scale.

In summary, we have shown that if Theorem 1.5 is false, then there must exist
a “worst possible” counter-example, and this counter-example must satisfy the
Convex Wolff Axioms at every scale. Finally, we arrive at a contradiction (and
thus complete the proof of Theorem 1.5) using a slight variant of the Sticky
Kakeya theorem from [16]. We state an informal version of this result below.

######  Theorem 1.8.

For all $\varepsilon>0$, there exists $\eta,\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let $\mathbb{T}$ be a set
of essentially distinct $\delta$-tubes in $\mathbb{R}^{3}$ that satisfy the
Convex Wolff Axioms at every scale. For each $T\in\mathbb{T}$, let
$Y(T)\subset T$ be a measurable set, and suppose that
$\sum_{\mathbb{T}}|Y(T)|\geq\delta^{\eta}\sum_{\mathbb{T}}|T|$. Then

$\Big{|}\bigcup_{T\in\mathbb{T}}Y(T)\Big{|}\geq\delta^{\varepsilon}.$ (1.3)

Theorem 1.8 is closely related to previous work by the authors from [16]. In
[16], the authors prove a variant of Inequality (1.3), in which the tubes
$\mathbb{T}$ satisfy slightly different hypotheses—the tubes are required to
point in different directions and satisfy a Frostman condition at exponent 2.
There are two places in [16] where one uses the hypothesis that the tubes
point in different directions, and this hypothesis can be replaced by the
Convex Wolff Axioms with minimal modification. A precise version of Theorem
1.8 is stated in Section 5. In Section 6 we briefly outline the proof of
Theorem 1.8, and how it may be obtained by modifying the arguments in [16].

###  1.2 Thanks

The authors would like to thank Jonathan Fraser and Terence Tao for comments
and suggestions on an earlier draft of this manuscript. Hong Wang is supported
by NSF CAREER DMS-2238818 and NSF DMS-2055544. Joshua Zahl is supported by a
NSERC Discovery Grant.

##  2 Notation and Definitions

We write $A\lesssim B$ or $A=O(B)$ to mean there is a constant $C$ (which
might depend on the ambient dimension $n$) so that $A\leq CB$. We write $A\sim
B$ if $A\lesssim B$ and $B\lesssim A$. If the constant $C$ is allowed to
depend on an additional parameter such as $\varepsilon$, we then we denote
this by $A\lesssim_{\varepsilon}B$. Many of our results will involve a small
positive parameter, which we will call $\delta$. We write $A\lessapprox B$ to
mean that there is a constant $C$ so that $A\leq C(\log 1/\delta)^{C}B$. As
above, if the constant $C$ is allowed to depend on an additional parameter
such as $\varepsilon$, we then we denote this by
$A\lessapprox_{\varepsilon}B$. Finally, during proof sketches and other
informal remarks, we will sometimes write $A\ll B$ to suggest to the reader
that that the real number $A$ is much smaller than the real number $B$.

###  2.1 Tubes and their shadings

Recall that a $\delta$-tube in $\mathbb{R}^{n}$ is the $\delta$-neighborhood
of a unit line segment. We will use $T$ to denote a $\delta$-tube, and
$\ell(T)$ to denote its coaxial line. If $T$ is a $\delta$-tube (or more
generally, the translation of a centrally symmetric convex set) and $R>0$, we
define $RT$ to be the $R$-fold dilate of $T$. We say two tubes $T,T^{\prime}$
are _essentially distinct_ if $T\not\subset 2T^{\prime}$ and
$T^{\prime}\not\subset 2T$.

We will sometimes be concerned with collections of tubes, plus shadings (i.e.
distinguished subsets) of these tubes. To this end, we introduce the following
definitions.

######  Definition 2.1.

Let $\mathcal{S}$ be a collection of subsets of $\mathbb{R}^{n}$, and let
$\tau\in(0,1]$. A collection of sets $\\{Y(S)\subset S\colon
S\in\mathcal{S}\\}$ is called a _$\tau$ -dense shading_ of $\mathcal{S}$ if
$\sum_{S\in\mathcal{S}}|Y(S)|\geq\tau\sum_{S\in\mathcal{S}}|S|$. $\\{Y(S)\\}$
is called a _uniformly $\tau$-dense shadings_ of $\mathcal{S}$ if
$|Y(S)|\geq\tau|S|$ for each $S\in\mathcal{S}$.

In practice, $\mathcal{S}$ will either be a collection of $\delta$-tubes, or a
collection of $s\times t\times 1$ rectangular prisms. We will use the notation
$(\mathbb{T},Y)_{\delta}$ to refer to a collection of $\delta$-tubes and an
associated shading $\\{Y(T)\colon T\in\mathbb{T}\\}$.

######  Definition 2.2.

Let $S\subset\mathbb{R}^{n}$, let $Y(S)\subset S$, and let $\delta>0$. We say
the shading $Y(S)$ is _regular_ at scales $\geq\delta$ if for each $x\in Y(S)$
and each $r\in[\delta,1]$, we have

$|Y(S)\cap B(x,r)|\geq(100\log(1/\delta))^{-1}|Y(S)|\Big{(}\frac{|B(x,r)\cap
S|}{|S|}\Big{)}.$ (2.1)

If the quantity $\delta$ is apparent from context, then we will omit it and
say that $Y(S)$ is regular.

The following is Lemma 2.7 from [10].

######  Lemma 2.3.

Let $S\subset\mathbb{R}^{n}$ and let $Y(S)$ be a shading. Then there is a
regular shading $Y^{\prime}(S)\subset Y(S)$ with
$|Y^{\prime}(S)|\geq\frac{1}{2}|Y(S)|$.

The proof in [10] considers the special case where $S$ is a unit line segment
(and hence $|B(x,r)\cap S|/|S|\sim r$), but the proof is identical. In brief,
we consider the quantity $|Y(S)\cap B(x,r)|$ for each dyadic scale
$r=2^{k}\delta$ and each ball $B(x,r)$ aligned to the dyadic grid. We delete
those balls for which $|Y(S)\cap B(x,r)|$ is too small (i.e. for which (2.1)
fails), and we denote the surviving subset of $Y(S)$ by $Y^{\prime}(S)$. At
each dyadic scale we have only deleted a small fraction of $Y(S)$, so at least
half of $Y(S)$ survives this process.

###  2.2 Wolff axioms and their generalizations

In this section we will define several anti-clustering conditions that we can
impose on our collection of tubes. In [17] (see also Definition 13.1 from
[7]), Wolff introduced what are now called the Wolff axioms; this is a non-
concentration condition that forbids a collection of $\delta$-tubes from
clustering near a fatter $\rho$-tube, and also forbids a collection of
$\delta$-tubes from concentrating on (affine) 2-planes. The precise definition
is as follows.

######  Definition 2.4.

Let $\mathbb{T}$ be a set of $\delta$-tubes in $\mathbb{R}^{n}$. We say
$\mathbb{T}$ satisfies the _Wolff axioms_ if:

  * •

Every $\rho$-tube can contain at most $C(\rho/\delta)^{n-1}$ tubes from
$\mathbb{T}$.

  * •

Every rectangular prism of dimensions $2\delta\times\rho\times 2$ can contain
at most $C(\rho/\delta)$ tubes from $\mathbb{T}$.

_Remarks_
1\. In the above definition, the constant $C$ is generally chosen to be a
large number depending only on the ambient dimension $n$.
2\. Every set of $\delta$-tubes pointing in $\delta$-separated directions will
satisfy the Wolff axioms with constant $C=100^{n}$.
3\. Every set of $\delta$-tubes contained in $B(0,1)\subset\mathbb{R}^{n}$
that satisfies the Wolff axioms has cardinality at most $2C\delta^{1-n}$.

In the arguments that follow, we will need several variants of the Wolff
axioms. We define these below.

######  Definition 2.5.

Let $C\geq 1$. We say a multi-set $\mathcal{S}$ of sets in $\mathbb{R}^{n}$
satisfies the _Convex Wolff Axioms_ with error $C$ if for all convex sets
$W\subset\mathbb{R}^{n}$, we have

$\\#\\{S\in\mathcal{S}\colon S\subset W\\}\leq C|W|(\\#\mathcal{S}).$ (2.2)

In practice we will have $n=3$, and the sets in $\mathcal{S}$ will be either
$\delta$-tubes, or $s\times t\times 1$ rectangular prisms (not necessarily
axis-parallel) for some $s,t\in[\delta,1]$.

######  Remark 2.6.

If the elements of $\mathcal{S}$ are convex, then $\\#\mathcal{S}\geq
C^{-1}\big{(}\inf_{S\in\mathcal{S}}|S|\big{)}^{-1}$. In particular, a set of
$\delta$-tubes in $\mathbb{R}^{3}$ satisfying the Convex Wolff Axioms must
have cardinality $\gtrsim\delta^{-2}$.

######  Remark 2.7.

Let $\mathcal{S}$ be a multi-set of subsets of $\mathbb{R}^{n}$. For each
$S\in\mathcal{S}$, let $T(S)\supset S$, and define the multi-set
$\mathcal{S}^{\prime}=\\{T(S)\colon S\in\mathcal{S}\\},$ i.e. the multi-sets
$\mathcal{S}$ and $\mathcal{S}^{\prime}$ have the same cardinality. If
$\mathcal{S}$ satisfies the Convex Wolff Axioms with error $C$, then
$\mathcal{S}^{\prime}$ also satisfies the Convex Wolff Axioms with error $C$.

In our arguments below, we will often consider a Kakeya set (or more
accurately a collection of $\delta$-tubes) at many different scales.
Typically, for a collection of $\delta$-tubes $\mathbb{T}$, we will want to
examine the coarsening of $\mathbb{T}$ at a larger scale $\rho$, and also
examine the collection of $\delta$-tubes from $\mathbb{T}$ that are contained
in inside each of these coarser $\rho$-tubes. As observed in Remark 2.7 above,
if $\mathbb{T}$ satisfies the Convex Wolff Axioms, then the coarsening of
$\mathbb{T}$ at a larger scale will also satisfy these axioms333In contrast,
the more traditional condition that the tubes in $\mathbb{T}$ point in
$\delta$-separated directions is _not_ preserved after coarsening. However,
the (rescaled) collection of $\delta$-tubes from $\mathbb{T}$ contained inside
each coarser $\rho$-tube might not satisfy the Convex Wolff Axioms. Our next
task is to introduce a slightly stricter variant of the Convex Wolff Axioms
that addresses this issue.

######  Definition 2.8.

Let $U$ be a convex set and let $S\subset U$. Let $U^{*}\supset U$ be the John
ellipsoid that circumscribes $U$, and let $\phi_{U}$ be an affine
transformation that sends $U^{*}$ to the unit ball. The _unit rescaling of $S$
relative to $U$_ is the set $\phi_{U}(S)$. If $\mathcal{S}$ be a multi-set of
subsets of $\mathbb{R}^{n}$ that are contained in $U$, then the _unit
rescaling of $\mathcal{S}$ relative to $U$_ is the multi-set
$\\{\phi_{U}(S)\colon S\in\mathcal{S}\\}$.

######  Remark 2.9.

In [16], the authors considered a similar definition in the special case where
$\mathcal{S}$ is a set of $\delta$-tubes and $U$ is a $\rho$-tube. In order
for this definition to be compatible with the definition from [16] of a tube’s
shading, the map $\phi_{U}$ was defined slightly differently, so that image of
$U^{*}$ was the ellipsoid $\\{Cx^{2}+Cy^{2}+z^{2}\leq 1\\}$ (for some fixed
constant $C$), rather than the unit ball.

######  Definition 2.10.

Let $\mathcal{A}$ be a set and $\mathcal{B}$ be a multi-set of convex subsets
of $\mathbb{R}^{n}$. We say that _$\mathcal{A}$ is a cover of $\mathcal{B}$_
if each set $B\in\mathcal{B}$ is contained in at least one set from
$\mathcal{A}$. For each $A\in\mathcal{A}$, we write $\mathcal{B}[A]$ to denote
the multi-set $\\{B\in\mathcal{B}\colon B\subset A\\}$. We say that
$\mathcal{A}$ is a _$K$ -uniform cover_ of $\mathcal{B}$ if
$\\#\mathcal{B}[A]\leq K(\\#\mathcal{B}[A^{\prime}])$ for each pair of sets
$A,A^{\prime}\in\mathcal{A}$. Finally, we say that $\mathcal{A}$ is a
_partitioning cover_ of $\mathcal{B}$ if $\mathcal{A}$ is a cover of
$\mathcal{B}$, and for every pair of distinct $A,A^{\prime}\in\mathcal{A}$,
the sets $\mathcal{B}[2A]$ and $\mathcal{B}[2A^{\prime}]$ are disjoint, where
$2A$ denotes the 2-fold dilate of $A$. In practice, $\mathcal{A}$ and
$\mathcal{B}$ will be collections of $\rho$-tubes and $\delta$-tubes
respectively, for some $\rho>\delta$.

######  Remark 2.11.

Note that if $\mathcal{B}$ satisfies the Convex Wolff Axioms with error $C$,
and if $\mathcal{A}$ is a $K$-uniform partitioning cover of $\mathcal{B}$,
then $\mathcal{A}$ satisfies the Convex Wolff Axioms with error $KC$.

######  Definition 2.12.

Let $C\geq 1,\delta>0$. We say a set $\mathbb{T}$ of $\delta$-tubes in
$\mathbb{R}^{n}$ satisfies the _Convex Wolff Axioms at every scale with error
$C$_ if the tubes in $\mathbb{T}$ are essentially distinct, and for every
$\rho_{0}\in[\delta,1]$, there exists $\rho\in[\rho_{0},C\rho_{0})$ and a set
of $\rho$-tubes $\mathbb{T}_{\rho}$ that satisfies the following properties.

  * (i)

$\mathbb{T}_{\rho}$ is a $C$-uniform partitioning cover of $\mathbb{T}$.

  * (ii)

For each $T_{\rho}\in\mathbb{T}_{\rho}$, the unit re-scaling of
$\mathbb{T}[T_{\rho}]$ relative to $T_{\rho}$ satisfies the Convex Wolff
Axioms with error $C$.

As the name suggests, Definition 2.12 is a multi-scale property. In
particular, if $\mathbb{T}$ satisfies the Convex Wolff Axioms at every scale,
then we can cover $\mathbb{T}$ by fatter $\rho$-tubes; both the $\rho$-tubes
and the (re-scaled) tubes from $\mathbb{T}$ inside each $\rho$-tube will again
satisfy the Convex Wolff Axioms at every scale. The precise statement is as
follows.

######  Lemma 2.13.

Let $\mathbb{T}$ be a set of $\delta$-tubes in $\mathbb{R}^{n}$ that satisfies
the Convex Wolff Axioms at every scale with error $C$. Let
$\rho_{0}\in[\delta,1]$. Then there exists $\rho\in[\rho_{0},C\rho_{0}]$ and a
set of $\rho$-tubes $\mathbb{T}_{\rho}$, so that $\mathbb{T}_{\rho}$ satisfies
the Convex Wolff Axioms at every scale with error $O(C)$, and for each
$T_{\rho}\in\mathbb{T}_{\rho}$, the unit rescaling of $\mathbb{T}[T_{\rho}]$
relative to $T_{\rho}$ satisfies the Convex Wolff Axioms at every scale with
error $O(C)$.

###### Proof.

Let $N$ be the largest integer with $C^{N+1}\leq\delta^{-1}$. For
$i=1,\ldots,N$, let $\mathbb{T}_{\tau_{i}},$ be a set of $\tau_{i}$-tubes,
with $\tau_{i}\in[\delta C^{i},\delta C^{i+1})$ that satisfy Items (i) and
(ii) from Definition 2.12. We claim that for $i<j$, if
$T_{\tau_{i}}\in\mathbb{T}_{\tau_{i}},T_{\tau_{j}}\in\mathbb{T}_{\tau_{j}}$,
and $\mathbb{T}[T_{\tau_{i}}]\cap\mathbb{T}[T_{\tau_{j}}]\neq\emptyset$, then
$\mathbb{T}[T_{\tau_{i}}]\subset\mathbb{T}[T_{\tau_{j}}]$. To see this,
observe that if the intersection contains at least one tube $T\in\mathbb{T}$,
then $2T_{\tau_{j}}\supset N_{\tau_{j}}(T)\supset T_{\tau_{i}}$. Since
$\mathbb{T}_{\tau_{j}}$ covers $\mathbb{T},$ and the sets
$\\{\mathbb{T}[2T_{\tau_{j}}^{\prime}]\colon
T_{\tau_{j}}^{\prime}\in\mathbb{T}_{\tau_{j}}\\}$ are disjoint, we conclude
that $\mathbb{T}[T_{\tau_{i}}]\subset\mathbb{T}[T_{\tau_{j}}]$.

The consequence of the above observation is the following: if we partially
order the sets $\\{\mathbb{T}[T_{\tau_{i}}]\colon
T_{\tau_{i}}\in\mathbb{T}_{\tau_{i}},\ i=1,\ldots,N\\}$ under inclusion, then
we actually obtain a total ordering, and this ordering forms a tree with $N$
levels—the vertices at level $i$ are precisely the sets
$\mathbb{T}[T_{\tau_{i}}]$. By Item (i) of Definition 2.12, for each
$i=1,\ldots,N$ the sets $\\{\mathbb{T}[T_{\tau_{i}}]\colon
T_{\tau_{i}}\in\mathbb{T}_{\tau_{i}}\\}$ have comparable cardinality, up to a
multiplicative factor of $C$. This tree is precisely what is needed to verify
Lemma 2.13: given $\rho_{0}\in[\delta,1]$, we select an index $i$ so that
$\tau_{i}\in[\rho_{0},C\rho_{0})$, and we choose $\rho=\tau_{i}$,
$\mathbb{T}_{\rho}=\mathbb{T}_{\tau_{i}}$; note that since $\mathbb{T}_{\rho}$
forms a partitioning cover of $\mathbb{T}$, the $\rho$-tubes in
$\mathbb{T}_{\rho}$ are essentially distinct. ∎

Finally, we will introduce two additional anti-concentration conditions that
will play a technical role in the arguments to follow.

######  Definition 2.14.

Let $C\geq 1$ and let $\mathbb{T}$ be a set of $\delta$-tubes in
$\mathbb{R}^{n}$. We say that $\mathbb{T}$ satisfies a _Frostman condition at
dimension $\sigma$ with error $C$_ if the tubes in $\mathbb{T}$ are
essentially distinct, and for every $\rho\geq\delta$ and every $\rho$-tube
$T_{\rho}$, we have $\\#\mathbb{T}[T_{\rho}]\leq
C\rho^{\sigma}(\\#\mathbb{T})$.

