# Sticky Kakeya sets and the sticky Kakeya conjecture

Hong Wang Department of Mathematics, University of California, Los Angeles,
CA<EMAIL_ADDRESS>Joshua Zahl Department of Mathematics, University
of British Columbia, Vancouver, BC<EMAIL_ADDRESS>
###### Abstract

A Kakeya set is a compact subset of $\mathbb{R}^{n}$ that contains a unit line
segment pointing in every direction. The Kakeya conjecture asserts that such
sets must have Hausdorff and Minkowski dimension $n$. There is a special class
of Kakeya sets, called sticky Kakeya sets. Sticky Kakeya sets exhibit an
approximate multi-scale self-similarity, and sets of this type played an
important role in Katz, Łaba, and Tao’s groundbreaking 1999 work on the Kakeya
problem. We propose a special case of the Kakeya conjecture, which asserts
that sticky Kakeya sets must have Hausdorff and Minkowski dimension $n$. We
prove this conjecture in three dimensions.

##  1 Introduction

A compact set $K\subset\mathbb{R}^{n}$ is called a _Kakeya set_ if it contains
a unit line segment pointing in every direction. A surprising construction by
Besicovitch [2] shows that such sets can have measure 0. The Kakeya set
conjecture asserts that every Kakeya set in $\mathbb{R}^{n}$ has Hausdorff and
Minkowski dimension $n$. There is also a slightly more technical, single-scale
variant of this conjecture, which is called the Kakeya maximal function
conjecture: let $\delta>0$ and let $\mathbb{T}$ be a set of $1\times\delta$
tubes in $\mathbb{R}^{n}$ whose coaxial lines point in $\delta$-separated
directions. Then the tubes must be almost disjoint, in the sense that for
every $1\leq d\leq n$ and every $\varepsilon>0$, there exists
$C=C(n,d,\varepsilon)$ (independent of $\delta$) so that

$\Big{\|}\sum_{T\in\mathbb{T}}\chi_{T}\Big{\|}_{\frac{d}{d-1}}\leq
C\Big{(}\frac{1}{\delta}\Big{)}^{\frac{n}{d}-1+\varepsilon}\Big{(}\sum_{T\in\mathbb{T}}|T|\Big{)}^{\frac{d-1}{d}}.$
(1.1)

The Kakeya set conjecture and Kakeya maximal conjecture were proved for $n=2$
by Davies [8] and Cordoba [7], respectively. The conjectures remain open in
three and higher dimensions.

The Kakeya conjecture is closely related to questions in Fourier analysis.
This connection was first explored by Fefferman [13], who used a variant of
Besicovitch’s construction to show that the ball multiplier
$\widehat{Tf}=\chi_{B}\hat{f}$ is unbounded on $L^{p}(\mathbb{R}^{2})$ when
$p\neq 2$. In [3], Bourgain obtained new estimates for Stein’s Fourier
restriction conjecture in $\mathbb{R}^{3}$ by first proving, and then using
estimates of the form (1.1) (for certain $d<3$) in three dimensions. In brief,
a function whose Fourier transform is supported on a curved manifold $M$ can
be decomposed into a sum of “wave packets,” each of which is supported on a
tube $T$. The curvature of $M$ ensures that many of these wave packets point
in different directions, and estimates of the form (1.1) can be used to
analyze the possible interection patterns of these wave packets.

Since Bourgain’s seminal work [3], Kakeya estimates have served as an input
when studying the Fourier restriction problem [5, 9, 30], and methods that
were developed in the context of the Kakeya problem have been successfully
applied to the Fourier restriction problem, Bochner-Riesz problem, and related
questions. Indeed, the modern renaissance in polynomial method techniques was
sparked by Zeev Dvir’s proof [11] of Wolff’s finite field Kakeya conjecture.
These polynomial method techniques have since revolutionized discrete math,
combinatorial geometry, and harmonic analysis. See e.g. [15, 26] for a modern
survey of these developments.

In this paper we will restrict our attention to the Kakeya set conjecture in
three dimensions. In [31], Wolff proved that every Kakeya set in
$\mathbb{R}^{3}$ has Hausdorff dimension at least $5/2$. In [17], Katz, Łaba,
and Tao made the following improvement: every Kakeya set in $\mathbb{R}^{3}$
has upper Minkowski dimension at least $5/2+c$, where $c>0$ is a small
absolute constant. Katz, Łaba, and Tao’s argument began by carefully analyzing
the structure of a (hypothetical) Kakeya set that had dimension exactly 5/2.
At scale $\delta$, such a set $K$ would contain a union of roughly
$\delta^{-2}$ $1\times\delta$ tubes that point in $\delta$-separated
directions, and the union of these tubes would have volume roughly
$\delta^{1/2}$. Next, Katz, Łaba, and Tao considered how these $1\times\delta$
tubes would arrange themselves at scale $\rho=\delta^{1/2}$. Since the
$1\times\delta$ tubes point in different directions, each $1\times\rho$ tube
can contain at most $\rho^{2}/\delta^{2}$ $1\times\delta$ tubes. If equality
(or near equality) holds (i.e. if the $1\times\delta$ tubes arrange themselves
into roughly $\rho^{-2}$ $1\times\rho$ tubes, each of which contain
$\rho^{2}/\delta$ $1\times\delta$ tubes) then we say the arrangement of
$1\times\delta$ tubes is _sticky_ at scale $\rho$. As a starting point for
their arguments in [17], Katz, Łaba, and Tao used a result by Wolff [33] to
prove that that if a Kakeya set $K$ has upper Minkowski dimension $5/2$, then
the corresponding set of $1\times\delta$ tubes must be sticky at every scale
$\rho\in(\delta,1)$.