######  Definition 2.15.

Let $C\geq 1$ and let $\mathbb{T}$ be a set of $\delta$-tubes in
$\mathbb{R}^{n}$. We say that $\mathbb{T}$ satisfies the _Self-Similar Convex
Wolff Axioms with error $C$_ if the tubes in $\mathbb{T}$ are essentially
distinct, and for every $\rho_{0}\in[\delta,1]$, there exists
$\rho\in[\rho_{0},C\rho_{0})$ and a set of $\rho$-tubes $\mathbb{T}_{\rho}$
that satisfies the following properties.

  * (i)

$\mathbb{T}_{\rho}$ is a $C$-uniform partitioning cover of $\mathbb{T}$.

  * (ii)

For each $T_{\rho}\in\mathbb{T}_{\rho}$, the unit re-scaling of
$\mathbb{T}[T_{\rho}]$ relative to $T_{\rho}$ satisfies the Convex Wolff
Axioms with error $C$.

  * (iii)

For each $T_{\rho}\in\mathbb{T}_{\rho}$, we have

$C^{-1}(\rho/\delta)^{\sigma}\leq\\#\mathbb{T}[T_{\rho}]\leq
C(\rho/\delta)^{\sigma},$ (2.3)

where $\sigma>0$ is the unique number satisfying
$\\#\mathbb{T}=\delta^{-\sigma}$.

Note that Items (i) and (ii) above are identical to their counterparts in
Definition 2.12. In particular, we have

$\textrm{Self-Similar CWA w.~{}err.~{}$C$}\implies\textrm{CWA at every scale
w.~{}err.~{}$C$}\implies\textrm{CWA w/~{}err.~{}$C$}.$ (2.4)

###  2.3 Discretized Assouad dimension

The conclusion of Theorem 1.5 involves a discretized analogue of Assouad
dimension. We formalize this as follows.

######  Definition 2.16.

Let $E\subset\mathbb{R}^{n}$, let $\beta\in[0,n],$ let $\delta>0$, and let
$A\geq 1$. We say that $E$ has _discretized Assouad dimension at least
$\beta$, at scale $\delta$ and scale separation $A$_ if there exist scales
$\delta\leq\rho\leq r\leq 1$ with $r\geq A\rho$, and a ball $B$ of radius $r$,
so that

$\big{|}B\cap N_{\rho}(E)\big{|}\geq(\rho/r)^{n-\beta}|B|.$

If the scale $\delta$ is apparent from context, then for berevity we may say
“$E$ has discretized Assouad dimension at least $\beta$ at scale separation
$A$.”

In this language, the conclusion of Theorem 1.5 says that the set
$\bigcup_{T\in\mathbb{T}}Y(T)$ has discretized Assouad dimension at least
$3-\varepsilon$ at scale separation $\delta^{-\eta}.$

##  3 Assouad-extremal Kakeya sets

In this section, we will suppose that Theorem 1.5 is false, and we seek to
construct a “worst possible” counter-example to Theorem 1.5. In this and later
sections, we will restrict attention to $\mathbb{R}^{3}$. Our example will be
worst possible in two respects. First, Inequality (1.1) will fail as
dramatically as possible, i.e. a reverse inequality will hold where the term
$(\rho/r)^{\varepsilon}$ on the RHS of (1.1) will be replaced by
$(\rho/r)^{\omega}$ for the largest possible value of $\omega>0$. Second, of
all collections of tubes for which such a reverse inequality holds for this
value of $\omega$, we will choose a set $\mathbb{T}$ of largest possible
cardinality. The hypotheses of Theorem 1.5 ensure that $\mathbb{T}$ has size
at least $\delta^{-2}$; we will choose a set of tubes of size
$\delta^{-\alpha}$, for the largest possible value of $\alpha$.

We now turn to the task of carefully defining the quantities $\omega$ and
$\alpha$; this will be done in Sections 3.1 and 3.2, respectively. This task
is complicated by the fact that Theorem 1.5 involves a sequence of
quantifiers, and we must unwind these quantifiers in the correct order.

###  3.1 Defining $\omega$

Let $(\mathbb{T},Y)_{\delta}$ be a set of tubes and their associated shading.
Let $\delta\leq\rho<r\leq 1$, and let $B$ be a ball of radius $r$ whose
intersection with $\bigcup Y(T)$ has positive measure. Define
$\zeta=\zeta(\mathbb{T},Y,B,\rho)$ to be the unique number satisfying

$\Big{|}B\cap
N_{\rho}\Big{(}\bigcup_{T\in\mathbb{T}}Y(T)\Big{)}\Big{|}=\Big{(}\frac{\rho}{r}\Big{)}^{\zeta}|B|.$
(3.1)

Next, for $\eta>0$, define

$\zeta(\mathbb{T},Y;\eta)=\inf_{B,\rho}\zeta(\mathbb{T},Y,B,\rho),$ (3.2)

where the infimum is taken over all pairs $(B,\rho)$, where
$\rho\in[\delta,\delta^{\eta}]$ and $B$ is a ball of radius
$r\in[\delta^{-\eta}\rho,1]$ whose intersection with $\bigcup Y(T)$ has
positive measure. Observe that as $\eta\searrow 0$, the set of admissible
pairs $(B,\rho)$ becomes larger, and hence $\zeta(\mathbb{T},Y;\eta)$ weakly
decreases, i.e. $\zeta(\mathbb{T},Y;\eta)$ is a weakly increasing function of
$\eta$.

For $s,\eta,\delta>0$ and $u\geq 2$, define

$\omega(\varepsilon,\eta,\delta;u)=\sup_{(\mathbb{T},Y)_{\delta}}\zeta(\mathbb{T},Y;\eta),$
(3.3)

where the supremum is taken over all choices of $(\mathbb{T},Y)_{\delta}$ with
the following properties:

  * •

$\mathbb{T}$ is a set of essentially distinct $\delta$-tubes contained in
$B(0,1)$ that satisfies the Convex Wolff Axioms with error
$\delta^{-\varepsilon}$, and $Y$ is a $\delta^{\varepsilon}$-dense shading.

  * •

$\\#\mathbb{T}\geq\delta^{-u+\varepsilon}$.

Again, $\omega(\varepsilon,\eta,\delta;u)$ is a weakly increasing function of
$\eta$. It is also a weakly increasing function of $\varepsilon$ and weakly
decreasing function of $u$, since as $\varepsilon$ become larger or $u$
becomes smaller, the set of admissible pairs $(\mathbb{T},Y)_{\delta}$ becomes
larger.

Define

$\omega(u)=\lim_{\varepsilon\to 0^{+}}\ \lim_{\eta\to 0^{+}}\
\limsup_{\delta\to
0^{+}}\omega(\varepsilon,\eta,\delta;u)=\inf_{\varepsilon,\eta>0}\limsup_{\delta\to
0^{+}}\omega(\varepsilon,\eta,\delta;u).$

######  Definition 3.1.

Define $\omega=\omega(2)$.

Unwrapping the above definitions, we have the following.

######  Lemma 3.2.

Let $\omega\geq 0$ be as in Definition 3.1. For all $\varepsilon>0$, there
exists $\eta,\delta_{0}>0$ so that the following holds for all
$\delta\in(0,\delta_{0}]$. Let $(\mathbb{T},Y)_{\delta}$ be a set of
essentially distinct tubes that satisfy the Convex Wolff Axioms with error
$\delta^{-\eta}$, with a $\delta^{\eta}$-dense shading.

Then $\bigcup_{T\in\mathbb{T}}Y(T)$ has discretized Assouad dimension at least
$3-\omega-\varepsilon$ at scale separation $\delta^{-\eta}$.

###### Proof.

Select $\eta>0$ sufficiently small so that $\limsup_{\delta\to
0^{+}}\omega(\eta,\eta,\delta;2)\leq\omega+\varepsilon/2.$ Then there exists
$\delta_{0}>0$ so that $\omega(\eta,\eta,\delta;2)\leq\omega+\varepsilon$ for
all $\delta\in(0,\delta_{0}]$. The conclusion of Lemma 3.2 now holds for this
choice of $\eta$ and $\delta_{0}$. ∎

In light of Lemma 3.2, in order to prove Theorem 1.5 it suffices to prove that
$\omega=0$.

###  3.2 Defining $\alpha$

Note that $\omega(u)$ is weakly monotone decreasing. We also have that
$\omega(4)=0$, since if $\mathbb{T}$ is a set of $\delta^{-4}$ essentially
distinct $\delta$-tubes in $B(0,1)$, then $\bigcup_{\mathbb{T}}T$ must have
volume $\gtrsim 1$. Define

$\alpha=\sup\\{u\in[2,4]\colon\omega(u)=\omega(2)\\}.$ (3.4)

The quantity $\alpha$ has the following interpretation. We have defined
$3-\omega$ to be the smallest possible discretized Assouad dimension of a
collection of essentially distinct tubes that satisfies the Convex Wolff
Axioms. Every such collection of tubes must have cardinality at least
$\delta^{-2}$, and in general we might expect that larger collections of tubes
should have larger discretized Assouad dimension. Any collection of
essentially distinct tubes of cardinality substantially larger than
$\delta^{-\alpha}$ that satisfies the Convex Wolff Axioms must have
discretized Assouad dimension larger than $3-\omega$.

###  3.3 The existence of Assouad-Extremal Kakeya sets

Now that we have defined $\omega$ and $\alpha$, we are ready to construct a
(hypothetical) “worst possible” counter-example to Theorem 1.5, in the sense
described at the beginning of Section 3.

######  Definition 3.3.

For $\varepsilon>0,$ we say a pair $(\mathbb{T},Y)_{\delta}$ is _$\varepsilon$
Assouad-extremal_ if it satisfies the following properties.

  * •

$\\#\mathbb{T}\geq\delta^{-\alpha+\varepsilon}$.

  * •

The tubes are contained in $B(0,1)$, essentially distinct, and satisfy the
Convex Wolff Axioms with error $\delta^{-\varepsilon}$.

  * •

The shading $Y$ is $\delta^{\varepsilon}$-dense.

  * •

For all $\rho,r\in[\delta,1]$ with $\rho\leq\delta^{\varepsilon}r$ and all
balls $B$ of radius $r$, we have

$\Big{|}B\cap
N_{\rho}\Big{(}\bigcup_{T\in\mathbb{T}}Y(T)\Big{)}\Big{|}\leq(\rho/r)^{\omega-\varepsilon}|B|.$
(3.5)

Note that if $\mathbb{T}$ is $\varepsilon$ Assouad-extremal, then it is also
$\varepsilon^{\prime}$ Assouad-extremal for all
$\varepsilon^{\prime}\geq\varepsilon$.

######  Lemma 3.4.

For all $\varepsilon,\delta_{0}>0$, there exists $\delta\in(0,\delta_{0}]$ and
an $\varepsilon$ Assouad-extremal set of tubes $(\mathbb{T},Y)_{\delta}.$

###### Proof.

Fix $u\geq\max(2,\alpha-\varepsilon/2)$ with $\omega(u)=\omega(2)=\omega$.
Since $\omega(s,\eta,\delta;u)$ is a weakly increasing function of $s$ and
$\eta$, we have $\limsup_{\delta\to
0^{+}}\omega(\varepsilon/2,\varepsilon,\delta;u)\geq\omega(u)$. Thus there
exists $\delta\in(0,\delta_{0}]$ with
$\omega(\varepsilon/2,\varepsilon,\delta;u)\geq\omega-\varepsilon/2$.

In light of (3.3), there exists a set of tubes and their associated shading
$(\mathbb{T},Y)_{\delta}$, so that (i)
$\\#\mathbb{T}\geq\delta^{-u+\varepsilon/2}\geq\delta^{-\alpha+\varepsilon}$;
(ii) the tubes are contained in $B(0,1)$, essentially distinct, and satisfy
the Convex Wolff Axioms with error $\delta^{-\varepsilon}$; (iii) $Y$ is a
$\delta^{\varepsilon}$-dense shading; and (iv) we have
$\zeta(\mathbb{T},Y;\eta)\geq\omega-\varepsilon$. Thus by (3.2), for every
pair of scales $\rho,r\in[\delta,1]$ with $\rho\leq\delta^{\varepsilon}r$ and
every ball $B$ of radius $r$, we have

$\zeta(\mathbb{T},Y,B,\rho)\geq\zeta((\mathbb{T},Y);\eta)\geq\omega-\varepsilon.$

But this is precisely (3.5). ∎

##  4 Convex Wolff Axioms and multi-scale structure

The purpose of this section is to prove the following result.

######  Proposition 4.1.

For all $\varepsilon>0$, there exists $\tau>0$, so that for all $\eta_{1}>0$,
there exists $\eta,\delta_{0}>0$ so that the following holds for all
$\delta\in(0,\delta_{0}]$. Let $\beta\geq 2$ and let $(\mathbb{T},Y)_{\delta}$
be a set of $\delta^{-\beta}$ essentially distinct $\delta$-tubes that satisfy
the Convex Wolff Axioms with error $\delta^{\eta}$, with $Y$ a $\delta^{\eta}$
uniformly dense shading. Then at least one of the following must be true.

  * (A)

$\bigcup Y(T)$ has discretized Assouad dimension at least $3-\varepsilon$ at
scale $\delta$, with scale separation $\delta^{-\eta}$.

  * (B)

There exists $\mathbb{T}^{\prime}\subset\mathbb{T}$ with
$\\#\mathbb{T}^{\prime}\geq\delta^{\eta_{1}}(\\#\mathbb{T})$, and
$\mathbb{T}^{\prime}$ satisfies the self-similar Convex Wolff Axioms with
error $\delta^{-\varepsilon}$.

  * (C)

There exists $\rho\in[\delta,\delta^{\varepsilon}]$ and a set
$\tilde{\mathbb{T}}$ of $\rho$-tubes, with the following properties.

    * (C.i)

The tubes in $\tilde{\mathbb{T}}$ are essentially distinct, and
$\\#\tilde{\mathbb{T}}\geq\rho^{-\beta-\tau}$.

    * (C.ii)

$\tilde{\mathbb{T}}$ satisfies the Convex Wolff Axioms with error
$\leq\rho^{-\eta_{1}}$.

    * (C.iii)

For each $T_{\rho}\in\tilde{\mathbb{T}}$, there exists $T\in\mathbb{T}$ with
$T\subset T_{\rho}$.

  * (D)

There exists $\rho\in[\delta^{1-\varepsilon},1]$, a $\rho$-tube $T_{\rho}$,
and a set $\tilde{\mathbb{T}}\subset\mathbb{T}[T_{\rho}]$, so that the
following holds.

    * (D.i)

$\\#\tilde{\mathbb{T}}\geq(\delta/\rho)^{-\beta-\tau}.$

    * (D.ii)

The unit rescaling of $\tilde{\mathbb{T}}$ relative to $T_{\rho}$ satisfies
the Convex Wolff Axioms with error $(\delta/\rho)^{-\eta_{1}}$.

######  Remark 4.2.

Conclusions (A) and (B) are somewhat self-explanatory. Conclusion (C) says
that at a coarser scale, we can find a set of tubes that is richer than our
original set $\mathbb{T}$. Conclusion (D) says that we can find a $\rho$-tube
so that the unit-rescaling of $\mathbb{T}[T_{\rho}]$ is richer than our
original set $\mathbb{T}$.

An important step in proving Proposition 4.1 is to study arrangements of tubes
that cluster into rectangular prisms. In the next section, we will show that
unions of rectangular prisms of dimensions $s\times t\times 1$ with
$s<\\!\\!<t$ have large Assouad dimension.

###  4.1 Arrangements of rectangular prisms

The next result is a straightforward consequence of Cauchy-Schwartz; it says
that a union of $\delta\times 1\times 1$ prisms satisfying the Convex Wolff
Axioms must have almost maximal volume.

######  Lemma 4.3.

Let $0<\delta\leq 1$ and $K\geq 1$. Let $\mathcal{R}$ be a multi-set of
$\delta\times 1\times 1$ rectangular prisms, and suppose that $\mathcal{R}$
satisfies the Convex Wolff Axioms with error $K$. Let
$\\{Y(R),R\in\mathcal{R}\\}$ be a $K^{-1}$-dense shading. Then

$\Big{|}\bigcup_{R\in\mathcal{R}}Y(R)\Big{|}\gtrsim K^{-3}|\log\delta|^{-1}.$
(4.1)

###### Proof.

For each prism $R\in\mathcal{R}$, there is an associated normal vector
$v(R)\in S^{2}$, which is well-defined up to uncertainty $O(\delta)$. For
$R,R^{\prime}\in\mathcal{R}$, define $\angle(R,R^{\prime})$ to be the
(unsigned) angle between $v(R)$ and $v(R^{\prime})$.

Fix $R\in\mathcal{R}$. We have

$\displaystyle\sum_{R^{\prime}\in\mathcal{R}}|R\cap R^{\prime}|$
$\displaystyle=\sum_{j=0}^{\log(1/\delta)}\sum_{\begin{subarray}{c}R^{\prime}\colon
R\cap R^{\prime}\neq\emptyset,\\\ \angle(R,R^{\prime})\sim
2^{j}\delta\end{subarray}}|R\cap
R^{\prime}|\lesssim\sum_{j=0}^{\log(1/\delta)}\Big{(}K(2^{j}\delta\\#\mathcal{R})\Big{)}\Big{(}\frac{\delta^{2}}{2^{j}\delta}\Big{)}$
$\displaystyle\lesssim K\delta^{2}(\\#\mathcal{R})\log(1/\delta).$

In the above computation, observe that if $R\cap R^{\prime}$ is non-empty and
$\angle(R,R^{\prime})=\theta\sim 2^{j}\delta$, then $R^{\prime}$ is contained
in the $O(\theta)\times 1\times 1$ rectangular prism concentric with $R$.
Since $\mathcal{R}$ obeys the Convex Wolff Axioms, at most
$K(2^{j}\delta)(\\#\mathcal{R})$ slabs $R^{\prime}\in\mathcal{R}$ can have
this property. We conclude that

$\Big{\|}\sum_{R\in\mathcal{R}}\chi_{R}\Big{\|}_{2}^{2}\lesssim
K\delta^{2}(\\#\mathcal{R})^{2}\log(1/\delta).$ (4.2)

Let $E=\bigcup_{R\in\mathcal{R}}Y(R)$. Since the shading $Y$ is $K^{-1}$
dense, by Cauchy-Schwartz we have

$K^{-1}\delta(\\#\mathcal{R})\leq\int\chi_{E}\cdot\sum_{R\in\mathcal{R}}\chi_{Y(R)}\leq\int\chi_{E}\cdot\sum_{R\in\mathcal{R}}\chi_{R}\leq|E|^{1/2}\Big{\|}\sum_{R\in\mathcal{R}}\chi_{R}\Big{\|}_{2}.$
(4.3)

Rearranging and using (4.2), we obtain (4.1). ∎

Next, we will establish a geometric argument showing that a union of $s\times
t\times 1$ prisms, $s\ll t$, satisfying the Convex Wolff Axioms must either
have nearly maximal discretized Assouad dimension, or the prisms must arrange
themselves into $s^{\prime}\times t^{\prime}\times 1$ prisms for $s^{\prime}$
and $t^{\prime}$ substantially larger than $s$ and $t$, respectively. The
precise statement is as follows.