Motivated by this observation, we introduce a special class of Kakeya sets,
which we call sticky Kakeya sets. Let $\mathcal{L}$ be the set of (affine)
lines in $\mathbb{R}^{n}$, equipped with the metric
$d(l,l^{\prime})=|p-p^{\prime}|+\angle(u,u^{\prime})$. Here $u$ (resp
$u^{\prime}$) is a unit vector parallel to $\ell$ and $p$ (resp. $p^{\prime}$)
is the unique point on $\ell$ with $p\perp u$. Since the direction map from
$\mathcal{L}$ to $\mathbb{R}\mathbb{P}^{n-1}$ is Lipschitz, if
$L\subset\mathcal{L}$ contains a line in every direction then $\dim_{P}(L)\geq
n-1$. (Here $\dim_{P}$ denotes packing dimension; similar statements hold for
other metric notions of dimension, but this is less relevant to the current
discussion). We say a Kakeya set is sticky if equality holds:

######  Definition 1.1.

A compact set $K\subset\mathbb{R}^{n}$ is called a _sticky Kakeya set_ if
there is a set of lines $L$ with packing dimension $n-1$ that contains at
least one line in each direction, so that $\ell\cap K$ contains a unit
interval for each $\ell\in L$.

Observe that if we drop the requirement that $L$ has packing dimension $n-1$,
then we have recovered the usual definition of a Kakeya set. In particular,
every sticky Kakeya set is also a (classical) Kakeya set. As we will see in
Section 2, when sticky Kakeya sets (in the sense of Definition 1.1) are
discretized at a small scale $\delta>0$, the corresponding collection of
$1\times\delta$ tubes is sticky in the sense of Katz, Łaba, and Tao. With
these definitions, we can introduce the sticky Kakeya set conjecture.

######  Conjecture 1.1.

Every sticky Kakeya set in $\mathbb{R}^{n}$ has Hausdorff and Minkowski
dimension $n$.

A defining feature of sticky Kakeya sets is that if $K$ is a (hypothetical)
counter-example to the sticky Kakeya conjecture, then $K$ must be roughly
self-similar at many different scales. This was first observed in [17] and
discussed in greater detail in [29]. A precise version of this principle is
stated in Proposition 3.2 below.

Since every sticky Kakeya set is also a (classical) Kakeya set, the Kakeya set
conjecture implies the sticky Kakeya conjecture, and partial progress toward
the Kakeya set conjecture implies the same partial progress for the sticky
Kakeya conjecture. In particular, Davies’ solution [8] to the Kakeya set
conjecture in the plane implies that Conjecture 1.1 holds when $n=2$. In this
paper, we will prove Conjecture 1.1 for $n=3$.

######  Theorem 1.1.

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

Our proof of Theorem 1.1 is inspired by, and partially follows, the arguments
recorded in Terence Tao’s blog entry [29]. That blog entry sketches an
approach to solving the Kakeya conjecture in $\mathbb{R}^{3}$ that was
explored by Nets Katz and Terence Tao in the early 2000s. In brief, [29]
begins by conjecturing that a (hypothetical) counter-example
$K\subset\mathbb{R}^{3}$ to the Kakeya conjecture in $\mathbb{R}^{3}$ must be
sticky, and must also have two additional structural properties, which are
called planiness and graininess. We will discuss this conjecture in Section
1.2 below. Next, the blog entry describes how these three properties can be
used to obtain multi-scale structural information about $K$. Finally, the blog
entry suggests how to use this structural information to produce a counter-
example to Bourgain’s discretized sum-product theorem [4], and hence obtain a
contradiction.

###  1.1 A sketch of the proof

We will briefly outline our approach to proving Theorem 1.1, and discuss how
it relates to the strategy from [29]. Our proof can be divided into three
major steps. The first step follows some of the arguments described in [29].
For steps 2 and 3 our arguments take a different path from those presented in
[29]; we do this in part because sticky Kakeya sets can have structural
properties that were not discussed in [29].

Step 1: Discretization and multi-scale self-similarity
In Section 2, we set up a discretization of the sticky Kakeya problem that
allows us to exploit multi-scale self-similarity. The technical details of
this procedure are new, and we hope that this setup will be useful when
studying the sticky Kakeya conjecture in higher dimensions.

In Section 3, we flesh-out the ideas from [29] to obtain multi-scale
structural information about sticky Kakeya sets. The main conclusion of this
section is as follows. Suppose that the sticky Kakeya conjecture in
$\mathbb{R}^{n}$ was false, i.e. $\inf\dim_{H}K=n-\sigma_{n}$, where the
infimum is taken over all sticky Kakeya sets in $\mathbb{R}^{n}$, and
$\sigma_{n}>0$. Let $K$ be a sticky Kakeya set with dimension
$n-\sigma_{n}+\varepsilon$ for some small $\varepsilon>0$ (we will call this
an $\varepsilon$-extremal sticky Kakeya set), and let $E$ be the
discretization of $K$ at a small scale $\delta>0$. Then $E$ contains a union
of $1\times\delta$-tubes, which are sticky in the sense of Katz, Łaba, and
Tao. Furthermore, this collection of $\delta$-tubes is coarsely self-similar:
for each intermediate scale $\rho\in(\delta,1)$, the $\rho$-thickening of $K$
again resembles an $\varepsilon$-extremal sticky Kakeya set, and the union of
$\delta$-tubes inside each $\rho$-tube resembles a (anisotropically re-scaled)
$\varepsilon$-extremal sticky Kakeya set. The precise statement is given by
Proposition 3.2.

In Section 4 we restrict attention to the case $n=3$ and again follow
arguments proposed by Katz and Tao in [29]. We show that extremal sticky
Kakeya sets must have a structural property called planiness. Specifically,
with $K$ and $E$ as above, if $\mathbb{T}$ is the set of $1\times\delta$ tubes
contained in $E$, then there is a function $V\colon E\to S^{2}$ so that for
each point $x\in E$, the tubes in $\mathbb{T}$ containing $x$ must make small
angle with the subspace $V(x)^{\perp}$. $V(x)$ is called a plane map, and we
use the multi-scale self-similarity of extremal Kakeya sets to show that $V$
must be Lipschitz. An important consequence is that inside each ball of radius
$\delta^{1/2}$, the set $E$ can be decomposed into a disjoint union of
parallel rectangular prisms of dimensions roughly
$\delta^{1/2}\times\delta^{1/2}\times\delta$; these rectangular prisms are
called grains. After an anisotropic re-scaling, we obtain a new collection of
$1\times\delta$ tubes whose union $E^{\prime}\subset\mathbb{R}^{3}$ satisfies
the following Property (P): For each $z_{0}\in[0,1]$, the slice
$E^{\prime}\cap\\{z=z_{0}\\}$ is a union of parallel $1\times\delta$
rectangles whose spacing forms an Ahlfors-regular set of dimension
$1-\sigma_{3}$; the direction of these rectangles is determined by a “slope
function” $f(z_{0})$, which is Lipschitz.

Step 2: Regularity of the slope function
In Section 5, we show that the slope function $f(z)$ described above is
$C^{2}$ (or more precisely, it agrees at scale $\delta$ with a function that
has controlled $C^{2}$ norm). To do this, it suffices to show that at every
scale $\rho\in(\delta,1)$ and for every interval $I$ of length $\rho^{1/2}$,
the graph of $f$ above $\rho$ can be contained in the $\rho$-neighborhood of a
line. We will briefly describe how this is done. Fix a scale $\rho$ and an
interval $I$, and let $F\subset[0,1]^{2}$ be a re-scaled copy of the graph of
$f$ above $I$. We will also construct a set $G\subset[0,1]^{2}$, which is the
graph of a Lipschitz function $g\colon[0,1]\to\mathbb{R}$ that encodes a local
analogue of the plane map $V$. We construct the set $G$ so that certain closed
paths inside $E^{\prime}$ are encoded by arithmetic operations on the sets $F$
and $G$. Specifically, whenever we select points $p\in F$ and $q,q^{\prime}\in
G$, we have that $p\cdot(q-q^{\prime})$ specifies the location of one of the
$1\times\delta$ rectangles in the set $E^{\prime}\cap\\{z=z_{0}\\}$ that was
described above in Property (P). Since the spacing of these rectangles form an
Ahlfors-regular set, we conclude that

$F\cdot(G-G)\quad\textrm{is contained in an Ahlfors regular set of dimension}\
1-\sigma_{3}.$ (1.2)

Recall that $F$ and $G$ are the graphs of Lipschitz functions, and thus
satisfy a non-concentration condition analogous to having Hausdorff dimension
1. Is it possible for (1.2) to occur? If $F$ and $G$ are contained in
orthogonal lines, then $F\cdot(G-G)$ is a point, so the answer is yes. The
next result says that this is the only way that (1.2) can occur.

######  Theorem 5.2, informal version.

Let $F$ and $G$ be one-dimensional sets that have been discretized at a small
scale $\rho>0$. Then either (A): (most of) $F$ and $G$ are contained in the
$\rho$-neighborhoods of orthogonal lines, or (B): there exists some scale
$\rho\leq t<\\!\\!<1$ so that the $t$-neighborhood of $F\cdot(G-G)$ contains
an interval.