######  Lemma 4.4.

For all $\varepsilon,\tau\in(0,1]$, there exists $\delta_{0},\eta>0$ so that
the following is true for all $\delta\in(0,\delta_{0}]$. Let $\delta\leq s\leq
t\leq 1$ with $s\leq\delta^{\varepsilon}t$. Let $\mathcal{R}$ be a multi-set
of $s\times t\times 1$ prisms contained in $B(0,1)$, and suppose that
$\mathcal{R}$ satisfies the Convex Wolff Axioms with error $\delta^{-\eta}$.
Let $\\{Y(R),R\in\mathcal{R}\\}$ be a $\delta^{\eta}$-dense shading.

Then at least one of the following two things must happen.
(A): There exists $r\in[\delta^{-\eta}s,1]$ and a $r$-ball $B$ so that

$\Big{|}B\cap\bigcup_{R\in\mathcal{R}}Y(R)\Big{|}\geq(s/r)^{\varepsilon}|B|.$
(4.4)

(B): There are scales
$s^{\prime}\in[\delta^{-\varepsilon^{2}/2}s,\delta^{\varepsilon/2}]$ and
$t^{\prime}=\min\big{(}1,t\frac{s^{\prime}}{s}\big{)}$, and a multi-set
$\mathcal{R}^{\prime}$ of $s^{\prime}\times t^{\prime}\times 1$ prisms that
satisfies the Convex Wolff Axioms with error $\delta^{-\tau}$, so that the
shading

$Y^{\prime}(R^{\prime})=R^{\prime}\cap
N_{s^{\prime}}\Big{(}\bigcup_{\begin{subarray}{c}R\in\mathcal{R}\\\ R\subset
2R^{\prime}\end{subarray}}Y(R)\Big{)}$ (4.5)

is $\delta^{\tau}$-dense.

######  Remark 4.5.

Note that as $\varepsilon\searrow 0$, Conclusion (A) becomes stronger, while
conclusion (B) becomes weaker (indeed, if $\varepsilon=0$ then Conclusion (B)
vacuously holds with $s^{\prime}=s$ and $t^{\prime}=t$). As $\tau\searrow 0$,
Conclusion (A) is unchanged, while Conclusion (B) becomes stronger.

###### Proof.

Step 1.

After dyadic pigeonholing, we can select a multiplicity $\mu$ and a
$\geq\frac{1}{10}|\log\delta|^{-1}\delta^{\eta}$-dense shading
$Y_{1}(R)\subset Y(R)$ so that
$\sum_{R\in\mathcal{R}}\chi_{Y_{1}(R)}(x)\sim\mu$ for all
$x\in\bigcup_{R\in\mathcal{R}}Y_{1}(R)$. Let $Y_{2}(R)\subset Y_{1}(R)$ be a
regular (in the sense of Definition 2.2) refinement of $Y_{1}$, with
$|Y_{2}(R)|\geq\frac{1}{2}|Y_{1}(R)|$ for each $R\in\mathcal{R}$.

If $\mu\leq\delta^{-\varepsilon^{2}+3\eta}(st\\#\mathcal{R})$, then

$\Big{|}\bigcup_{R\in\mathcal{R}}Y(R)\Big{|}\geq\mu^{-1}\sum_{R\in\mathcal{R}}|Y_{1}(R)|\gtrapprox\delta^{\varepsilon^{2}-3\eta}\delta^{\eta}\frac{\sum_{R\in\mathcal{R}}|R|}{st(\\#\mathcal{R})}\geq\delta^{\varepsilon^{2}-\eta}\geq\delta^{-\eta}s^{\varepsilon},$

where the second-last inequality used the hypothesis that $\mathcal{R}$
satisfies the Convex Wolff Axioms with error $\delta^{-\eta}$ (and hence
$\sum|R|\geq\delta^{\eta}$) and the final inequality used the hypothesis that
$s\leq\delta^{\varepsilon}t$ and hence $s\leq\delta^{\varepsilon}$. If
$\delta_{0}$ is sufficiently small depending on $\eta$, then Conclusion (A)
holds with $r=1$, and we are done.

Step 2.

Suppose instead that
$\mu\geq\delta^{-\varepsilon^{2}+3\eta}(st\\#\mathcal{R})$. Each $s\times
t\times 1$ prism $R\in\mathcal{R}$ is the image of the set
$[-s/2,s/2]\times[-t/2,t/2]\times[-1/2,1/2]$ under a rigid motion $\tau$. We
define $\ell=\ell(R)$ to be the image of the $z$-axis under $\tau$, and we
define $\Pi(R)$ to be the image of the $yz$-plane under $\tau$.

For each $x\in\bigcup_{R\in\mathcal{R}}Y_{2}(R)$, define the “plane spread”

$\operatorname{P-spread}(x)=\max\Big{\\{}\angle\big{(}\Pi(R),\Pi(R^{\prime})\big{)}\Big{\\}},$

where the maximum is taken over all pairs $R,R^{\prime}$ for which $x\in
Y_{2}(R)$ and $x\in Y_{2}(R^{\prime})$, and where $\angle(\Pi,\Pi^{\prime})$
denotes the unsigned angle between the normal vectors of $\Pi$ and
$\Pi^{\prime}$. Since $\Pi(R)$ and $\Pi(R^{\prime})$ are defined up to
uncertainty $O(s/t)$, $\operatorname{P-spread}(x)$ is meaningfully defined if
it is substantially larger than $s/t$. If
$x\not\in\bigcup_{R\in\mathcal{R}}Y_{2}(R)$, define
$\operatorname{P-spread}(x)=0$.

Step 3. Suppose there exists a prism $R_{0}\in\mathcal{R}$ and a number
$\theta\geq\delta^{-4\eta/\varepsilon}\frac{s}{t}$ so that

$\big{|}\\{x\in
R_{0}\colon\theta\leq\operatorname{P-spread}(x)<2\theta\\}\big{|}\geq\frac{1}{100}|\log\delta|^{-2}\delta^{\eta}|R_{0}|.$
(4.6)

Since $R_{0}$ is a $s\times t\times 1$ prism, for each $x_{0}\in R_{0}$ we
have $|R_{0}\cap B(x_{0},t)|\sim st^{2}$. By pigeonholing, we can select a
point $x_{0}\in R_{0}$ so that

$\big{|}\\{x\in R_{0}\cap
B(x_{0},t)\colon\theta\leq\operatorname{P-spread}(x)<2\theta\\}\big{|}\gtrapprox\delta^{\eta}st^{2}.$
(4.7)

After a harmless translation and rotation, we may suppose that $\Pi(R_{0})$ is
the $yz$-plane, i.e. $\operatorname{span}\\{{\bf e}_{2},{\bf e}_{3}\\}$, and
that $R_{0}\cap B(x_{0},t)$ is contained in the “square”
$S=[-s,s]\times[-t,t]\times[-t,t]$ (note as well that $|S|\sim|R_{0}\cap
B(x_{0},t)|$). Let $\tilde{S}$ denote the thickened square
$[-10t\theta,10t\theta]\times[-10t,10t]\times[-10t,10t]$, i.e. $\tilde{S}$ is
comparable to the $t\theta$ neighborhood of $S$.

Denote the set on the LHS of (4.7) by $E$; then for each $x\in E$ there is a
prism $R_{x}\in\mathcal{R}$ with $x\in Y_{2}(R)$ and
$\angle(\Pi(R_{x}),\operatorname{span}\\{{\bf e}_{2},{\bf
e}_{3}\\})\in[\theta,2\theta)$. This implies that $R_{x}\cap B(x,t)\subset
R_{x}\cap B(x_{0},10t)\subset\tilde{S}$. Since the shading $Y_{2}(R_{x})$ is
regular, we also have

$|Y_{2}(R_{x})\cap B(x_{0},2t)|\gtrapprox|Y_{2}(R_{x})|\frac{|R_{x}\cap
B(x_{0},2t)|}{|R_{x}|}\gtrsim\delta^{\eta}st^{2}.$ (4.8)

Define the linear map $A(x,y,z)=\Big{(}\frac{x}{10t\theta},\ \frac{y}{10t},\
\frac{z}{10t}\Big{)}$, so $A(\tilde{S})=[-1,1]^{3}$. Define

$\hat{S}=A(S)=[-\frac{s}{10t\theta},\frac{s}{10t\theta}]\times[-1/10,1/10]\times[-1/10,1/10],$

and let $\hat{E}=A(E)\subset\hat{S}$. For each $x\in E$, let
$\hat{R}_{x}=A(R_{x}\cap\tilde{S})$; this is comparable to a rectangular prism
of dimensions roughly $\frac{s}{t\theta}\times 1\times 1$, and this prism
makes angle $\sim 1$ with the $yz$-plane, i.e. the prism has a normal
direction ${\bf n}(\hat{R}_{x})$ (defined up to uncertainty
$O(\frac{s}{t\theta})$) with $|{\bf n}(\hat{R}_{x})\wedge{\bf e}_{1}|\gtrsim
1$. By (4.8) we have

$|\hat{Y}_{2}(\hat{R}_{x})|\gtrapprox\delta^{\eta}\frac{s}{t\theta}.$ (4.9)

By pigeonholing, we can find a unit vector $e\in\mathbb{R}^{3}$ and a set
$\hat{E}^{\prime}\subset\hat{E}$ with $|\hat{E}^{\prime}|\sim|\hat{E}|$, so
that $\angle\big{(}e,{\bf n}(\hat{R}_{x})\big{)}\leq\frac{1}{100}$ for each
$x\in\hat{E}^{\prime}$. In particular, this implies $|e\wedge{\bf
e}_{1}|\gtrsim 1$, and hence the projection of $e$ to the $yz$-plane has
magnitude $\sim 1$; denote the image of $e$ under this projection by
$e^{\prime}$.

(4.7) says that $|E|\gtrapprox\delta^{\eta}st^{2}$ and hence
$|\hat{E}^{\prime}|\gtrsim|\hat{E}|\gtrapprox\delta^{\eta}\frac{s}{t\theta}$.
On the other hand, $\hat{E}^{\prime}\subset\hat{S}$, and the latter set has
volume $\sim\frac{s}{t\theta}$. Since
$\hat{S}=[-\frac{s}{10t\theta},\frac{s}{10t\theta}]\times[-1/10,1/10]\times[-1/10,1/10]$
and $e^{\prime}$ is contained in the $yz$ plane, by Fubini we can select a
line $\ell$ pointing in direction $e^{\prime}$ that satisfies
$|\ell\cap\hat{E}^{\prime}|\gtrapprox\delta^{\eta}$.

Select a $\frac{s}{t\theta}$-separated set of points $x_{1},\ldots,x_{N}$,
with $N\gtrapprox\delta^{\eta}\frac{t\theta}{s}$, from
$|\ell\cap\hat{E}^{\prime}|$; we will index these points so that
$|x_{i}-x_{j}|\geq\frac{s}{t\theta}|i-j|$. Note that $R_{x_{i}}\cap\ell$ is a
line segment of length between $\frac{s}{t\theta}$ and $10\frac{s}{t\theta}$,
and hence if $i,j$ are a pair of indices with $|x_{i}-x_{j}|\geq
100\frac{s}{t\theta}$, then
$\operatorname{dist}(R_{x_{i}}\cap\ell,R_{x_{j}}\cap\ell)\gtrsim\frac{s}{t\theta}|i-j|$.
Since $\ell$ points in direction $e^{\prime}$, and $|e^{\prime}\cdot{\bf
n}(\hat{R}_{x_{j}})|\sim 1$, we conclude that
$\operatorname{dist}(x_{i},R_{x_{j}})\gtrsim\frac{s}{t\theta}|i-j|$, so in
particular $R_{x_{i}}$ contains a point that has distance
$\gtrsim\frac{s}{t\theta}|i-j|$ from $R_{x_{j}}$. Thus either $R_{x_{i}}$ and
$R_{x_{j}}$ do not intersect, or if they do intersect then their normal
vectors satisfy $\angle\big{(}{\bf n}(\hat{R}_{x_{i}}),\ {\bf
n}(\hat{R}_{x_{j}})\big{)}\gtrsim\frac{s}{t\theta}|i-j|.$ In either case, we
have

$|\hat{R}_{x_{i}}\cap\hat{R}_{x_{j}}|\lesssim\frac{(\frac{s}{t\theta})^{2}}{\frac{s}{t\theta}(1+|i-j|)}=\frac{s}{t\theta(1+|i-j|)}.$
(4.10)

The above inequality vacuously holds if $i,j$ are a pair of indices with
$|x_{i}-x_{j}|<100\frac{s}{t\theta}$, since in this case $|i-j|\sim 1$ and we
can use the trivial bound
$|\hat{R}_{x_{i}}\cap\hat{R}_{x_{j}}|\leq|\hat{R}_{x_{i}}|$.

We conclude that

$\Big{\|}\sum_{i=1}^{N}\chi_{\hat{Y}_{2}(\hat{R}_{i})}\Big{\|}_{2}^{2}\leq\sum_{i,j=1}^{N}|\hat{R}_{i}\cap\hat{R}_{j}|\lesssim
N\frac{s}{t\theta}\big{|}\log\big{(}\frac{s}{t\theta}\big{)}\big{|},$ (4.11)

and hence

$\Big{|}\bigcup_{i=1}^{N}\hat{Y}_{2}(\hat{R}_{i})\Big{|}\geq\frac{\Big{(}\sum_{i}|\hat{Y}_{2}(\hat{R}_{i})|\Big{)}^{2}}{\Big{\|}\sum_{i}\chi_{\hat{Y}_{2}(\hat{R}_{i})}\Big{\|}_{2}^{2}}\gtrapprox\frac{\Big{(}N\delta^{\eta}\frac{s}{t\theta}\Big{)}^{2}}{N\frac{s}{t\theta}\big{|}\log\big{(}\frac{s}{t\theta}\big{)}\big{|}}\gtrapprox\delta^{2\eta}N\frac{s}{t\theta}\gtrapprox\delta^{3\eta},$

where the second inequality used (4.9) and (4.11). Undoing the scaling
transformation $A$, we conclude that

$\Big{|}\tilde{S}\cap\bigcup_{R\in\mathcal{R}}Y(R)\Big{|}\gtrapprox\delta^{3\eta}|\tilde{S}|.$

Recall that $\tilde{S}$ is a rectangular prism of dimensions roughly
$t\theta\times t\times t$. Define $r=t\theta$; then by pigeonholing, we can
find a ball $B\subset\tilde{S}$ of radius $r$ so that

$\Big{|}B\cap\bigcup_{R\in\mathcal{R}}Y(R)\Big{|}\gtrapprox\delta^{3\eta}\
|B|.$

By $\delta_{0}$ sufficiently small depending on $\eta$ and the implicit
constants in the above quasi-inequality, we can ensure that

$\Big{|}B\cap\bigcup_{R\in\mathcal{R}}Y(R)\Big{|}\geq\delta^{4\eta}|B|,$
(4.12)

and since $\theta\geq\delta^{-4\eta/\varepsilon}\frac{s}{t}$, we have
$\delta^{4\eta}\geq(s/r)^{\varepsilon}$. Thus (4.12) implies (4.4), and hence
Conclusion (A) holds.

Step 4. Suppose instead that for every prism $R_{0}\in\mathcal{R}$ and every
$\theta\geq\delta^{-4\eta/\varepsilon}\frac{s}{t}$, (4.6) fails. Let
$s_{1}=\delta^{-4\eta/\varepsilon}s$. Since $s\leq\delta^{\varepsilon}t$, if
we select $\eta$ sufficiently small then $s_{1}\leq t$. Select a
$\gtrapprox\delta^{\eta}$ dense shading $Y_{3}(R)\subset Y_{2}(R)$ so that
$\operatorname{P-spread}(x)\leq\frac{s_{1}}{t}$ for each $x\in Y_{3}(R)$. Let
$\mathcal{R}_{0}\subset\mathcal{R}$ consist of those $R\in\mathcal{R}$ with
$|Y_{3}(R)|\gtrapprox\delta^{\eta}|R|$; we will choose the implicit constant
and implicit power of $|\log\delta|$ so that
$\\#\mathcal{R}_{0}\gtrapprox\delta^{\eta}(\\#\mathcal{R}$).

For each $x\in\bigcup_{R\in\mathcal{R}_{0}}Y_{3}(R)$, define the “line spread”

$\operatorname{L-spread}(x)=\max\Big{\\{}\angle\big{(}\ell(R),\ell(R^{\prime})\big{)}\Big{\\}},$
(4.13)

where the maximum is taken over all pairs $R,R^{\prime}$ for which $x\in
Y_{3}(R)$ and $x\in Y_{3}(R^{\prime})$, and where $\angle(\ell,\ell^{\prime})$
denotes the unsigned angle between the lines $\ell$ and $\ell^{\prime}$. Since
$\ell(R)$ and $\ell(R^{\prime})$ are defined up to uncertainty $O(t)$,
$\operatorname{L-spread}(x)$ is meaningfully defined if it is substantially
larger than $t$. If $x\not\in\bigcup_{R\in\mathcal{R}_{0}}Y_{3}(R)$, define
$\operatorname{L-spread}(x)=0$.