The precise version of Theorem 5.2 is stated in Section 5 and proved in
Section 8. Theorem 5.2 appears to be a result in additive combinatorics, but
our proof of Theorem 5.2 uses recent ideas from projection theory that were
developed in the context of the Falconer distance problem [23, 27, 24]. In
brief, if $G$ is not contained in a line, then by an analogue of Beck’s
theorem due to Orponen, Shmerkin, and the first author [24], we might expect
$G$ to span a two-dimensional set of lines, and hence a one-dimensional set of
directions, i.e. there is a one-dimensional set $\Theta\subset S^{1}$ so that
for each $\theta\in\Theta$, there are points $q,q^{\prime}\in G$ with
$\theta=\frac{q-q^{\prime}}{|q-q^{\prime}|}$. If this happens, then Kaufman’s
projection theorem says that there exists a direction $\theta\in\Theta$ for
which $F\cdot\theta$ (and hence $F\cdot(q-q^{\prime})$ ) has dimension 1. The
actual proof of Theorem 5.2 must overcome several difficulties when executing
the above strategy. First, Theorem 5.2 has weaker hypotheses than the ones
stated above; the set $G\cdot(F-F)$ is replaced by a smaller set, where the
differences and dot products are taken along a sparse (but not too sparse!)
subset of $F\times G\times G$. Second, the proof sketch above supposes a
dichotomy, where either $G$ is contained in a line, or it spans a one-
dimensional set of directions. In the discretized setting, however, this
dichotomy is less apparent because $G$ can behave differently at different
scales.

Returning to the proof of Theorem 1.1, if Item (B) from Theorem 5.2 holds,
then by an observation of Dyatlov and the second author [10, Proposition
6.13], $F\cdot(G-G)$ cannot be contained in an Ahlfors-regular set of
dimension strictly smaller than 1. Thus Theorem 5.2 implies that the graph of
$f$ above $I$ must be contained in the $\rho$ neighborhood of a line. Since
this holds for every interval $I$ at every scale $\rho$, we conclude that $f$
is $C^{2}$.

Step 3: Twisted projections
In Section 6 we exploit the fact that lines in a Kakeya set point in different
directions to show that the slope function $f$ has (moderately) large
derivative, i.e. $|f^{\prime}|$ is bounded away from 0.

In Section 7, we begin by observing the following consequence of Property (P)
from Step 1 above. Since each slice $E^{\prime}\cap\\{z=z_{0}\\}$ is a union
of parallel $1\times\delta$ rectangles whose spacing forms an Ahlfors regular
set and whose slope is given by $f(z_{0})$, if we define
$\pi_{f}(x,y,z)=\big{(}x+yf(z),\ z\big{)}$, then
$\pi_{f}(E^{\prime})\subset[0,1]^{2}$ has measure roughly
$\delta^{\sigma_{3}}$. This is because each slice
$\pi_{f}(E^{\prime})\cap\\{z=z_{0}\\}$ is a union of $\delta$-intervals that
are arranged like a $1-\sigma_{3}$ dimensional Ahlfors regular set.

If $T$ is a $1\times\delta$ tube, then $\pi_{f}(T)$ is the
$\delta$-neighborhood of a $C^{2}$ curve. Since the tubes in $\mathbb{T}$
point in different directions and since $|f^{\prime}|$ is bounded away from 0,
the corresponding curves $\pi_{f}(T)$ satisfy a separation property related to
Sogge’s cinematic curvature condition [21, 28]. The next result says that the
union of these curves must be large

######  Proposition 7.1, informal version.

Let $\mathbb{T}$ be a set of $1\times\delta$ tubes pointing in
$\delta$-separated directions, with $\\#\mathbb{T}\gtrsim\delta^{-2}$. Let
$f\colon[0,1]\to\mathbb{R}$ with $\|f\|_{C^{2}}\lesssim 1$ and
$|f^{\prime}|\sim 1$. Then for all $\varepsilon>0$, there exists
$c_{\varepsilon}>0$ so that $\bigcup_{T\in\mathbb{T}}\pi_{f}(T)$ has measure
at least $c_{\varepsilon}\delta^{\varepsilon}$.

The precise version of Proposition 7.1 is a maximal function estimate
analogous to (1.1). It is proved using a variant of Wolff’s $L^{3}$ bound for
the Wolff circular maximal function [32], where circles are replaced by a
family of $C^{2}$ curves that satisfy Sogge’s cinematic curvature condition.
This latter result was recently proved by Pramanik, Yang, and the second
author [25]. Since $E^{\prime}$ contains about $\delta^{-2}$ tubes that point
in $\delta$-separated directions, Proposition 7.1 says that
$\pi_{f}(E^{\prime})$ must have measure at least
$c_{\varepsilon}\delta^{\varepsilon}$, for each $\varepsilon>0$. On the other
hand, Property (P) says that $\pi_{f}(E^{\prime})$ has measure at most
$\delta^{\sigma_{3}}$. We conclude that $\sigma_{3}=0$, which finishes the
proof.