We claim there is a number $\theta\in[\delta^{-\varepsilon^{2}/2}t,\pi/4]$ so
that the shading

$Y_{4}(R)=\\{x\in
Y_{3}(R)\colon\operatorname{L-spread}(x)\in[\theta,2\theta)\\}$ (4.14)

is $\gtrapprox\delta^{\eta}$ dense. We argue as follows: if this was not the
case, then

$\displaystyle\sum_{R\in\mathcal{R}_{0}}|\\{x\in Y_{3}(R)\colon$
$\displaystyle\operatorname{L-spread}(x)<\delta^{-\varepsilon^{2}/2}t\\}|\gtrapprox\sum_{R\in\mathcal{R}_{0}}|Y_{3}(R)|$
$\displaystyle\gtrapprox\delta^{2\eta}\sum_{R\in\mathcal{R}}|Y_{1}(R)|\gtrsim\delta^{2\eta}\mu\Big{|}\bigcup_{R\in\mathcal{R}}Y_{1}(R)\Big{|}$
$\displaystyle\geq\delta^{2\eta}\mu\Big{|}\bigcup_{R\in\mathcal{R}_{0}}Y_{3}(R)\Big{|},$

and hence there must exist a point $x\in\bigcup_{R\in\mathcal{R}_{0}}Y_{3}(R)$
with

$\\#\\{R\in\mathcal{R}_{0}\colon x\in
Y_{3}(R)\\}\gtrapprox\delta^{2\eta}\mu\geq\delta^{-\varepsilon^{2}+5\eta}(st\\#\mathcal{R}),\
\ \operatorname{L-spread}(x)\leq\delta^{-\varepsilon^{2}/2}t,\ \
\operatorname{P-spread}(x)\leq\frac{s_{1}}{t}.$ (4.15)

(the last condition follows from the fact that $x\in\bigcup Y_{3}(R)$). But
this means that the rectangular prisms $\\{R\in\mathcal{R}_{0}\colon x\in
Y_{3}(R)\\}$ are contained in the $O(1)$ dilate of a rectangular prism of
dimensions $s_{1}\times\delta^{-\varepsilon^{2}/2}t\times 1$. Since the prisms
in $\mathcal{R}$ satisfy the Convex Wolff Axioms with error $\delta^{-\eta}$,
we have

$\\#\\{R\in\mathcal{R}_{0}\colon x\in
Y_{3}(R)\\}\lesssim\delta^{-\eta}\Big{(}s_{1}\cdot\delta^{-\varepsilon^{2}/2}t\Big{)}(\\#\mathcal{R})=\delta^{-3\eta/\varepsilon-\eta-\varepsilon^{2}/2}(st\\#\mathcal{R}),$
(4.16)

Comparing (4.15) and (4.16), we obtain a contradiction if $\delta_{0}$ and
$\eta$ are sufficiently small.

Step 5. At this point we can fix $\theta\geq\delta^{-\varepsilon^{2}/2}t$, and
we may suppose that the corresponding shading $Y_{4}$ from (4.14) is
$\gtrapprox\delta^{\eta}$ dense. For each prism $R\in\mathcal{R}_{0}$, let
$\hat{R}$ be the prism of dimensions $\frac{s_{1}\theta}{t}\times\theta\times
1$ obtained by anisotropically dilating $R$. We claim that if
$R_{0},R\in\mathcal{R}_{0}$ are prisms and if $x\in Y_{4}(R_{0})\cap
Y_{3}(R)$, then $R\subset 2\hat{R}_{0}$. This follows from the fact that
$\angle(\Pi(R_{0}),\Pi(R))\leq\frac{s_{1}}{t}$, and
$\angle(\ell(R_{0}),\ell(R))\leq 2\theta$. In fact more is true—if we define
$w=\max\big{(}\frac{s_{1}\theta}{t},t\big{)}$, then $N_{w}(R)$ is comparable
to a prism of dimensions $\frac{s_{1}\theta}{t}\times w\times 1$, and this
prism is contained in $2\hat{R}_{0}$. Furthermore, $N_{w}(R)\cap R_{0}$ is
comparable to a rectangular prism of dimensions $s\times t\times w/\theta$.

Let $\mathcal{R}_{1}\subset\mathcal{R}_{0}$ consist of those rectangular
prisms for which $|Y_{4}(R)|\gtrapprox\delta^{2\eta}|R|$. Let
$R_{0}\in\mathcal{R}_{1}$; recall that $R_{0}$ is a $s\times t\times 1$
rectangular prism; for notational simplicity we will suppose
$R_{0}=[-s/2,s/2]\times[-t/2,t/2]\times[-1/2,1/2]$, and that the intersection
of $Y_{4}(R_{0})$ with the $z$-axis has measure $\gtrapprox\delta^{\eta}$.
Define
$\tilde{R}_{0}=[-\frac{s_{1}\theta}{t},\frac{s_{1}\theta}{t}]\times[-\theta,\theta]\times[-1,1]$.

Select $\frac{w}{\theta}$-separated points $z_{1},\ldots,z_{N}\in
Y_{4}(R_{0})$ on the $z$-axis, with
$N\gtrapprox\delta^{\eta}\frac{\theta}{w}$. We will label the points so that
$|z_{i}-z_{j}|\geq\frac{w}{\theta}|i-j|$. For each index $i$, select a prism
$R_{i}\in\mathcal{R}_{0}$ with $z_{i}\in Y_{3}(R_{i})$ and
$\angle((0,0,1),\ell(R))\in[\theta,2\theta)$.

For each index $i$, let $R_{i}^{\prime}$ be the $\frac{s_{1}\theta}{t}$
neighborhood of $R_{i}$, and let
$Y_{3}^{\prime}(R_{i}^{\prime})=N_{\frac{s_{1}\theta}{t}}(Y_{3}(R_{i}))$.
Since $R_{i}\in\mathcal{R}_{0}$ (and hence
$|Y_{3}(R_{i})\gtrapprox\delta^{\eta}|R_{i}|$), we have

$|Y_{3}^{\prime}(R_{i}^{\prime})|\gtrapprox\delta^{\eta}|R_{i}^{\prime}|.$
(4.17)

$R_{i}^{\prime}$ is comparable to a prism of dimensions
$\frac{s_{1}\theta}{t}\times w\times 1$ whose coaxial line makes angle
$\sim\theta$ with the $z$ axis. We have

$|R_{i}^{\prime}\cap
R_{j}^{\prime}|\lesssim\big{(}\frac{s_{1}\theta}{t}\big{)}\frac{w}{1+|i-j|},$

and hence

$\Big{\|}\sum_{i=1}^{N}\chi_{Y_{3}^{\prime}(R_{i}^{\prime})}\Big{\|}_{2}^{2}\leq\sum_{i,j=1}^{N}|R_{i}^{\prime}\cap
R_{j}^{\prime}|\lesssim N\frac{s_{1}\theta
w}{t}\big{|}\log(\frac{\theta}{w}\big{)}\big{|}.$

We conclude that

$\Big{|}\bigcup_{i=1}^{N}Y_{3}^{\prime}(R_{i}^{\prime})\Big{|}\geq\frac{\Big{(}\sum|Y_{3}^{\prime}(R_{i}^{\prime})|\Big{)}^{2}}{\Big{\|}\sum\chi_{Y_{3}^{\prime}(R_{i}^{\prime})}\Big{\|}_{2}^{2}}\gtrapprox\frac{\Big{(}N\delta^{\eta}\frac{s_{1}\theta}{t}w\Big{)}^{2}}{N\frac{s_{1}\theta
w}{t}}=N\delta^{2\eta}\frac{s_{1}\theta}{t}w\gtrapprox\delta^{3\eta}\frac{s_{1}}{t}.$
(4.18)

On the other hand, each set $R_{i}^{\prime}$ is contained in $\hat{R}_{0}$,
and thus

$\Big{|}\hat{R}_{0}\cap
N_{\frac{s_{1}\theta}{t}}\Big{(}\bigcup_{\begin{subarray}{c}R\in\mathcal{R}\\\
R\subset\hat{R}_{0}\end{subarray}}Y(R)\Big{)}\Big{|}\gtrapprox\delta^{3\eta}|\hat{R}_{0}|.$
(4.19)

Recall that $\hat{R}_{0}$ is a prism of dimensions
$2\frac{s_{1}\theta}{t}\times 2\theta\times 2$, and
$s_{1}=\delta^{-3\eta/\varepsilon}s$. Define
$s^{\prime}=\frac{s_{1}\theta}{t}$ and
$t^{\prime}=\min(1,t\frac{s^{\prime}}{s})=\min(1,\delta^{-3\eta/\varepsilon}\theta)$.
Then $s^{\prime}\in[\delta^{-\varepsilon^{2}/2}s,\frac{s_{1}}{t}]$; since
$s\leq\delta^{\varepsilon}t$, we can select $\eta$ sufficiently small so that
$\frac{s_{1}}{t}\leq\delta^{\varepsilon/2}$, and hence
$s^{\prime}\in[\delta^{-\varepsilon^{2}/2}s,\delta^{\varepsilon/2}]$.

By (4.19) we conclude that for each $R_{0}\in\mathcal{R}_{1}$, if
$\hat{R}_{0}^{\prime}$ is the corresponding $s^{\prime}\times t^{\prime}\times
1$ prism and $\eta$ is selected sufficiently small compared to $\varepsilon$
and $\tau$, then the shading $\hat{Y}^{\prime}(\hat{R}_{0}^{\prime})$ given by
(4.5) satisfies

$|\hat{Y}^{\prime}(\hat{R}_{0}^{\prime})|\gtrapprox\delta^{3\eta}|\hat{R}_{0}|\geq\delta^{3\eta+3\eta/\varepsilon}|\hat{R}_{0}^{\prime}|\geq\delta^{\tau}|\hat{R}_{0}^{\prime}|.$
(4.20)

To finish the proof, define

$\mathcal{R}^{\prime}=\\{\hat{R}_{0}^{\prime}\colon\hat{R}_{0}\in\mathcal{R}_{1}\\}.$

(4.20) ensures that $|Y^{\prime}(R^{\prime})|\geq\delta^{\tau}|R^{\prime}|$
for each $R^{\prime}\in\mathcal{R}^{\prime}.$ Since $\mathcal{R}$ satisfies
the Convex Wolff axioms with error $\delta^{-\eta}$ and
$\\#\mathcal{R}_{1}\gtrapprox\delta^{2\eta}\\#\mathcal{R}_{1}$, we have that
$\mathcal{R}_{1}$ satisfies the Convex Wolff axioms with error
$\lessapprox\delta^{-3\eta}$, and hence by Remark 2.7 we have that
$\mathcal{R}^{\prime}$ satisfies the Convex Wolff Axioms with error
$\lessapprox\delta^{-3\eta}$; if $\delta_{0}$ and $\eta$ is selected
sufficiently small then this is $\leq\delta^{-\tau}$. ∎

The culmination of Section 4.1 is the following result, which is obtained by
iterating Lemma 4.4.

######  Proposition 4.6.

For all $\varepsilon>0$, there exists $\eta,\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let $s,t\in[\delta,1]$ with
$s\leq\delta^{\varepsilon}t$. Let $\mathcal{R}$ be a multi-set of $s\times
t\times 1$ prisms contained in $B(0,1)$, and suppose that $\mathcal{R}$
satisfies the Convex Wolff Axioms with error $\delta^{-\eta}$. Let
$\\{Y(R),R\in\mathcal{R}\\}$ be a $\delta^{\eta}$-dense shading.

Then $\bigcup_{R\in\mathcal{R}}Y(R)$ has discretized Assouad dimension at
least $3-\varepsilon$ at scale $\delta$ and scale separation $\delta^{-\eta}$.

###### Proof.

Fix $\varepsilon>0$ and let let $\eta,\delta_{0}$ be small quantities to be
determined below. Let $\delta\in(0,\delta_{0}]$, $s,t\in[\delta,1]$ with
$s\leq\delta^{\varepsilon}t$; let $\mathcal{R}$ be a set of $s\times t\times
1$ prisms that satisfy the Convex Wolff Axioms with error $\delta^{-\eta}$;
and let $Y(R)$ be a $\delta^{\eta}$-dense shading.

Define $\mathcal{R}_{1}=\mathcal{R}$ and define the shading $Y_{1}=Y$. Let
$s_{1}=s$ and $t_{1}=t$. Let
$\eta=\tau_{1}<\tau_{2}<\tau_{3}<\ldots<\tau_{N},$ $N=\lfloor
2/\varepsilon^{2}\rfloor$ be small quantities to be specified below. We will
perform the following iterative process for each $i=1,2,\ldots,N$. At the
beginning of step $i$, the following things will be true

  * •

We have a set $\mathcal{R}_{i}$ of $s_{i}\times t_{i}\times 1$ prisms, with
$s_{i}\leq\delta^{\varepsilon}t_{i}$.

  * •

$\mathcal{R}_{i}$ satisfies the Convex Wolff axioms with error
$\delta^{-\tau_{i}}$.

  * •

We have a shading $\\{Y_{i}(R)\colon R\in\mathcal{R}_{i}\\}$ that is
$\delta^{\tau_{i}}$ dense.

  * •

For each $R_{i}\in\mathcal{R}_{i}$, we have

$Y_{i}(R_{i})\subset N_{s_{i}}\Big{(}\bigcup_{R\in\mathcal{R}}Y(R)\Big{)}.$
(4.21)

Apply Lemma 4.4 to $(\mathcal{R}_{i},Y_{i})$ with $\varepsilon$ as above and
$\tau_{i+1}$ in place of $\tau$. If $\tau_{i}>0$ is sufficiently small
depending on $\varepsilon$ and $\tau_{i+1}$, then the Lemma can be applied
(indeed, $\tau_{i}$ must be less than or equal to the quantity “$\eta$” from
the conclusion of Lemma 4.4). The quantity $\delta_{0}$ (and hence $\delta$)
will be chosen sufficiently small so that Lemma 4.4 can be applied at each
step.

Case (A). Suppose that Conclusion (A) from Lemma 4.4 holds, i.e. there exists
$r\in[\delta^{-\tau_{i}}s_{i},1]$ and an $r$-ball $B$ so that

$\Big{|}B\cap\bigcup_{R\in\mathcal{R}_{i}}Y_{i}(R_{i})\Big{|}\geq(s_{i}/r)^{\varepsilon}|B|.$

Using (4.21), we conclude that

$\Big{|}B\cap
N_{s_{i}}\Big{(}\bigcup_{R\in\mathcal{R}}Y(R)\Big{)}\Big{|}\geq(s_{i}/r)^{\varepsilon}|B|,$

i.e. $\bigcup_{R\in\mathcal{R}}Y(R)$ has discretized Assouad dimension $\geq
3-\varepsilon$ at scale $\delta$ and scale separation
$\delta^{-\tau_{i}}\geq\delta^{-\eta}$. If this occurs, then we halt the
iteration and conclude the proof.

Case (B). Next, suppose that Conclusion (B) holds. Let $s_{i+1}=s^{\prime}$;
$t_{i+1}=t_{i}^{\prime}$; let $\mathcal{R}_{i+1}$ be the set of $s_{i+1}\times
t_{i+1}\times 1$ prisms, with shading $Y_{i+1}=Y_{i}^{\prime}$, as described
by Lemma 4.4. We have that $\mathcal{R}_{i+1}$ satisfies the Convex Wolff
Axioms with error $\delta^{-\tau_{i+1}}$, and $Y_{i+1}$ is
$\delta^{\tau_{i+1}}$ dense. Furthermore, the shading $Y_{i+1}$ satisfies
(4.21) with $i+1$ in place of $i$.

Case (B.i). Suppose that Conclusion (B) holds and $t_{i+1}=1$. Recall that
$s_{i+1}\leq\delta^{\varepsilon/2}.$ Thus we can apply Lemma 4.3 with
$s_{i+1}$ in place of $\delta$. We have

$\Big{|}N_{s_{i+1}}\Big{(}\bigcup_{R\in\mathcal{R}}Y(R)\Big{)}\Big{|}\geq\Big{|}\bigcup_{R\in\mathcal{R}_{i+1}}Y_{i+1}(R)\Big{|}\gtrsim\delta^{3\tau_{i+1}}|\log
s_{i+1}|^{-1}\gtrsim s_{i+1}^{7\tau_{i+1}/\varepsilon}.$ (4.22)

We can select $\tau_{i+1}$ and $\delta_{0}$ sufficiently small so that the LHS
of (4.22) has size at least $s_{i+1}^{\varepsilon}|B(0,1)|$. Since the set on
the LHS of (4.22) is contained in $B(0,1)$, we conclude that
$\bigcup_{R\in\mathcal{R}}Y(R)$ has discretized Assouad dimension $\geq
3-\varepsilon$ at scale $\delta$ and scale separation
$\delta^{-\varepsilon/2}\geq\delta^{-\eta}.$ If this occurs, then we halt the
iteration and conclude the proof.

Case (B.ii). Suppose $t_{i+1}<1$; then $s_{i+1}/t_{i+1}=s_{i}/t_{i}$, so in
particular $s_{i+1}\leq\delta^{\varepsilon}t_{i+1}$. Thus we have verified all
of the properties required to continue the iteration, with $i+1$ in place of
$i$. Note that $s_{i+1}\geq\delta^{-\varepsilon^{2}/2}s_{i}$;
$t_{i+1}\geq\min(1,\delta^{\varepsilon}s_{i+1})$, and the iterative process
halts if $t_{i+1}=1$. We conclude that this iterative process will halt after
at most $N=\lfloor 2/\varepsilon^{2}\rfloor$ steps. ∎

###  4.2 Arrangements of tubes that cluster into rectangular prisms

In Section 4.1, we showed that unions of rectangular prisms that are far from
being tubes (i.e. prisms of dimensions $s\times t\times 1$, with $s$ much
smaller than $t$) have large discretized Assouad dimension. In this section,
we will use these results to show that if an arrangement of $\delta$-tubes
clusters into prisms that are far from being (fatter) tubes, then the union of
these tubes must have large discretized Assouad dimension. This is made
precise in Lemmas 4.8 and 4.10 below. We begin with a technical lemma.

######  Lemma 4.7.

For all $\varepsilon>0$, there exists $\eta,\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let $\rho\in[20\delta,1]$
and let $K\geq 1$. Let $(\mathbb{T},Y)_{\delta}$ be a set of essentially
distinct $\delta$-tubes, with $\\#\mathbb{T}\geq K\rho/\delta$, and $Y$ a
uniformly $\delta^{\eta}$-dense shading (see Definition 2.1). Let $R_{0}$ be a
$10\delta\times\rho\times 1$ rectangle, and suppose each $T\in\mathbb{T}$ is
contained in $R_{0}$.

Then there exists a number $K\delta\lesssim\omega\leq\rho$ and a set
$\mathcal{R}$ of essentially distinct $10\delta\times\omega\times 1$ prisms,
so that

$\\#\Big{(}\bigcup_{R\in\mathcal{R}}\\{T\in\mathbb{T}\colon T\subset
R\\}\Big{)}\gtrapprox\\#\mathbb{T},$ (4.23)

and for each prism $R\in\mathcal{R}$ we have

$\Big{|}\bigcup_{\begin{subarray}{c}T\in\mathbb{T}\\\ T\subset
R\end{subarray}}Y(T)\Big{|}\geq\delta^{\varepsilon}|R|.$ (4.24)

###### Proof.

For each $x\in\mathbb{R}^{3}$, define the “line spread”

$\operatorname{L-spread}(x)=\max\big{\\{}\angle(T,T^{\prime})\big{\\}},$

where the maximum is taken over all pairs of tubes $T,T^{\prime}$ intersecting
$x$. We define $\operatorname{L-spread}(x)=0$ if at most one tube intersects
$x$. Let $C$ be a large number to be determined below. After dyadic
pigeonholing, we can select $\theta\in[\frac{K\delta}{C},\rho]$ so that the
shading

$Z(T)=\\{x\in T\colon\operatorname{L-spread}(x)\in[\theta,2\theta)\\}$

satisfies

$\sum_{T\in\mathbb{T}}|Z(T)|\geq(C|\log\delta|)^{-1}\delta^{2}(\\#\mathbb{T}).$
(4.25)

(Note that the shading $Z(T)$ is not necessarily a subset of $Y(T)$). Our
argument that such a shading $Z(T)$ must exist proceeds as follows: if we
could not select a $\theta\in[\frac{K\delta}{C},\rho]$ so that (4.25) holds,
then the shading

$Z^{\prime}(T)=\big{\\{}x\in
T\colon\operatorname{L-spread}(x)\leq\frac{K\delta}{C}\big{\\}}$

satisfies
$\sum_{T\in\mathbb{T}}|Z^{\prime}(T)|\geq\frac{1}{100}\delta^{2}(\\#\mathbb{T})$.
But for each $x\in\mathbb{R}^{3}$, the set of tubes $T\in\mathbb{T}$ with
$x\in Z(T)$ are contained in the $10\delta$-neighborhood of a plane (since
they are contained in $R_{0}$), and also make angle $\leq\frac{K\delta}{C}$
with a common unit vector; since the tubes in $\mathbb{T}$ are essentially
distinct, we conclude that $O(K/C)$ tubes $T\in\mathbb{T}$ can satisfy $x\in
Z^{\prime}(T)$, and hence

$\Big{|}\bigcup_{T\in\mathbb{T}}T\Big{|}\geq\Big{|}\bigcup_{T\in\mathbb{T}}Z^{\prime}(T)\Big{|}\gtrsim(C/K)\sum_{T\in\mathbb{T}}|Z^{\prime}(T)|\gtrsim
C\rho\delta.$ (4.26)

If $C$ is selected sufficiently large then this is impossible, since the set
on the LHS of (4.26) is contained in $R_{0}$, which has volume $10\rho\delta$.

At this point we have established the existence of an angle
$\theta\in[\frac{K\delta}{C},\rho]$ so that the associated shading $Z(T)$
satisfies (4.25). In the arguments that follow, the quantity $C$ has been
fixed, and all implicit constants are allowed to depend on $C$. Let

$\mathbb{T}_{1}=\\{T\in\mathbb{T}\colon|Z(T)|\gtrsim|\log\delta|^{-1}|T|\\}.$
(4.27)

Selecting the implicit constant in (4.27) appropriately, we can ensure that
$\\#\mathbb{T}_{1}\gtrapprox\\#\mathbb{T}$. For each $T_{1}\in\mathbb{T}_{1}$,
let $x_{2},\ldots,x_{N},\ N\gtrapprox\theta/\delta$ be a set of
$\delta/\theta$-separated points from $Y_{1}(T_{1})$. For each $x_{i}$, select
$T_{i}\in\mathbb{T}$ with $x_{i}\in T_{i}$ and
$\angle(T_{1},T_{i})\in[\theta,2\theta)$. A $L^{2}$ argument (see e.g. (4.11))
plus the fact that $|Y(T_{i})|\geq\delta^{\eta}$ shows that

$\Big{|}\bigcup_{i=2}^{N}Y(T_{i})\Big{|}\gtrapprox\delta^{2\eta}\sum_{i=2}^{N}|T_{i}|\gtrsim\delta^{2\eta}\delta\theta.$
(4.28)

On the other hand, the set on the LHS of (4.28) is contained in $R_{0}\cap
N_{2\theta}(T_{1})$, which is essentially a rectangular prism of dimensions
$10\delta\times 4\theta\times 1$. Denote this prism by $R(T_{1})$. We have
just shown that

$\Big{|}\bigcup_{\begin{subarray}{c}T\in\mathbb{T}\\\ T\subset
R(T_{1})\end{subarray}}Y(T)\Big{|}\gtrapprox\delta^{2\eta}|R(T_{1})|.$ (4.29)

To conclude the proof, let $\omega=4\theta$ and let
$\mathcal{R}_{1}=\\{R(T_{1})\colon T_{1}\in\mathbb{T}_{1}\\}$. We claim that
(4.23) holds for $\mathcal{R}_{1}$. Indeed, this follows from the fact that
$T_{1}\subset R(T_{1})$ for each $T_{1}\in\mathbb{T}_{1}$, and
$\\#\mathbb{T}_{1}\gtrapprox\\#\mathbb{T}$. By (4.29) we see that (4.24) holds
for each $R\in\mathcal{R}_{1}$, provided we select $\eta$ and $\delta_{0}$
sufficiently small. To conclude the proof, greedily select a set
$\mathcal{R}\subset\mathcal{R}_{1}$ of essentially distinct rectangular
prisms; we can perform this selection so that (4.23) continues to hold. ∎

######  Lemma 4.8.

For all $\varepsilon>0$, there exists $\eta,\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let $\mathbb{T}$ be a set
of essentially distinct $\delta$-tubes that satisfies the Convex Wolff Axioms
with error $\delta^{-\eta}$. Let $Y$ be a shading with
$|Y(T)|\geq\delta^{\eta}|T|$ for each tube $T$. Then at least one of the
following must occur.

  * (A)

There is a set $\mathbb{T}^{\prime}\subset\mathbb{T}$ with
$\\#\mathbb{T}^{\prime}\geq\frac{1}{2}(\\#\mathbb{T})$ so that for every
$\delta\leq\rho\leq 1$ and every $\delta\times\rho\times 1$ rectangle $R$, at
most $\delta^{-\varepsilon}(\rho/\delta)$ tubes from $\mathbb{T}^{\prime}$ are
contained in $2R$.

  * (B)

$\bigcup Y(T)$ has discretized Assouad dimension at least $3-\varepsilon$ at
scale $\delta$ and scale separation $\delta^{-\eta}$.

######  Remark 4.9.

Since $\mathbb{T}$ satisfies the Convex Wolff Axioms, we have that at most
$\delta^{-\eta}(\rho\delta)(\\#\mathbb{T})$ tubes are contained in each
$2\delta\times 2\rho\times 2$ rectangle. If $\\#\mathbb{T}$ is substantially
larger than $\delta^{-2}$, then the estimate from Conclusion (A) is stronger.

###### Proof.

We say a $2\delta\times 2\rho\times 2$ rectangle is “heavy” if it contains
more than $\delta^{-\varepsilon}(\rho/\delta)$ tubes from $\mathbb{T}$. If
fewer than $\frac{1}{2}(\\#\mathbb{T})$ of the tubes are contained in heavy
rectangles, let $\mathbb{T}^{\prime}$ be the set of tubes that are not
contained in heavy rectangles. Then Conclusion (A) holds, and we are done.

Suppose instead that at least $\frac{1}{2}(\\#\mathbb{T})$ tubes are contained
in heavy rectangles. Since the tubes in $\mathbb{T}$ are essentially distinct,
each $2\delta\times 2\rho\times 2$ rectangle can contain at most
$O\big{(}(\rho/\delta)^{2}\big{)}$ tubes from $\mathbb{T}$. Thus if a
$2\delta\times 2\rho\times 2$ rectangle is heavy, then we must have
$\rho\gtrsim\delta^{1-\varepsilon}$. Let $\mathcal{R}_{0}$ denote this set of
heavy rectangles (note that the rectangles in $\mathcal{R}_{0}$ can have
different dimensions). Let $\varepsilon_{1}$ be a small quantity to be chosen
below. Apply Lemma 4.7 to each rectangle $R_{0}\in\mathcal{R}_{0}$, with
$\varepsilon_{1}$ in place of $\varepsilon$, and $K=\delta^{-\varepsilon}$. We
obtain a number $\omega(R_{0})\gtrsim\delta^{1-\varepsilon}$ and a set
$\mathcal{R}(R_{0})$ of $\delta\times\omega(R_{0})\times O(1)$ rectangles. We
have

$\\#\Big{(}\bigcup_{R_{0}\in\mathcal{R}_{0}}\bigcup_{R\in\mathcal{R}(R_{0})}\\{T\in\mathbb{T}\colon
T\subset R\\}\Big{)}\gtrapprox\\#\mathbb{T}.$ (4.30)

After dyadic pigeonholing, we can select
$\omega_{0}\in[\delta^{1-\varepsilon},1]$ so that if we define
$\mathcal{R}_{1}=\\{R_{0}\in\mathcal{R}_{0}\colon\omega(R_{0})\in[\omega_{0},2\omega_{0})\\}$,
then (4.30) continues to hold with $\mathcal{R}_{1}$ in place of
$\mathcal{R}_{0}$.

Define the multi-set

$\mathcal{R}_{2}=\bigsqcup_{R_{1}\in\mathcal{R}_{1}}\mathcal{R}(R_{1}).$

(In the above, $\bigsqcup$ is used to suggest that if the same rectangle is
present in multiple sets $\mathcal{R}(R_{1})$, then this rectangle should
occur multiple times in $\mathcal{R}_{2}$; i.e. the union might create a
multi-set). Let

$\mathbb{T}_{2}=\bigcup_{R\in\mathcal{R}_{2}}\\{T\in\mathbb{T}\colon T\subset
R\\}.$

Since $\mathbb{T}$ satisfies the Convex Wolff Axioms with error
$\delta^{-\eta}$, by (4.30) we have that the set (i.e. not multi-set)
$\mathbb{T}_{2}$ satisfies the Convex Wolff Axioms with error
$\lessapprox\delta^{-\eta}$. Finally, for each $T\in\mathbb{T}_{2}$, let
$R(T)$ be a rectangle in $\mathcal{R}_{2}$ with $T\subset R$ (if more than one
$R\in\mathcal{R}_{2}$ contains $T$, simply select one of them). Define the
multi-set $\mathcal{R}=\\{R(T)\colon T\in\mathbb{T}_{2}\\}$, i.e.
$\mathcal{R}$ is a multi-set with the same cardinality as $\mathbb{T}_{2}$. By
Remark 2.7, we conclude that $\mathcal{R}$ satisfies the Convex Wolff Axioms
with error $\lessapprox\delta^{-\eta}$.

For each $R\in\mathcal{R}$, define the shading

$Y(R)=R\cap\bigcup_{T\in\mathbb{T}}Y(T).$

By Lemma 4.7, we have $|Y(R)|\geq\delta^{\varepsilon_{1}}|R|$ for each
$R\in\mathcal{R}$. To conclude the proof, apply Proposition 4.6 with
$\varepsilon$ as above, $s=\delta$, and $t=\omega_{0}$; to ensure that
Proposition 4.6 can be applied, we must select $\varepsilon_{1}$ sufficiently
small depending on $\varepsilon$ (our choice of $\varepsilon_{1}$ depends on
the quantity “$\eta$” from Proposition 4.6), and then $\eta$ sufficiently
small depending on $\varepsilon_{1}$. Since $\bigcup Y(T)\subset\bigcup Y(R),$
we conclude that Conclusion (B) holds. ∎

Lemma 4.8 says that either a union of $\delta$-tubes has large discretized
Assouad dimension, or these tubes cannot concentrate into
$\delta\times\rho\times 1$ rectangular prisms. The next lemma is similar,
except the latter conclusion is strengthened: either a union of $\delta$-tubes
has large discretized Assouad dimension, or these tubes cannot concentrate
into $s\times t\times 1$ rectangular prisms, for any $\delta\leq s\leq t\leq
1$. To prove this result, we apply Lemma 4.8 to the $s$-thickening of the
tubes in our arrangement, for many different values of $s$. The precise
statement is as follows.

######  Lemma 4.10.

For all $\varepsilon=1/N>0$, there exists $\eta,\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let
$(\mathbb{T},Y)_{\delta}$ be a set of essentially distinct $\delta$-tubes that
satisfies the Convex Wolff Axioms with error $\delta^{-\eta}$, and let $Y$ be
a uniformly $\delta^{\eta}$-dense shading. Then at least one of the following
must occur.

  * (A)

There is a set $\mathbb{T}^{\prime}\subset\mathbb{T}$ with
$\\#\mathbb{T}^{\prime}\geq\delta^{\varepsilon}(\\#\mathbb{T})$ so that the
following holds. For every pair of numbers $\delta\leq s\leq t\leq 1$ with $s$
of the form $\delta^{i\varepsilon}$, there is a 1-uniform partitioning cover
$\mathbb{T}_{s}^{\prime}$ of $\mathbb{T}^{\prime}$ (see Definition 2.10),
where $\mathbb{T}_{s}^{\prime}$ is a set of essentially distinct $s$-tubes.
Furthermore, for every $s\times t\times 1$ rectangular prism $R$, at most
$\delta^{-\varepsilon}\frac{t}{s}\frac{(\\#\mathbb{T}^{\prime})}{(\\#\mathbb{T}^{\prime}_{s})}$
tubes from $\mathbb{T}^{\prime}$ are contained in $R$.

  * (B)

$\bigcup Y(T)$ has discretized Assouad dimension at least $3-\varepsilon$ at
scale $\delta$ and scale separation $\delta^{-\eta}$.

###### Proof.

Step 1.
Our first step is to regularize the set $\mathbb{T}$ at scales of the form
$s_{i}=\delta^{i/N}$, with $i=1,\ldots,N$ (we do not need to consider
$s_{0}=1$, since in that case there is nothing to prove). After repeated
pigeonholing, we can select a set $\mathbb{T}^{\dagger}\subset\mathbb{T}$ with
$\\#\mathbb{T}^{\dagger}\gtrsim(C|\log\delta|)^{N}(\\#\mathbb{T})$ and sets of
$s_{i}$-tubes $\mathbb{T}_{s_{i}}^{\dagger},\ i=0,\ldots,N$, so that
$\mathbb{T}_{s_{N}}^{\dagger}=\mathbb{T}^{\dagger}$, and each set
$\mathbb{T}_{s_{i}}^{\dagger}$ is a partitioning cover of
$\mathbb{T}_{s_{i+1}}^{\dagger}$ (and hence also a partitioning cover of
$\mathbb{T}_{s_{j}}^{\dagger}$ for all $j>i$). See e.g. Lemma 3.4 from [11]
for an explicit description of this iterated pigeonholing process. We will
choose $\delta_{0}$ sufficiently small so that
$\\#\mathbb{T}^{\dagger}\geq\delta^{\eta/2}(\\#\mathbb{T})$. In particular, by
Remark 2.11 this means that $\mathbb{T}^{\dagger}$ and each set
$\mathbb{T}^{\dagger}_{s_{i}},\ i=1,\ldots,N$ satisfies the Convex Wolff
Axioms with error $\delta^{-2\eta}$.

Step 2.
We would like to apply Lemma 4.8 to each set
$\mathbb{T}^{\dagger}_{s_{N}},\ldots,\mathbb{T}^{\dagger}_{s_{1}}$ in turn. If
Conclusion (B) of Lemma 4.8 holds at any step, then we conclude that
Conclusion (B) of Lemma 4.10 holds and we halt. If not, then few tubes from
$\mathbb{T}^{\dagger}_{s_{i}}$ can cluster into a $s_{i}\times t\times 1$
prism; if each tube from $\mathbb{T}^{\dagger}_{s_{i}}$ contains the same
number of tubes from $\mathbb{T}$, then Conclusion (A) of Lemma 4.10 holds,
and we are done. The difficulty is that conclusion (A) of Lemma 4.10 only
holds for a subset of the tubes in $\mathbb{T}^{\dagger}_{s_{i}}$; this
complication introduces additional pigeonholing at each step, which in turn
forces us to re-label all of the relevant objects. We now turn to the
unpleasant details.

Define $\mathbb{T}^{(1)}_{s_{N}}=\mathbb{T}^{{\dagger}}_{s_{N}}$ and apply
Lemma 4.8 with $\varepsilon/2$ in place of $\varepsilon$ to
$\mathbb{T}^{(1)}_{s_{N}}$, with the shading $Y$ inherited from $\mathbb{T}$
(recall that this shading is uniformly $\delta^{\eta}$ dense). If Conclusion
(B) of Lemma 4.8 holds, then Conclusion (B) of Lemma 4.10 holds, and we are
done.

If Conclusion (A) of Lemma 4.8 holds, then we obtain a set
$(\mathbb{T}^{(1)}_{s_{N}})^{\prime}\subset\mathbb{T}^{(1)}_{s_{N}}$ with
$\\#(\mathbb{T}^{(1)}_{s_{N}})^{\prime}\geq\frac{1}{2}(\\#\mathbb{T}^{(1)}_{s_{N}})$,
so that for each $t\in[\delta,1]$ and each $2s_{N}\times 2t\times 2$ rectangle
$R$ (recall $s_{N}=\delta$), at most $\delta^{-\varepsilon/2}(t/\delta)$ tubes
from $(\mathbb{T}^{(1)}_{s_{N}})^{\prime}$ are contained in $R$. After dyadic
pigeonholing and refining $(\mathbb{T}^{(1)}_{s_{N}})^{\prime}$, we can select
sets $\mathbb{T}^{(2)}_{s_{N}}\subset(\mathbb{T}^{(1)}_{s_{N}})^{\prime}$ and
$\mathbb{T}^{(2)}_{s_{N-1}}\subset\mathbb{T}^{{\dagger}}_{s_{N-1}}$ so that
$\\#\mathbb{T}^{(2)}_{s_{N}}\gtrsim|\log\delta|^{-1}(\mathbb{T}^{(1)}_{s_{N}})^{\prime}$,
and $\\#(\mathbb{T}^{(2)}_{s_{N}}[T_{s_{N-1}}]$ is the same for each
$T_{s_{N-1}}\in\mathbb{T}^{(2)}_{s_{N-1}}$, i.e. $\mathbb{T}^{(2)}_{s_{N-1}}$
is a 1-balanced partitioning cover of $\mathbb{T}^{(2)}_{s_{N}}$.

Next, we will describe a slightly more elaborate version of the above process.
Let $2\leq k\leq N$, and suppose we have constructed sets
$\mathbb{T}_{s_{N}}^{(k)},\ldots,\mathbb{T}_{s_{N-k+1}}^{(k)}$, that have the
following properties:

  * (a)

For $N-k+1\leq i<N$, $\mathbb{T}_{s_{i}}^{(k)}$ is a 1-balanced partitioning
cover of $\mathbb{T}_{s_{i+1}}^{(k)}$ (and hence also a 1-balanced
partitioning cover of $\mathbb{T}_{s_{j}}^{(k)}$ for $j>i$).