###  1.2 The sticky Kakeya conjecture versus the Kakeya conjecture

In this section, we will informally discuss what Theorem 1.1 tells us about
the Kakeya conjecture. In particular, how close does Theorem 1.1 get us to
proving the Kakeya conjecture in $\mathbb{R}^{3}$? Katz and Tao’s arguments
from [29] begin by conjecturing that a (hypothetical) counter-example
$K\subset\mathbb{R}^{3}$ to the Kakeya conjecture in $\mathbb{R}^{3}$ must
have the structural properties stickiness, planiness, and graininess. The
results from [1] and [14] show that such a counter-example must be plainy and
grainy, but it is unclear whether $K$ must be sticky. To reformulate the
remaining part of Katz and Tao’s conjecture in our language, we need the
following definition.

######  Definition 1.2.

A Kakeya set $K\subset\mathbb{R}^{n}$ is $\varepsilon$-close to being a sticky
if there is a set of lines $L$ with packing dimension at most
$n-1+\varepsilon$ that contains at least one line in each direction, so that
$\ell\cap K$ contains a unit interval for each $\ell\in L$.

In particular, a sticky Kakeya set is $\varepsilon$-close to being sticky for
every $\varepsilon>0$. The following version of Katz and Tao’s conjecture says
that Kakeya sets that are nearly extremal are close to being sticky.

######  Conjecture 1.2.

Suppose the Kakeya conjecture in $\mathbb{R}^{n}$ is false, i.e.
$\inf\dim_{H}K=d<n$, where the infimum is taken over all Kakeya sets in
$\mathbb{R}^{n}$. Then for all $\varepsilon>0$, there exists $\eta>0$ so that
the following holds. Let $K\subset\mathbb{R}^{n}$ be a Kakeya set with
$\dim_{H}(K)\leq d+\eta$. Then $K$ is $\varepsilon$-close to being sticky.

If Conjecture 1.2 is true for $n=3$, then the following generalization of
Theorem 1.1 would imply the Kakeya conjecture in $\mathbb{R}^{3}$.

######  Theorem 1.1′.

For all $\varepsilon>0$, there exists $\eta>0$ so that the following holds.
Let $K\subset\mathbb{R}^{3}$ be a Kakeya set that is $\eta$-close to being
sticky. Then $\dim_{H}K\geq 3-\varepsilon$.

######  Remark 1.1.

A consequence of Theorem 1.1′ is that if an X-ray estimate holds in
$\mathbb{R}^{3}$ at dimension $d<3$, then there exists $c>0$ so that every
Kakeya set in $\mathbb{R}^{3}$ must have upper Minkowski dimension at least
$d+c$.

One piece of evidence in favor of Conjecture 1.2 at the time it was formulated
in [29] was that the only known example of a Kakeya-like object with Hausdorff
dimension less than 3 was the (second) Heisenberg group

$\mathbb{H}=\\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}\colon\operatorname{Im}(z_{3})=\operatorname{Im}(z_{1}\bar{z}_{2})\\}.$

$\mathbb{H}$ is a counter-example to a strong form of the Kakeya conjecture,
and it has (complex analogues of) the structural properties stickiness,
planiness, and graininess.

More recently, however, Katz and the second author [20] discovered a second
Kakeya-like object that has small volume at certain scales—this is analogous
to having Hausdorff dimension less than 3. This object is called the $SL_{2}$
example, and it is not sticky. The $SL_{2}$ example is not a counter-example
to Conjecture 1.2 (nor to the Kakeya conjecture) because it is not a Kakeya
set in $\mathbb{R}^{3}$. Nonetheless, the existence of the $SL_{2}$ example
suggests that in order to resolve the Kakeya conjecture in $\mathbb{R}^{3}$,
it will be necessary to study Kakeya sets that are far from sticky.

As a starting point for this latter program, it seems reasonable to show that
that the $SL_{2}$ example cannot be realized in $\mathbb{R}^{3}$. To make this
statement precise, define $\mathcal{L}_{SL_{2}}$ be the set of lines in
$\mathbb{R}^{3}$ that can either be written in the form
$(a,b,0)+\mathbb{R}(c,d,1)$ with $ad-bc=1$, or $(0,0,0)+\mathbb{R}(c,d,0)$.
Equivalently, $\mathcal{L}_{SL_{2}}$ is the set of horizontal lines in the
first Heisenberg group.

We conjecture that lines from this set cannot be used to construct a counter-
example to the Kakeya conjecture.

######  Conjecture 1.3.

Let $K\subset\mathbb{R}^{3}$ be compact, and suppose there is a set of lines
$L\subset\mathcal{L}_{SL_{2}}$ that contains at least one line in each
direction, so that $\ell\cap K$ contains a unit interval for each $\ell\in L$.
Then $K$ has Hausdorff and Minkowski dimension 3.

We will not discuss Conjectures 1.2 or 1.3 further. The remainder of this
paper will be devoted to proving Theorem 1.1 and 1.1′.