  * (b)

$\\#\mathbb{T}_{s_{N}}^{(k)}\gtrsim_{k}|\log\delta|^{-k}(\\#\mathbb{T}_{s_{N}}^{(1)})$.

  * (c)

For $N-k+2\leq i\leq N$ and $s_{i}\leq t\leq 1$, each $s_{i}\times t\times 1$
rectangle $R$ contains
$\lesssim_{k}|\log\delta|^{k}\delta^{-\varepsilon/2}\frac{\\#\mathbb{T}_{s_{N}}^{(k)}}{\\#\mathbb{T}_{s_{i}}^{(k)}}$
tubes from $\mathbb{T}_{N}^{(k)}$.

We construct sets
$\mathbb{T}_{s_{N}}^{(k+1)},\ldots,\mathbb{T}_{s_{N-k}}^{(k+1)}$ as follows.
For each $T_{s_{N-k+1}}\in\mathbb{T}^{(k)}_{s_{N-k+1}}$, define the shading

$Y(T_{N-k+1})=T_{s_{N-k+1}}\cap
N_{s_{N-k+1}}\Big{(}\bigcup_{T\in\mathbb{T}}Y(T)\Big{)}.$

Since $\mathbb{T}[T_{s_{N-k+1}}]$ is non-empty and each tube $T\in\mathbb{T}$
has a $\delta^{\eta}$-dense shading, we conclude that
$|Y(T_{s_{N-k+1}})|\gtrsim\delta^{\eta}|T_{s_{N-k+1}}|$, i.e. the shading is
uniformly $\gtrsim\delta^{\eta}$-dense. Since $s_{N-k+1}\geq\delta^{1/N}$,
this means that the shading is uniformly $s_{N-k+1}^{\eta N}$ dense. We will
select $\eta$ and $\delta_{0}$ sufficiently small so that Lemma 4.8 can be
applied to the pair $(\mathbb{T}^{(k)}_{s_{N-k+1}},Y)_{s_{N-k+1}}$ with
$s_{N-k+1}$ in place of $\delta$, and $\varepsilon/2$ in place of
$\varepsilon$. Let $\eta_{1}=\eta_{1}(\varepsilon)>0$ be the output of that
lemma.

If Conclusion (B) of Lemma 4.10 holds, then
$\bigcup_{T_{s_{N-k+1}}\in\mathbb{T}^{(k)}_{s_{N-k+1}}}Y(T_{s_{N-k+1}})$ has
discretized Assouad dimension at least $3-\varepsilon/2$ at scale $s_{N-k+1}$
and scale separation $s_{N-k+1}^{-\eta_{1}}$. But this means that
$\bigcup_{T\in\mathbb{T}}Y(T)$ has has discretized Assouad dimension at least
$3-\varepsilon/2$ at scale $\delta$ and scale separation
$\delta^{-\eta_{1}/N}$; if we select $\eta\leq\eta_{1}/N$, then Conclusion (B)
of Lemma 4.10 holds, and we are done.

Suppose instead that Conclusion (A) of Lemma 4.10 holds. Then there exists a
set
$(\mathbb{T}^{(k)}_{s_{N-k+1}})^{\prime}\subset\mathbb{T}^{(k)}_{s_{N-k+1}}$
with
$\\#(\mathbb{T}^{(k)}_{s_{N-k+1}})^{\prime}\geq\frac{1}{2}(\\#\mathbb{T}^{(k)}_{s_{N-k+1}})$,
so that for each $s_{N-k+1}\leq t\leq 1$ and each $2s_{N-k+1}\times 2t\times
2$ rectangle $R$, at most
$s_{N-k+1}^{-\varepsilon/2}\frac{t}{s_{N-k+1}}\leq\delta^{-\varepsilon/2}\frac{t}{s_{N-k+1}}$
tubes from $(\mathbb{T}^{(k)}_{s_{N-k+1}})^{\prime}$ are contained in $R$.

After dyadic pigeonholing and refining
$(\mathbb{T}^{(k)}_{s_{N-k+1}})^{\prime}$, we can select sets
$\mathbb{T}^{(k+1)}_{s_{N-k+1}}\subset(\mathbb{T}^{(k)}_{s_{N-k+1}})^{\prime}$
and $\mathbb{T}^{(k+1)}_{s_{N-k}}\subset\mathbb{T}^{{\dagger}}_{s_{N-k}}$ so
that
$\\#\mathbb{T}^{(k+1)}_{s_{N-k+1}}\gtrsim|\log\delta|^{-1}(\mathbb{T}^{(k)}_{s_{N-k+1}})^{\prime}$,
and $\mathbb{T}^{(k+1)}_{s_{N-k}}$ is a 1-balanced partitioning cover of
$\mathbb{T}^{(k+1)}_{s_{N-k+1}}$. For each $N-k+1<i\leq N$, define
$\mathbb{T}^{(k+1)}_{s_{i}}$ to be those tubes from $\mathbb{T}^{(k)}_{s_{i}}$
that are contained in some tube from $\mathbb{T}^{(k+1)}_{s_{N-k+1}}$; it is
still the case that $\mathbb{T}_{s_{i}}$ is a 1-balanced partitioning cover of
$\mathbb{T}_{s_{i+1}}$, for $N-k+1\leq i<N$, and by construction it is also
the case that $\mathbb{T}^{(k+1)}_{s_{N-k}}$ is a 1-balanced partitioning
cover of $\mathbb{T}^{(k+1)}_{s_{N-k+1}}$.

Furthermore,
$\\#\mathbb{T}^{(k+1)}_{s_{N}}\gtrsim|\log\delta|^{-1}(\\#\mathbb{T}^{(k)}_{s_{N}})$
(this is because the cardinality of $\mathbb{T}^{(k)}_{s_{N-k+1}}$ was reduced
by a factor of $\lesssim|\log\delta|^{-1}$, and $\mathbb{T}^{(k)}_{s_{N-k+1}}$
was a 1-balanced partitioning cover of $\mathbb{T}^{(k)}_{s_{N}}$), and thus
Item (c) remains true for $k+1$ in place of $j$, and Item (c) remains true for
$N-k+2\leq i\leq N$, with $k+1$ in place of $k$ (indeed, the cardinality of
$\mathbb{T}_{s_{N}}^{(k+1)}$ has decreased by at most a multiplicative factor
of $O(|\log\delta|)$ compared to the cardinality of
$\mathbb{T}_{s_{N}}^{(k)}$, and the cardinality of
$\mathbb{T}_{s_{i}i}^{(k+1)}$ is no larger than that of
$\mathbb{T}_{s_{i}i}^{(k)}$)

All that remains is to verify Item (c) above for $i=N-k+1$. Let $s_{N-k+1}\leq
t\leq 1$, let $R$ be a $s_{N-k+1}\times t\times 1$ rectangle, and let
$R^{\prime}=2R$. We already know that at most
$\delta^{-\varepsilon/2}\frac{t}{s_{N-k+1}}$ tubes from
$\mathbb{T}^{(k+1)}_{s_{N-k+1}}$ can be contained in the $R^{\prime}$. On the
other hand, if $T$ is a $\delta$-tube that is contained in $R$ and also
contained in a $s_{n-k+1}$ tube $T_{s_{N-k+1}}$, then $T_{s_{N-k+1}}\subset
2R$. Since $\mathbb{T}^{(k+1)}_{s_{N-k+1}}$ is a 1-balanced cover of
$\mathbb{T}^{(k+1)}_{s_{N}}$, each tube in $\mathbb{T}^{(k+1)}_{s_{N-k+1}}$
contains
$\frac{\\#\mathbb{T}^{(k+1)}_{s_{N}}}{\\#\mathbb{T}^{(k+1)}_{s_{N-k+1}}}$
tubes from $\mathbb{T}^{(k+1)}_{s_{N}}$. We conclude that
$\lesssim\delta^{-\varepsilon/2}\frac{t}{s_{N-k+1}}\frac{\\#\mathbb{T}^{(k+1)}_{s_{N}}}{\\#\mathbb{T}^{(k+1)}_{s_{N-k+1}}}$
tubes from $\mathbb{T}^{(k+1)}_{s_{N}}$ are contained in $R$. The completes
the $k\to k+1$ step of the above iteration.

Step 3.
If the iterative process described above does not halt before the $N$-th,
step, then define $\mathbb{T}^{\prime}=\mathbb{T}^{(N)}_{s_{N}}$. By selecting
$\delta_{0}$ sufficiently small we ensure that
$\\#\mathbb{T}^{\prime}\geq\delta^{-\varepsilon}(\\#\mathbb{T})$. For each
$s=\delta^{i/N},\ i=1,\ldots,N$, we have the 1-balanced partitioning cover
$\mathbb{T}^{\prime}_{s}=\mathbb{T}^{(N)}_{s_{i}}$. If we select $\delta_{0}$
sufficiently small depending on $\varepsilon$, then Item (c) above ensures
that for every $t\in[s,1]$ and every $s\times t\times 1$ rectangular prism
$R$, at most
$\delta^{-\varepsilon}\frac{t}{s}\frac{(\\#\mathbb{T}^{\prime})}{(\\#\mathbb{T}^{\prime}_{s})}$
tubes from $\mathbb{T}^{\prime}$ are contained in $R$. ∎

###  4.3 Proof of Proposition 4.1

We are now ready to prove Proposition 4.1. For the reader’s convenience, we
reproduce it here

######  Proposition 4.1.

For all $\varepsilon>0$, there exists $\tau>0$, so that for all $\eta_{1}>0$,
there exists $\eta,\delta_{0}>0$ so that the following holds for all
$\delta\in(0,\delta_{0}]$. Let $\beta\geq 2$ and let $(\mathbb{T},Y)_{\delta}$
be a set of $\delta^{-\beta}$ essentially distinct $\delta$-tubes that satisfy
the Convex Wolff Axioms with error $\delta^{\eta}$, with $Y$ a $\delta^{\eta}$
uniformly dense shading. Then at least one of the following must be true.

  * (A)

$\bigcup Y(T)$ has discretized Assouad dimension at least $3-\varepsilon$ at
scale $\delta$, with scale separation $\delta^{-\eta}$.

  * (B)

There exists $\mathbb{T}^{\prime}\subset\mathbb{T}$ with
$\\#\mathbb{T}^{\prime}\geq\delta^{\eta_{1}}(\\#\mathbb{T})$, and
$\mathbb{T}^{\prime}$ satisfies the self-similar Convex Wolff Axioms with
error $\delta^{-\varepsilon}$.

  * (C)

There exists $\rho\in[\delta,\delta^{\varepsilon}]$ and a set
$\tilde{\mathbb{T}}$ of $\rho$-tubes, with the following properties.

    * (C.i)

The tubes in $\tilde{\mathbb{T}}$ are essentially distinct, and
$\\#\tilde{\mathbb{T}}\geq\rho^{-\beta-\tau}$.

    * (C.ii)

$\tilde{\mathbb{T}}$ satisfies the Convex Wolff Axioms with error
$\leq\rho^{-\eta_{1}}$.

    * (C.iii)

For each $T_{\rho}\in\tilde{\mathbb{T}}$, there exists $T\in\mathbb{T}$ with
$T\subset T_{\rho}$.

  * (D)

There exists $\rho\in[\delta^{1-\varepsilon},1]$, a $\rho$-tube $T_{\rho}$,
and a set $\tilde{\mathbb{T}}\subset\mathbb{T}[T_{\rho}]$, so that the
following holds.

    * (D.i)

$\\#\tilde{\mathbb{T}}\geq(\delta/\rho)^{-\beta-\tau}.$

    * (D.ii)

The unit rescaling of $\tilde{\mathbb{T}}$ relative to $T_{\rho}$ satisfies
the Convex Wolff Axioms with error $(\delta/\rho)^{-\eta_{1}}$.

###### Proof.

Step 1.
Let $\eta_{2}=1/N>0$ be a small quantity to be chosen below. Apply Lemma 4.10
with $\eta_{2}$ in place of $\varepsilon$. We conclude that either Conclusion
(A) of Proposition 4.1 holds, there exists a set
$\mathbb{T}^{\prime}\subset\mathbb{T}$ that satisfies Conclusion (B) of Lemma
4.10.

Step 2.

Suppose there exists $\rho\in[\delta,\delta^{\varepsilon}]$ of the form
$\rho=\delta^{i/N}$ so that
$\\#\mathbb{T}^{\prime}_{\rho}\geq\delta^{-\beta-\tau}$ (here
$\\#\mathbb{T}^{\prime}_{\rho}$ is a 1-uniform partitioning cover of
$\mathbb{T}^{\prime}$, coming from Conclusion (B) of Lemma 4.10). Then by
Remark 2.11, $\mathbb{T}^{\prime}_{\rho}$ satisfies the Convex Wolff Axioms
with error $\delta^{-2\eta}\leq\rho^{-2\eta/\varepsilon}$; this is at most
$\rho^{-\eta_{1}}$, provided we choose $\eta\leq\varepsilon\eta_{1}/2$. Thus
Conclusion (C) holds, with $\tilde{\mathbb{T}}=\mathbb{T}^{\prime}_{\rho}$.

Step 3. Our goal in this step is to show that either Conclusion (D) of
Proposition 4.1 holds, or for every $\rho\in[\delta,\delta^{\varepsilon}]$ of
the form $\rho=\delta^{i/N}$, we have

$\rho^{-\beta+\varepsilon/3}\leq\\#\mathbb{T}^{\prime}_{\rho}\leq\rho^{-\beta-\tau},\quad\rho\in[\delta,\delta^{\varepsilon}].$
(4.31)

Note that the RHS of the above inequality was already established in Step 2.
First, we consider the case where $\beta\leq 2+\varepsilon/4$; in this case
(4.31) will always hold. Indeed, let $\rho\in[\delta,\delta^{\varepsilon}]$ be
of the form $\rho=\delta^{i/N}$. Then for each
$T_{\rho}\in\mathbb{T}_{\rho}^{\prime}$, we have

$\frac{\\#\mathbb{T}^{\prime}}{\\#\mathbb{T}^{\prime}_{\rho}}=\\#\mathbb{T}^{\prime}[T_{\rho}]\leq\delta^{-2\eta}\rho^{2}\\#\mathbb{T}^{\prime},$

where the second inequality follows from the fact that $\mathbb{T}^{\prime}$
satisfies the Convex Wolff Axioms with error $\delta^{-2\eta}$. We conclude
that $\\#\mathbb{T}^{\prime}_{\rho}\geq\delta^{2\eta}\rho^{-2}$. In
particular, if $\beta\leq 2+\varepsilon/4$, and if we select $\eta$
sufficiently small, then

$\\#\mathbb{T}^{\prime}_{\rho}\geq\delta^{2\eta}\rho^{-2}\geq\rho^{2\eta/\varepsilon+\varepsilon/4}\rho^{-\beta}\geq\rho^{-\beta+\varepsilon/3},$
(4.32)

and hence (4.31) holds.

Next we consider the case where $\beta>2+\varepsilon/4$. Suppose there exists
some $\rho\in[\delta,\delta^{\varepsilon}]$ of the form $\rho=\delta^{i/N}$
for which $\\#\mathbb{T}^{\prime}_{\rho}<\rho^{-\beta+\varepsilon/4}.$ Let
$\rho\in[\delta,\delta^{\varepsilon}]$ be the smallest number of the form
$\delta^{iN}$ for which this is true. Note that
$\rho\geq\delta^{1-\varepsilon/20}$, since the tubes in $\mathbb{T}^{\prime}$
are essentially distinct, and thus
$\\#\mathbb{T}^{\prime}_{\rho}\geq(\delta/\rho)^{4}(\\#\mathbb{T}^{\prime})\geq(\delta/\rho)^{4}\delta^{-\beta+2\eta}$.

Let $T_{\rho}\in\mathbb{T}_{\rho}$. By Conclusion (B) of Lemma 4.10, for each
$s\in[\delta,\rho]$ of the form $\delta^{jN}$, each $t\in[s,\rho]$ and each
$s\times t\times 1$ rectangular prism $R\subset T_{\rho}$, we have

$\displaystyle\\#\\{T\in\mathbb{T}^{\prime}[T_{\rho}]\colon T\subset R\\}$
$\displaystyle\leq\delta^{-\eta_{2}}\frac{t}{s}\frac{(\\#\mathbb{T}^{\prime})}{(\\#\mathbb{T}^{\prime}_{s})}\leq\delta^{-\eta_{2}}\frac{t}{s}\frac{\delta^{-\beta}}{\rho^{-\beta+\varepsilon/4}}\Big{(}\frac{s}{\rho}\Big{)}^{\beta-\varepsilon/4}$
$\displaystyle\leq\delta^{-\eta_{2}}\frac{t}{s}\frac{\delta^{-\beta}}{\rho^{-\beta+\varepsilon/4}}\Big{(}\frac{s}{\rho}\Big{)}^{2}\leq\delta^{-2\eta_{2}}\frac{t}{s}\frac{(\\#\mathbb{T}^{\prime})}{(\\#\mathbb{T}_{\rho})}\Big{(}\frac{s}{\rho}\Big{)}^{2}\leq\delta^{-2\eta_{2}}\frac{st}{\rho^{2}}(\\#\mathbb{T}^{\prime}[T_{\rho}]),$

where the second inequality used the fact that $\\#\mathbb{T}_{s}^{\prime}\geq
s^{-\beta+\varepsilon/4}$ and the minimality of $\rho$, the third inequality
used the assumption $\beta-\varepsilon/4\geq 2$, and the fourth inequality
used the assumptions $\\#\mathbb{T}^{\prime}\geq\delta^{-\beta+\eta_{1}}$ and
$\\#\mathbb{T}_{\rho}\leq\rho^{-\beta+\varepsilon/4}$.

Define $\tilde{\mathbb{T}}=\mathbb{T}^{\prime}[T_{\rho}]$, and let $W\subset
T_{\rho}$ be a convex set that contains at least one tube from
$\tilde{\mathbb{T}}$. Then $W$ is comparable to a $s\times t\times 1$ prism,
for some $\delta\leq s\leq t\leq\rho$; increasing $s$ (and possibly $t$ by a
factor of at most $\delta^{-1/N}$, we may suppose that $s$ is of the form
$\delta^{jN}$. Thus we have

$\\#\\{T\in\tilde{\mathbb{T}}\colon T\subset
W\\}\lesssim\delta^{-4\eta_{2}}\frac{|W|}{|T_{\rho}|}(\\#\tilde{\mathbb{T}})\leq(\delta/\rho)^{-40\eta_{2}/\varepsilon}\frac{|W|}{\rho^{2}}(\\#\tilde{\mathbb{T}}).$

If we select $\eta_{1}<40\eta_{2}\varepsilon$, then our set
$\tilde{\mathbb{T}}$ satisfies Conclusion (D) from Proposition 4.1 and we are
done.