###  1.3 Thanks

The authors would like to thank Larry Guth, Nets Katz, Pablo Shmerkin, and
Terence Tao for helpful conversations and suggestions during the preparation
of this manuscript. Hong Wang was supported by NSF Grant DMS-2055544. Joshua
Zahl was supported by a NSERC Discovery grant.

##  2 Discretization

###  2.1 Lines, tubes and shadings

We begin by recalling some standard definitions and terminology associated
with the Kakeya problem. Through this section, we will fix an integer $n\geq
2$; all implicit constants may depend on $n$. Slightly abusing the notation we
introduced in the introduction, we will define $\mathcal{L}_{n}$ to be the set
of lines in $\mathbb{R}^{n}$ of the form $(\underline{p},0)+\mathbb{R}v$,
where $\underline{p}\in[-\frac{1}{n},\frac{1}{n}]^{n-1}$ and $v\in S^{n-1}$
has final coordinate $v_{n}\geq 1/2$. We define
$d(\ell,\ell^{\prime})=|\underline{p}-\underline{p}^{\prime}|+\angle(v,v^{\prime})$.
This definition of distance is comparable to the definition from Section 1. We
define the measure $\lambda_{n}$ on $\mathcal{L}_{n}$ to be the product
measure of $(n-1)$-dimensional Euclidean measure and normalized surface
measure on $S^{n-1}$ (the latter we denote by $\nu_{n-1}$). With these
definitions of distance and measure, the packing dimension of a set
$L\subset\mathcal{L}_{n}$ agrees with its upper modified box dimension.

For $\ell\in\mathcal{L}_{n}$ and $\delta>0$, we define the $\delta$-tube $T$
with coaxial line $\ell$ to be $N_{2n\delta}(\ell)\cap[-1,1]^{n}$, where
$N_{t}(X)$ denotes the $t$-neighborhood of $X$. This definition is slightly
nonstandard, since we use the $2n\delta$-neighborhood rather than the
$\delta$-neighborhood of $\ell$; we do this so that a cube of side-length
$\delta$ intersecting $\ell$ will be contained in the associated tube $T$.
Observe that any tube of this type has volume comparable to $\delta^{n-1}$. If
$T$ is a $\delta$-tube and $C>0$, we will use $CT$ to denote the
$C\delta$-tube with the same coaxial line. We say two $\delta$-tubes
$T,T^{\prime}$ are _essentially identical_ if their respective coaxial lines
satisfy $d(\ell,\ell^{\prime})\leq\delta$; otherwise $T$ and $T^{\prime}$ are
_essentially distinct_. We say that $T$ and $T^{\prime}$ are _essentially
parallel_ if their respective coaxial lines satisfy
$|\operatorname{dir}(\ell)-\operatorname{dir}(\ell^{\prime})|\leq\delta$,
where $\operatorname{dir}(\ell)$ is the unit vector $v$ described above.

A $\delta$-_cube_ is a set of the form $[0,\delta)^{n}+p$, where
$p\in(\delta\mathbb{Z})^{n}.$ A _shading_ of a $\delta$-tube $T$ is a set
$Y(T)\subset T$ that is a union of $\delta$-cubes. We will write
$(\mathbb{T},Y)_{\delta}$ to denote a set of $\delta$ tubes and their
associated shadings $\\{Y(T)\colon T\in\mathbb{T}\\}$. If
$(\mathbb{T},Y)_{\delta}$ is a set of tubes and their shadings, we write
$E_{\mathbb{T}}$ to denote the set $\bigcup_{T\in\mathbb{T}}Y(T)$; the shading
$Y$ will always be apparent from context.

We say that $(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ is a _sub-collection_
of $(\mathbb{T},Y)_{\delta}$ if $\mathbb{T}^{\prime}\subset\mathbb{T}$ and
$Y^{\prime}(T^{\prime})\subset Y(T^{\prime})$ for each
$T^{\prime}\in\mathbb{T}^{\prime}$. We say that
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ is a _refinement_ of
$(\mathbb{T},Y)_{\delta}$ if in addition,
$\sum_{T^{\prime}\in\mathbb{T}^{\prime}}|Y^{\prime}(T)|\geq(\log(1/\delta))^{-C}\sum_{T\in\mathbb{T}}|Y(T)|$,
where $C\geq 0$ is a constant that depends only on $n$. We will denote this by
$\sum_{T^{\prime}\in\mathbb{T}^{\prime}}|Y^{\prime}(T)|\gtrapprox_{\delta}\sum_{T\in\mathbb{T}}|Y(T)|.$
In practice, we will only study collections of tubes $(\mathbb{T},Y)_{\delta}$
with $\\#\mathbb{T}\leq\delta^{-100n}$. By dyadic pigeonholing, every
collection of tubes with this property has a refinement
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ with
$\sum_{T\in\mathbb{T}^{\prime}}\chi_{Y^{\prime}(T)}=\mu\chi_{E_{\mathbb{T}^{\prime}}}$
for some integer $\mu\geq 1$; this is called a _constant multiplicity_
refinement of $(\mathbb{T},Y)_{\delta}$.

Let $T$ be a $\delta$-tube and let $\tilde{T}$ be a $\rho$-tube, with
$\delta\leq\rho$. We say that $\tilde{T}$ _covers_ $T$ if their respective
coaxial lines satisfy $d(\ell,\tilde{\ell})\leq\rho/2$. If $\mathbb{T}$ is a
collection of $\delta$-tubes and $\tilde{\mathbb{T}}$ is a collection of
$\rho$-tubes, we say $\tilde{\mathbb{T}}$ covers $\mathbb{T}$ if every tube in
$\mathbb{T}$ is covered by some tube in $\tilde{\mathbb{T}}$. If this is the
case, we write $\mathbb{T}[\tilde{T}]$ to denote the set of tubes in
$\mathbb{T}$ that are covered by $\tilde{T}$. A similar definition holds for
collections of tubes and their associated shadings: we say that
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ _covers_ $(\mathbb{T},Y)_{\delta}$ if
$\tilde{\mathbb{T}}$ covers $\mathbb{T}$, and for each
$\tilde{T}\in\tilde{\mathbb{T}}$ and each $T\in\mathbb{T}[\tilde{T}]$ we have
$Y(T)\subset\tilde{Y}(\tilde{T})$. Such a cover is called _balanced_ if
$|E_{\mathbb{T}}\cap Q|$ is the same for each $\rho$-cube $Q\subset
E_{\tilde{\mathbb{T}}}$

###  2.2 Defining $\sigma_{n}$

In this section we will define a quantity, called $\sigma_{n}$, that will
allow us to formulate a discretized version of the sticky Kakeya conjecture.

######  Definition 2.1.

For $s,t,\delta\in(0,1)$, define

$M(s,t,\delta)=\inf_{\mathbb{T},Y}\Big{|}\bigcup_{T\in\mathbb{T}}Y(T)\Big{|}.$
(2.1)

Here, the infimum is taken over all pairs $(\mathbb{T},Y)_{\delta}$ with the
following properties.

  1. (a)

The tubes in $\mathbb{T}$ are essentially distinct.

  2. (b)

For each $\delta\leq\rho\leq 1$, $\mathbb{T}$ can be covered by a set of
$\rho$ tubes, at most $\delta^{-t}$ of which are pairwise parallel.

  3. (c)

$\sum_{T\in\mathbb{T}}|Y(T)|\geq\delta^{s}$

Next, let

$N(s,t)=\limsup_{\delta\to 0^{+}}\frac{\log M(s,t,\delta)}{\log\delta}.$ (2.2)

$N(s,t)$ is defined for all $(s,t)\in(0,1)^{2}$. Observe that for
$\delta\in(0,1)$, if $s\leq s^{\prime}$ and $t\leq t^{\prime}$ and, then
$M(s,t,\delta)\geq M(s^{\prime},t^{\prime},\delta)$, and hence $N(s,t)\leq
N(s^{\prime},t^{\prime})$. In particular, since $0\leq N(s,t)\leq n$, we have
that

$\lim_{(s,t)\to(0,0)}N(s,t)=\inf_{(s,t)\in(0,1)^{2}}N(s,t).$

Denote this common value by $\sigma_{n}$. Our definition of $\sigma_{n}$ was
chosen so that two properties hold. First, $\sigma_{n}$ allows us to bound the
Hausdorff dimension of sticky Kakeya sets. This will be described in Section
2.3. Second, collections $(\mathbb{T},Y)_{\delta}$ that nearly extremize the
infimum in (2.1) will have useful structural properties at many scales. This
will be described in Section 3.

###  2.3 $\sigma_{n}$ and the dimension of sticky Kakeya sets

In this section we will relate $\sigma_{n}$ to the dimension of sticky Kakeya
sets. Our goal is to prove the following.

######  Proposition 2.1.

For all $\varepsilon>0$, there exists $\eta>0$ so that the following holds.
Let $K\subset\mathbb{R}^{n}$ be a Kakeya set that is $\eta$-close to being
sticky. Then $\dim_{H}K\geq n-\sigma_{n}-\varepsilon$. In particular, if $K$
is a sticky Kakeya set then $\dim_{H}K\geq n-\sigma_{n}$.

Our first task is to analyze families of lines in $\mathbb{R}^{n}$ that point
in many directions and have packing dimension close to $n-1$. For each
$\rho>0$ and $\varepsilon>0$, define the class of “Quantitatively Sticky” sets
$\operatorname{QStick}(n,t,\rho)$ to be the collection of sets
$L\subset\mathcal{L}_{n}$ that satisfy

$\Big{|}\big{\\{}\underline{p}\in\mathbb{R}^{n-1}\colon(\underline{p},v)\in
N_{\delta}(L)\big{\\}}\Big{|}\leq\delta^{n-1-t}\quad\textrm{for all}\ v\in
S^{n-1},\ \delta\in(0,\rho),$

where in the above expression, $N_{\delta}(L)$ denotes the
$\delta$-neighborhood of $L$ in the metric space $\mathcal{L}_{n}$.