Step 4. At this point, we have reduced to the case where (4.31) holds for all
$\rho\in[\delta,\delta^{\varepsilon}]$ of the form $\rho=\delta^{i/N}$. We
will show that Conclusion (B) holds. To begin, note that Items (i) and (iii)
from Definition 2.15 hold for $\mathbb{T}^{\prime}$: for each
$\rho_{0}\in[\delta,\delta^{\varepsilon}]$, we can select
$\rho\in[\rho_{0},\delta^{-\varepsilon}\rho_{0}]$ of the form
$\rho=\delta^{i/N}$; $\mathbb{T}_{\rho}$ is a partitioning cover of
$\mathbb{T}^{\prime}$, so Item (i) holds, and Item (iii) holds by (4.31) (if
$\rho_{0}>\delta^{\varepsilon}$ then we can select $\rho=1$ and there is
nothing to prove).

It remains to show that Item (ii) holds for $\mathbb{T}^{\prime}$; as
explained above it suffices to consider $\rho$-tubes with
$\rho\in[\delta,\delta^{\varepsilon}]$ of the form $\rho=\delta^{i/N}$, and
convex sets $R$ that are $s\times t\times 1$ prisms, with $s\in[\delta,\rho]$
of the form $s=\delta^{j/N}$ and $t\in[s,\rho]$. We would like to estimate the
number of tubes from $\mathbb{T}^{\prime}[T_{\rho}]$ contained in $R$.

By Conclusion (B) of Lemma 4.10, we have

$\displaystyle\\#\\{T\in\mathbb{T}^{\prime}[T_{\rho}]$ $\displaystyle\colon
T\subset
R\\}\leq\delta^{-\eta_{2}}\frac{t}{s}\frac{(\\#\mathbb{T}^{\prime})}{(\\#\mathbb{T}^{\prime}_{s})}\lesssim\delta^{-\eta_{2}-\varepsilon/4}ts(s^{\beta-2}\delta^{-\beta})$
$\displaystyle\lesssim\delta^{-\eta_{2}-\varepsilon/4}ts\rho^{\beta-2}\delta^{-\beta}=\delta^{-\eta_{2}-\varepsilon/4}\frac{ts}{\rho^{2}}\frac{\delta^{-\beta}}{\rho^{-\beta}}\leq\delta^{-2\eta_{2}-\varepsilon/4}\frac{ts}{\rho^{2}}(\\#\mathbb{T}^{\prime\prime}[T_{\rho}]).$

Selecting $\eta_{2}$ sufficiently small, we obtain

$\\#\\{T\in\mathbb{T}^{\prime}[T_{\rho}]\colon T\subset
R\\}\leq\delta^{-\varepsilon}\frac{|R|}{|T_{\rho}|}(\\#\mathbb{T}^{\prime}[T_{\rho}]).$

We conclude that $\mathbb{T}$ satisfies Condition (B) from Definition 2.15,
and hence Conclusion (B) of Proposition 4.1 holds. ∎

##  5 Sticky Kakeya and the Self-Similar Wolff Axioms

Our goal in this section is to prove Theorem 1.5. Our first task is to analyze
the structure of extremal collections of tubes. If $\omega>0$, then such
collections must satisfy the Self-Similar Convex Wolff Axioms.

###  5.1 Extremal Kakeya sets satisfy the Self-Similar Convex Wolff Axioms

######  Proposition 5.1.

Suppose $\omega$ from Definition 3.1 is positive. Then for all
$\varepsilon>0$, there exists $\eta,\delta_{0}>0$ so that the following holds
for all $\delta\in(0,\delta_{0}]$. Let $(\mathbb{T},Y)_{\delta}$ be $\eta$
Assouad Extremal. Then there exists $\mathbb{T}^{\prime}\subset\mathbb{T}$ so
that $(\mathbb{T}^{\prime},Y)_{\delta}$ is $\varepsilon$ Assouad extremal, and
$\mathbb{T}^{\prime}$ satisfies the Self-Similar Convex Wolff axioms with
error $\delta^{-\varepsilon}$.

###### Proof.

Step 1. By decreasing $\varepsilon$ if necessary, we can suppose that
$\varepsilon<\omega/2$. Let $\tau>0$ be the value from Proposition 4.1, with
this choice of $\varepsilon$. Let $\eta_{1}$ be a small quantity to be chosen
below ($\eta_{1}$ will depend on $\omega$ and $\varepsilon$), and let
$\eta^{\prime},\delta_{0}^{\prime}$ be the values from Proposition 4.1 for
this choice of $\varepsilon$ and $\eta_{1}$.

Let $\eta$ and $\delta_{0}$ be small positive quantities to be chosen below.
$\eta$ and $\delta_{0}$ will depend on the following quantities:
$\omega,\varepsilon,\tau,\eta_{1},\eta^{\prime}$. In addition, $\delta_{0}$
will depend on $\delta_{0}^{\prime}$.

Let $\delta\in(0,\delta_{0}]$ and let $(\mathbb{T},Y)_{\delta}$ be $\eta$
Assouad Extremal. By pigeonholing, we can select a subset
$\mathbb{T}_{1}\subset\mathbb{T}$ with
$\\#\mathbb{T}_{1}\gtrsim\delta^{\eta}(\\#\mathbb{T})$ so that
$|Y(T)|\geq\frac{1}{2}\delta^{\eta}|T|$ for each $T\in\mathbb{T}_{1}$. Then
$\\#\mathbb{T}_{1}=\delta^{-\beta}$ for some $\beta\geq\alpha-2\eta$ (recall
that $\alpha$ is defined in (3.4)), and $\mathbb{T}_{1}$ satisfies the Convex
Wolff Axioms with error $\delta^{-2\eta}$. Furthermore, for all
$\rho_{1},r_{1}$ with $\rho_{1}\leq\delta^{\eta}r_{1}$, and all balls $B$ of
radius $r_{1}$, we have

$\Big{|}B\cap
N_{\rho_{1}}\Big{(}\bigcup_{T\in\mathbb{T}_{1}}Y(T)\Big{)}\Big{|}\leq(\rho_{1}/r_{1})^{\omega-\eta}|B|.$
(5.1)

Step 2. Apply Proposition 4.1 to $(\mathbb{T}_{1},Y)_{\delta}$. We will show
that Conclusion (A) from Proposition 4.1 cannot hold. Suppose to the contrary
that Conclusion (A) holds, i.e. there exists $\rho,r\in[\delta,1]$ with
$\rho\leq\delta^{\eta^{\prime}}r$ and a $r$-ball $B$ such that

$\Big{|}B\cap
N_{\rho}\Big{(}\bigcup_{T\in\mathbb{T}_{1}}Y(T)\Big{)}\Big{|}\geq(\rho/r)^{\varepsilon}|B|.$
(5.2)

If we select $\eta\leq\min(\eta^{\prime},\omega_{2})$ (recall that
$\varepsilon\leq\omega/2$), then (5.2) contradicts (5.1). Hence Conclusion (A)
cannot hold.

Step 3. We will show that Conclusion (C) from Proposition 4.1 cannot hold.
Suppose to the contrary that Conclusion (C) holds, i.e. there exists
$\rho\in[\delta,\delta^{\varepsilon}]$ and a set of $\rho$-tubes
$\tilde{\mathbb{T}}$ satisfying Items (C.i), (C.ii) and (C.iii) from
Conclusion (C). By Item (C.iii), we have that the shading

$\tilde{Y}(\tilde{T})=\tilde{T}\cap
N_{\rho}\Big{(}\bigcup_{T\in\mathbb{T}_{1}}Y(T)\Big{)},\quad\tilde{T}\in\tilde{\mathbb{T}}$

satisfies
$|\tilde{Y}(\tilde{T})|\gtrsim\delta^{\eta}|\tilde{T}|\gtrsim\rho^{\eta/\varepsilon}|\tilde{T}|$
for each $\tilde{T}\in\tilde{\mathbb{T}}$; we will select
$\eta<\eta^{\prime}\varepsilon$ and $\delta_{0}$ sufficiently small that
$|\tilde{Y}(\tilde{T})|\leq\rho^{\eta^{\prime}}|\tilde{T}|$ for each
$\tilde{T}\in\tilde{\mathbb{T}}$.

Define

$\gamma=\omega-\omega(\beta+\tau).$ (5.3)

(In the above equation, $\omega(\beta+\tau)$ refers to the function $\omega$,
evaluated at the (positive) number $\beta+\tau$). Provided $\eta_{1}$ is
selected sufficiently small (depending on $\tau$), we have
$\beta+\tau\geq\alpha-2\eta_{1}+\tau>\alpha$, and hence by the definition of
$\alpha$ from (3.4), we have $\gamma>0$. Hence if $\eta_{1}$ is selected
sufficiently small, then

$\limsup_{x\searrow
0}\omega(\eta_{1},\eta_{1},x,\beta+\tau)\leq\omega-\gamma/2,$

i.e. there exists $x_{0}>0$ so that

$\omega(\eta_{1},\eta_{1},x,\beta+\tau)\leq\omega-\gamma/4\quad\textrm{for
all}\ x\in(0,x_{0}].$

Select $\delta_{0}$ sufficiently small so that $\delta_{0}^{\varepsilon}\leq
x_{0}$, and thus $\rho\in[\delta,\delta^{\varepsilon}]$ implies $\rho\leq
x_{0}$. In particular,
$\omega(\eta_{1},\eta_{1},\rho,\beta+\tau)\leq\omega-\gamma/4,$ i.e. there
exists $\rho_{1},r_{1}\in[\rho,1]$ with $\rho_{1}\leq\rho^{\eta_{1}}r_{1}$,
and a $r_{1}$-ball $B$ such that

$\Big{|}B\cap
N_{\rho_{1}}\Big{(}\bigcup_{\tilde{T}\in\tilde{\mathbb{T}}}\tilde{Y}(\tilde{T})\Big{)}\Big{|}\geq(\rho_{1}/r_{1})^{\omega-\gamma/8}|B|.$
(5.4)

But if we select $\eta\leq\min(\eta_{1}\varepsilon,\gamma/9)$, then
$\rho_{1}\leq\delta^{\eta}r_{1}$, and (5.4) contradicts (5.1). Hence
Conclusion (C) cannot hold.

Step 4. We will show that Conclusion (D) from Proposition 4.1 cannot hold.
Suppose to the contrary that Conclusion (D) holds, i.e. there exists
$\rho\in[\delta^{1-\varepsilon},1]$, a $\rho$-tube $T_{\rho}$, and a set
$\tilde{\mathbb{T}}\subset\mathbb{T}_{1}[T_{\rho}]$ satisfying Items (D.i) and
(D.ii) from Conclusion (D). Let
$\tilde{\delta}=\delta/\rho\in[\delta,\delta^{\varepsilon}]$, and let
$\tilde{\mathbb{T}}_{\tilde{\delta}}$ be unit rescaling of
$\tilde{\mathbb{T}}$ relative to $T_{\rho}$. Let
$\tilde{Y}_{\tilde{\delta}}(\tilde{T}_{\tilde{\delta}})$ be the image of
$\tilde{Y}(\tilde{T})$ under this same rescaling. We have
$|\tilde{Y}_{\tilde{\delta}}(\tilde{T})|\gtrsim\delta^{\eta}|\tilde{T}_{\tilde{\delta}}|\geq\tilde{\delta}^{\eta/\varepsilon}|\tilde{T}_{\tilde{\delta}}|$,
and we will select $\eta$ and $\delta_{0}$ sufficiently small so that
$|\tilde{Y}_{\tilde{\delta}}(\tilde{T}_{\tilde{\delta}})|\geq\tilde{\delta}^{\eta_{1}}|\tilde{T}_{\tilde{\delta}}|$
for all $\tilde{T}_{\tilde{\delta}}\in\tilde{\mathbb{T}}_{\tilde{\delta}}$.
The tubes in $\tilde{\mathbb{T}}_{\tilde{\delta}}$ are essentially distinct;
by Item (D.i) we have
$\\#\tilde{\mathbb{T}}_{\tilde{\delta}}\geq\tilde{\delta}^{-\beta-\tau}$, and
by Item (D.ii) we have that $\tilde{\mathbb{T}}_{\tilde{\delta}}$ satisfies
the Convex Wolff Axioms with error $\lesssim\tilde{\delta}^{-\eta_{1}}$.

Arguing as in Step 3, we conclude that there exists
$\rho_{2},r_{2}\in[\tilde{\delta},1]$ with
$\rho_{2}\leq\tilde{\delta}^{\eta_{1}}r_{2}$, and a $r_{2}$-ball $B_{2}$ such
that

$\Big{|}B_{2}\cap
N_{\rho_{2}}\Big{(}\bigcup_{\tilde{T}_{\tilde{\delta}}\in\tilde{\mathbb{T}}_{\tilde{\delta}}}\tilde{Y}_{\tilde{\delta}}(\tilde{T}_{\tilde{\delta}})\Big{)}\Big{|}\geq(\rho_{2}/r_{2})^{\omega-\gamma/8}|B_{2}|.$

Undoing the unit rescaling described above, we conclude there exists an
ellipsoid $E$ of dimensions $\rho r_{2}\times\rho r_{2}\times\rho$ so that

$\Big{|}E\cap
N_{\rho\rho_{2}}\Big{(}\bigcup_{\tilde{T}\in\tilde{\mathbb{T}}}\tilde{Y}(\tilde{T})\Big{)}\Big{|}\geq(\rho_{2}/r_{2})^{\omega-\gamma/8}|E|.$

Indeed, the above inequality follows from the observation that the image of a
$\rho_{2}$-ball centered at $x$ under the above rescaling is an ellipse of
dimensions $\rho\rho_{2}\times\rho\rho_{2}\times\rho_{2}$, and this ellipse
contains a $\rho\rho_{2}$-ball centered at $x$. In particular, we can find a
$\rho r_{2}$-ball $B$ with $|B\cap E|\sim|B|$, so that

$\Big{|}B\cap
N_{\rho\rho_{2}}\Big{(}\bigcup_{\tilde{T}\in\tilde{\mathbb{T}}}\tilde{Y}(\tilde{T})\Big{)}\Big{|}\gtrsim(\rho_{2}/r_{2})^{\omega-\gamma/8}|B|.$
(5.5)

Finally, observe that $\frac{\rho\rho_{2}}{\rho
r_{2}}=\frac{\rho_{2}}{r_{2}}\leq\rho^{\eta_{1}}\leq\delta^{\eta_{1}\varepsilon}\leq\delta^{\eta}$
(provided we select $\eta\leq\eta_{1}\varepsilon$). Thus if we choose
$\eta\leq\gamma/9$ and $\delta_{0}$ sufficiently small, then (5.5) contradicts
(5.1). Hence Conclusion (D) cannot hold.

Step 5. At this point, we shown that Conclusions (A), (C), and (D) from
Proposition 4.1 cannot hold. Thus Conclusion (B) must hold. This completes the
proof of Proposition 5.1. ∎

###  5.2 Proof of Theorem 1.5

Our proof of Theorem 1.5 follows by combining two main ingredients. The first
is Proposition 5.1. The second is a variant of the Sticky Kakeya Theorem,
which was proved by the authors in [16]. The precise statement we need is as
follows.

######  Theorem 5.2.

For all $\varepsilon>0$, there exists $\eta,\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let $\mathbb{T}$ be a set
of $\delta$-tubes that satisfy the Convex Wolff Axioms at every scale with
error $\delta^{-\eta}$, and let $Y(T)$ be a $\delta^{\eta}$ dense shading.
Then

$\Big{|}\bigcup_{T\in\mathbb{T}}Y(T)\Big{|}\geq\delta^{\varepsilon}.$ (5.6)

As noted above, Theorem 5.2 is similar to the Sticky Kakeya Theorem proved by
the authors in [16]. In Section 6 we will discuss the proof of Theorem 5.2,
and how it differs from the arguments in [16]. Assuming this result for now,
we can combine these two ingredients to prove Theorem 1.5 as follows

###### Proof of Theorem 1.5.

It suffices to show that $\omega=0$. Suppose instead that $\omega>0$. Let
$\eta_{0},\delta_{0}$ be the output from Theorem 5.2, with
$\varepsilon=\omega/3$. By Proposition 5.1, there exists
$\delta\in(0,\delta_{0}]$ and a set $(\mathbb{T}^{\prime},Y)_{\delta}$ with
discretized Assouad dimension $\leq 3-\omega/3$ that satisfies the Self-
Similar Convex Wolff Axioms (and hence the Convex Wolff Axioms at every scale)
with error $\delta^{-\eta_{0}}$, and for which $\\{Y(T)\\}$ is a
$\delta^{\eta_{0}}$ dense shading. But then (6.2) says that $\bigcup Y(T)$ has
discretized Assouad dimension at least $3-\omega/3$, which is a contradiction.
∎

Thus we have completed the proof of Theorem 1.5, except that it remains to
prove Theorem 5.2; we will do so in the next section.

##  6 Kakeya estimates for tubes satisfying the Convex Wolff Axioms at all
scales

Our goal in this section is to prove Theorem 5.2. Theorem 5.2 is closely
related to the discretized analogue of [16, Theorem 1.1], and specifically the
estimate $\sigma_{3}=0$ from that paper; the latter result says that if
$\mathbb{T}$ is a collection of $\delta$-tubes in $\mathbb{R}^{3}$ of
cardinality roughly $\delta^{-2}$ that point in $\delta$-separated directions,
and if $\mathbb{T}$ satisfies a property called “stickiness,” then the union
of the tubes must have volume $\gtrapprox 1$. Stickiness is a technical
property, which roughly speaking says that for each $\rho\in[\delta,1]$, the
tubes in $\mathbb{T}$ can be covered by a collection of $\rho$-tubes that
point in $\rho$-separated directions (more accurately, the tubes point in
$\rho$-separated directions up to multiplicity $\lessapprox 1$).

Stickiness is a multi-scale property, in the following sense: if $\mathbb{T}$
is sticky, then for each $\rho\geq\delta$ we can cover $\mathbb{T}$ by a
sticky collection of $\rho$-tubes. This multi-scale property was exploited
throughout the arguments in [16]. Theorem 5.2 differs from [16, Theorem 1.1],
because the stickiness hypothesis has been replaced by the hypothesis that the
tubes satisfy the Convex Wolff axioms at every scale. As we will see below,
however, this latter hypothesis is sufficient in order to repeat the arguments
from [16]. Indeed, with two exceptions that will be explained below, the
arguments from [16] can be repeated mutatis mutatis to obtain Theorem 5.2.

The proof of [16, Theorem 1.1] is divided into several sections. The paper
begins by supposing that the result is false, and studying the structure of a
(hypothetical) counter-example. In Section 2, the authors define what it means
for a pair $(\mathbb{T},Y)$ of tubes and their associated shading to be a
worst possible counter-example to [16, Theorem 1.1]; this is called an
$\varepsilon$-extremal pair. In our discussion below, we will call it a
$\varepsilon$-extremal pair. In [16], Section 3, it is shown that an
$\varepsilon$-extremal pair $(\mathbb{T},Y)$ must look coarsely self similar
at every intermediate scale $\rho\in[\delta,1]$. More precisely, for every
such $\rho$ it is possible to cover the tubes in $\mathbb{T}$ by a collection
of $\rho$-tubes, so that the corresponding collection of $\rho$-tubes is a
worst possible counter-example to [16, Theorem 1.1] at scale $\rho$, and the
$\delta$-tubes inside each $\rho$-tube also a form a (re-scaled) worst-
possible counter-example at scale $\delta/\rho$. The analogous statement and
proof in our setting are nearly identical; we will state the former and
briefly sketch the latter.

In Sections 3, 4, and 5 of [16], the authors use the multi-scale self-
similarity established in Section 2 of [16] to show that an extremal pair
$(\mathbb{T},Y)$ must have certain structural properties called planiness and
graininess, and in particular there must exist local grains that are described
by a Lipschitz plane map, and global grains that are described by a $C^{2}$
slope function $f$; this will be described in greater detail below. The
arguments from [16] can be repeated without modification to establish the same
conclusions in the present setting. In particular, the hypothesis from [16]
that the tubes point in $\delta$-separated directions has not been used up
until this step.

In Section 6 of [16], the authors show that the derivative $f^{\prime}$ of the
slope function has magnitude roughly 1 (and in particular, this quantity is
bounded away from 0). It is during this step that the authors use the
hypothesis that the tubes point in $\delta$-separated directions. With some
small modifications, we can instead use the hypothesis that the tubes satisfy
the Convex Wolff axioms at every scale; we will elaborate on this step in
Section 6.4.

Finally, in Section 7 of [16], the authors recall an analogue of Wolff’s
circular maximal function estimate [19, 14], and they show how this estimate
contradicts the assumption that there exits a collection of sticky tubes whose
union has volume much smaller than 1, thereby completing the proof of the
theorem. This section of [16] uses the assumption that the tubes point in
different directions, in order to obtain a rich planar arrangement of curves
to which Wolff’s circular maximal theorem can be applied. With some
modifications, the direction-separated assumption can be replaced with a
weaker ball condition; we will explain this in detail in Sections 6.5 and 7.

###  6.1 Relevant definitions from [16]

Before describing the arguments in [16], it will be helpful to recall a few
key definitions. First, a minor technical annoyance: tubes and cubes in [16]
are defined slightly differently than in the present paper. In [16], a
$\delta$-tube in $\mathbb{R}^{3}$ is defined to be a set of the form
$N_{6\delta}(\ell)\cap[-1,1]^{3}$, where $\ell$ is a line in $\mathbb{R}^{3}$.
This definition is adopted for technical reasons, and for consistency with the
results in [16] we will use this definition throughout Section (6); the
distinction between this definition and our previous definition in Section 1
is not important in the arguments that follow. Second, in [16], a shading
$Y(T)\subset T$ is defined to be a union of axis-aligned $\delta$-cubes
contained in $T$. Again, the distinction between this definition and our
previous one from Section 2.1 is harmless: replace each tube $T$ (in the sense
of Section 1) and a each shading $Y(T)$ (in the sense of Section 2.1) with the
corresponding tube in the sense of [16] and the the shadings consisting of the
union of axis-aligned $\delta$-cubes $Q$ that satisfy $|Q\cap
Y(T)|\geq\frac{1}{100}|Y(T)|/|T|$. This gives us a collection of tubes and
shadings in the sense of [16], and if the conclusion of Theorem 1.5 holds for
this latter collection of tubes and shadings, then it also holds for the
original collection of tubes and shadings, except Inequality (1.1) has been
weakened by an additional (harmless) factor of $\delta^{\eta}$.

#### Tubes, covers, rescaling.

We will often refer to a collection of $\delta$-tubes $\mathbb{T}$, and a
shading $Y(T)$ of these tubes. We will refer to this pair as
$(\mathbb{T},Y)_{\delta}$. We use $E_{\mathbb{T}}$ to denote the set
$\bigcup_{T\in\mathbb{T}}Y(T)$; the shading $Y$ will always be apparent from
context. For $p\in\mathbb{R}^{3}$, we define

$\mathbb{T}(p)=\\{T\in\mathbb{T}\colon p\in Y(T)\\}.$ (6.1)

We say a pair $(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ is a _refinement_ of
$(\mathbb{T},Y)_{\delta}$ if $\mathbb{T}^{\prime}\subset\mathbb{T}$;
$Y^{\prime}(T^{\prime})\subset Y(T^{\prime})$ for each
$T\in\mathbb{T}^{\prime}$; and
$\sum_{\mathbb{T}^{\prime}}|Y^{\prime}(T^{\prime})|\geq
C^{-1}|\log\delta|^{-C}\sum_{\mathbb{T}}|Y(T)|$ for some absolute constant $C$
(in practice, $C=100$ will always suffice).

Let $0<\delta\leq\rho\leq 1$. In [16], the authors say that a $\rho$-tube
$\tilde{T}$ _covers_ a $\delta$-tube $T$ if their respective coaxial lines
satisfy $d(\ell,\tilde{\ell})\leq\rho/2$, where $d(\cdot,\cdot)$ is an
appropriately chosen metric on the affine Grassmannian of lines in
$\mathbb{R}^{3}$; see Section 2 of [16] for details. Note that if $\tilde{T}$
covers $T$, then $T\subset\tilde{T}$, and hence this definition is consistent
with ours from Definition 2.10 (the definition in [16] is slightly stronger
than Definition 2.10, but the distinction is harmless).

If $(\mathbb{T},Y)_{\delta}$ and $(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ are
pairs of $\delta$ (resp. $\rho$) tubes and their associated shadings, then we
say $(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ _covers_ $(\mathbb{T},Y)_{\delta}$
if each $T\in\mathbb{T}$ is covered by at least one
$\tilde{T}\in\tilde{\mathbb{T}}$, and for each
$\tilde{T}\in\tilde{\mathbb{T}}$ covering $T$, we have
$Y(T)\subset\tilde{Y}(\tilde{T})$. If this is the case, we write
$\mathbb{T}[\tilde{T}]$ to denote the set of tubes in $\mathbb{T}$ covered by
$\tilde{T}$. We say that $(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ is a
_balanced cover_ of $(\mathbb{T},Y)_{\delta}$ if it is a cover, and in
addition $|E_{\mathbb{T}}\cap Q|$ is the same for each $\rho$-cube $Q\subset
E_{\tilde{\mathbb{T}}}$.

Let $(\mathbb{T},Y)_{\delta}$ be a set of $\delta$-tubes and their associated
shading. Let $\tilde{T}$ be a $\rho$-tube, and suppose each tube in
$\mathbb{T}$ is covered by $\tilde{T}$. We define a new pair
$(\hat{\mathbb{T}},\hat{Y})_{\delta/\rho}$, which we call the _unit rescaling
of $(\mathbb{T},Y)_{\delta}$ relative to $\tilde{T}$_. Informally, our
definition is as follows: consider the linear transformation that sends the
coaxial line of $\tilde{T}$ to the $z$ axis, and dilates the $x$ and $y$
directions by $\rho^{-1}$; the image of $\tilde{T}$ under this transformation
is comparable to the unit ball. The tubes in $\hat{\mathbb{T}}$ are those
$(\delta/\rho)$-tubes whose coaxial lines are the images of the coaxial lines
of the tubes in $\mathbb{T}$ under this transformation. The shadings
$\hat{Y}(\hat{T})$ consist of the union of axis-aligned $\delta/\rho$-cubes
that intersect the image of $Y(T)$ under the above transformation. See Section
3 of [16] for details. As above, the definition of unit rescaling in [16] is
slightly different than Definition 2.8, but the distinction is harmless.

#### The plane map.

In the arguments in [16], the authors show that certain collections of tubes
possess an important structural property called planiness. Let
$(\mathbb{T},Y)_{\delta}$ be a set of tubes and their associated shading. We
say a function $V\colon E_{\mathbb{T}}\to S^{2}$ is a _plane map_ for
$(\mathbb{T},Y)_{\delta}$ if

$|\operatorname{dir}(T)\cdot V(p)|\leq\delta\quad\textrm{for all}\
(T,p)\in\mathbb{T}\times\mathbb{R}^{3}\ \textrm{with}\ p\in Y(T),$

where $\operatorname{dir}(T)\in S^{2}$ is a unit vector parallel to the line
coaxial with $T$.

#### Discretized Ahlfors-David regular sets.

We say a set $E\subset\mathbb{R}$ is a $(\delta,\alpha,C)$-ADset if for all
$\rho\geq\delta$, all $r\geq\rho$, and all intervals $I$ of length $r$, we
have $\mathcal{E}_{\rho}(E\cap I)\leq C(r/\rho)^{\alpha}$, where
$\mathcal{E}_{\rho}(E\cap I)$ means the $\rho$-covering number of $E\cap I$.
In practice we will have $\alpha\in(0,1]$, and $C$ will be a small power of
$1/\delta$. See Section 4 of [16] for further discussion and motivation for
this definition.

###  6.2 Extremal collection of tubes, and multi-scale structure

We are ready to begin the proof of Theorem 5.2. Suppose to the contrary that
the result was false. Then there exists $\sigma^{\prime}>0$ so that for all
$\eta,\delta_{0}>0$, there exists $\delta\in(0,\delta_{0}]$; a set
$\mathbb{T}$ of $\delta$-tubes that satisfy the Convex Wolff Axioms at every
scale with error $\delta^{-\eta}$; and a $\delta^{\eta}$-dense shading $Y(T)$,
so that

$\Big{|}\bigcup_{T\in\mathbb{T}}Y(T)\Big{|}\leq\delta^{\sigma^{\prime}}.$
(6.2)

Let $\sigma>0$ be the supremum of all numbers $\sigma^{\prime}$ with the above
property; this supremum exists, since by hypothesis the set of admissible
$\sigma^{\prime}$ is non-empty, and it is bounded above by 3. Our goal is to
show that we must in fact have $\sigma=0$, and thereby obtain a contradiction.

The following definition is the analogue of [16, Definition 3.1].

######  Definition 6.1.

Let $\varepsilon,\delta>0$. We say a pair $(\mathbb{T},Y)_{\delta}$ is
$\varepsilon$-extremal if

  1. 1.

$\mathbb{T}$ satisfies the Convex Wolff Axioms at all scales with error
$\delta^{-\varepsilon}$.

  2. 2.

$Y$ is $\delta^{\varepsilon}$-dense.

  3. 3.

$\Big{|}\bigcup_{T\in\mathbb{T}}Y(T)\Big{|}\leq\delta^{\sigma-\varepsilon}.$

It is immediate from the above definitions that for every
$\varepsilon,\delta_{0}>0$, there exists a $\varepsilon$-extremal collection
of tubes $(\mathbb{T},Y)_{\delta}$ for some $\delta\in(0,\delta_{0}]$; cf.
[16, Lemma 3.1].

Our next task is to prove an analogue of [16, Proposition 3.2]. The precise
statement is as follows.

######  Proposition 6.2.

For all $\varepsilon>0$, there exists $\eta>0$ and $\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let
$(\mathbb{T},Y)_{\delta}$ be a $\eta$-extremal collection of tubes, and let
$\rho\in[\delta^{1-\varepsilon},\delta^{\varepsilon}]$. Then there is a
refinement $(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ of
$(\mathbb{T},Y)_{\delta}$ and a balanced cover
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ of
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ with the following properties.

  1. (i)

$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ is $\varepsilon$-extremal.

  2. (ii)

For each $\tilde{T}\in\tilde{\mathbb{T}}$, the unit rescaling of
$(\mathbb{T}^{\prime}[\tilde{T}],Y^{\prime})_{\delta}$ relative to $\tilde{T}$
is $\varepsilon$-extremal.

  3. (iii)

For each $p\in\mathbb{R}^{3}$,
$\\#\tilde{\mathbb{T}}(p)\leq\rho^{2-\sigma-\varepsilon}(\\#\tilde{\mathbb{T}}).$

  4. (iv)

For each $p\in\mathbb{R}^{3}$ and each $\tilde{T}\in\tilde{\mathbb{T}}$,
$\\#\mathbb{T}^{\prime}[\tilde{T}](p)\leq(\delta/\rho)^{2-\sigma-\varepsilon}(\\#\mathbb{T}^{\prime}[\tilde{T}]).$

###### Proof.

The proof is nearly identical to the proof of the corresponding statement in
[16, Proposition 3.2]. In [16], the first major step was to establish the
existence of a cover $\mathbb{T}_{\rho}$ of $\mathbb{T}$ (or more accurately,
a large subset $\mathbb{T}^{\prime}\subset\mathbb{T}$), with the property that
both $\mathbb{T}_{\rho}$ and the the re-scaled sets
$\mathbb{T}^{\prime}[T_{\rho}]$ are sticky. In our case, this step was handled
by Lemma 2.13. The second major step was to establish a shading on
$\mathbb{T}_{\rho}$ and a refinement of the shadings on the sets
$\mathbb{T}^{\prime}[T_{\rho}]$ so that not too many tubes pass through each
point (and as a consequence, the corresponding collections are
$\varepsilon$-extremal).

Next we will give a slightly more detailed sketch, though we refer the reader
to the proof of [16, Proposition 3.2] for full details. We begin by reducing
to the case where each $T\in\mathbb{T}$ satisfies
$|Y(T)|\geq\frac{1}{2}\delta^{\eta}|T|$. This can be done by discarding those
tubes with $|Y(T)|<\frac{1}{2}\delta^{\eta}|T|$ and then repeatedly
pigeonholing to re-establish a collection of tubes that satisfies the Convex
Wolff Axioms at every scale (with a slightly worse constant, say
$\delta^{-2\eta}$); after this process, the cardinality of our collection of
tubes has been reduced by at most a factor of $\delta^{\eta}$.

Next, we apply Lemma 2.13 with $\rho_{0}=\delta^{2\eta}\rho$; we get a number
$\rho^{\prime}\in[\delta^{2\eta}\rho,\rho]$; and a collection
$\mathbb{T}_{\rho^{\prime}}$ of $\rho^{\prime}$-tubes, as described by that
Lemma. Of course, we may replace each $\rho^{\prime}$ tube by corresponding
$\rho$-tube, and denote the resulting collection by $\tilde{\mathbb{T}}$.
After pigeonholing to re-establish the property that the tubes are essentially
distinct, it is still the case that $\tilde{\mathbb{T}}$ satisfies the Convex
Wolff Axioms at every scale with error $\delta^{-O(\eta)}$, and each of the
re-scaled collections $\mathbb{T}[\tilde{T}]$ satisfies the Convex Wolff
Axioms at every scale with error $\delta^{-O(\eta)}$.

After refining the shadings $Y(T)$, we may suppose that for each collection
$\mathbb{T}[\tilde{T}]$, the same number of tubes pass through each
$\delta$-cube in $E_{\mathbb{T}[\tilde{T}]}$ in the sense of (6.1); after
further pigeonholing we may suppose that this number is the same (up to a
multiplicative factor of 2) for each set $\mathbb{T}[\tilde{T}]$; call this
number $\mu_{\operatorname{fine}}$. Since $\mathbb{T}[\tilde{T}]$ is a (re-
scaled) collection of tubes satisfying the Convex Wolff Axioms at every scale,
if $\eta$ and $\delta_{0}$ are chosen sufficiently small then we must have
$\mu_{\operatorname{fine}}\leq(\delta/\rho)^{2-\sigma-\varepsilon/10}(\\#\mathbb{T}^{\prime}[\tilde{T}])$;
this is Conclusion (iv) above.

Our next task is to define a shading $\tilde{Y}$ on the tubes of
$\tilde{\mathbb{T}}$. To begin, for each $\tilde{T}$ we consider the union of
all $\rho$-cubes $\tilde{Q}\subset\tilde{T}$ that intersect
$E_{\mathbb{T}[\tilde{T}]}$. After pigeonholing, we can refine this shading so
that each $\rho$-cube intersects $E_{\mathbb{T}[\tilde{T}]}$ in a set of
roughly the same size, and the same number of tubes from $\tilde{\mathbb{T}}$
pass through each cube of $E_{\tilde{\mathbb{T}}}$; call this number
$\mu_{\operatorname{coarse}}$. We refine the corresponding shadings $Y(T),\
T\in\mathbb{T}$ so that $Y(T)\subset\tilde{Y}(\tilde{T})$ whenever
$T\in\mathbb{T}[\tilde{T}]$; in particular
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ is now a balanced cover of
$(\mathbb{T},Y)_{\delta}$. Note that Conclusion (iv) remains true after this
procedure. Since $\tilde{\mathbb{T}}$ satisfies the Convex Wolff Axioms with
error $\delta^{-O(\eta)}$, if $\eta$ and $\delta_{0}$ are chosen sufficiently
small then we must have
$\mu_{\operatorname{coarse}}\leq\rho^{2-\sigma-\varepsilon/10}(\tilde{\mathbb{T}})$;
this is Conclusion (iii) above.

Finally, observe that
$\mu_{\operatorname{fine}}\mu_{\operatorname{coarse}}\leq\delta^{2-\sigma-\varepsilon/5}(\\#\mathbb{T})$,
but on the other hand since $\mathbb{T}$ was $\eta$-extremal, we must have
$\mu_{\operatorname{fine}}\mu_{\operatorname{coarse}}\gtrsim\delta^{2-\sigma+\eta}(\\#\mathbb{T})$.
We conclude that

$\displaystyle(\delta/\rho)^{2-\sigma+\varepsilon/10}(\\#\mathbb{T}^{\prime}[\tilde{T}])\leq$
$\displaystyle\mu_{\operatorname{fine}}\leq(\delta/\rho)^{2-\sigma-\varepsilon/10}(\\#\mathbb{T}^{\prime}[\tilde{T}]),\quad\tilde{T}\in\tilde{\mathbb{T}},$
$\displaystyle\rho^{2-\sigma+\varepsilon/10}(\\#\tilde{\mathbb{T}})\leq$