For example, if
$L=\\{(\underline{p},v)\in\mathcal{L}_{n}\colon\underline{p}=f(v)\\}$ for some
Lipschitz function $f$, then for each $t>0$ we have
$L\in\operatorname{QStick}(n,t,\rho)$ for all sufficiently small $\rho$. The
next lemma explains how $\operatorname{QStick}(n,t,\rho)$ is related to sticky
Kakeya sets.

######  Lemma 2.2.

Let $L\subset\mathcal{L}_{n}$ with $\nu_{n-1}(\operatorname{dir}(L))>0$. Then
for all $t>\dim_{P}(L)-(n-1)$, there exists $L^{\prime}\subset L$ and $\rho>0$
with $\nu_{n-1}(\operatorname{dir}(L^{\prime}))>0$ and
$L^{\prime}\in\operatorname{QStick}(n,t,\rho)$.

###### Proof.

Let $d=\dim_{P}L$, and let $\varepsilon=t-d+(n-1)$. Since
$d=\dim_{P}(L)=\overline{\dim}_{MB}(L)$, we can write $L$ as a countable union
of sets $\\{L_{i}\\}$, with $\overline{\dim}_{M}L_{i}\leq d+\varepsilon/4$ for
each $i$. Since
$\nu_{n-1}\big{(}\bigcup\operatorname{dir}(L_{i})\big{)}=\nu_{n-1}(\operatorname{dir}(L))>0$,
there exists an index $i$ so that $\nu_{n-1}(\operatorname{dir}L_{i})>0$. Re-
indexing if necessary, we can suppose that
$\nu_{n-1}(\operatorname{dir}L_{1})>0$. Since $\overline{\dim}_{M}L_{1}\leq
d+\varepsilon/4$, we can select $\rho_{1}>0$ sufficiently small so that

$\lambda_{n}\big{(}N_{\delta}(L_{1})\big{)}\leq\delta^{(2n-2)-d-\varepsilon/2}\quad\textrm{for
all}\ \delta\in(0,\rho_{1}).$ (2.3)

We say a direction $v\in S^{n-1}$ is _over-represented_ at scale $\delta$ if

$\Big{|}\big{\\{}\underline{p}\in\mathbb{R}^{n-1}\colon(\underline{p},v)\in
N_{2\delta}(L_{1})\big{\\}}\Big{|}>\delta^{n-1-t}=\delta^{(2n-2)-\delta-\varepsilon}.$

By (2.3) and Fubini (recall that $\lambda_{n}$ is a product measure on
$\mathbb{R}^{n-1}\times S^{n-1}$), for each $\delta\in(0,\rho_{1})$, we have

$\nu_{n-1}\big{(}\\{v\in S^{n-1}\colon v\ \textrm{is over-represented at
scale}\ \delta\\}\big{)}\lesssim\delta^{\varepsilon/2}.$ (2.4)

Thus for each integer $k_{1}$ with $2^{-k_{1}}\leq\rho_{1}$, we have

$\sum_{k\geq k_{1}}\nu_{n-1}\big{(}\\{v\colon v\ \textrm{is over-represented
at scale}\ 2^{-k}\\}\big{)}\lesssim k_{1}^{-\varepsilon/2}.$ (2.5)

Selecting $k_{1}$ appropriately, we can ensure that the LHS of (2.5) is at
most $\nu_{n-1}(\operatorname{dir}(L_{1}))/2$. Note that if $v\in S^{n-1}$ is
not over-represented at any dyadic scale $2^{-k}\leq 2^{-k_{1}}$, then

$\Big{|}\big{\\{}\underline{p}\in\mathbb{R}^{n-1}\colon(\underline{p},v)\in
N_{\delta}(L_{1})\big{\\}}\Big{|}\leq\delta^{n-1-t}\quad\textrm{for all}\ \
\delta\in(0,2^{-k_{1}}).$

To conclude the proof, let $\rho=2^{-k_{1}}$ and let

$L^{\prime}=L_{1}\ \backslash\
\bigcup_{k=k_{1}}^{\infty}\operatorname{dir}^{-1}\big{(}\\{v\colon v\
\textrm{is over-represented at scale}\ 2^{-k}\\}\big{)}.\qed$

With this lemma, Proposition 2.1 now follows from standard discretization
arguments. The details are as follows.

###### Proof of Proposition 2.1.

Let $\varepsilon>0$ be given. By the definition of $\sigma_{n}$, there exists
$s,t\in(0,1)$ so that

$N(s,t)\leq\sigma_{n}+\varepsilon/2.$ (2.6)

We will show that Proposition 2.1 holds with $\eta=t/4$.

Let $K$ be a Kakeya set that is $\eta$-close to being sticky. Since $K$ is
compact, after a translation and dilation we can suppose that
$K\subset[-\frac{1}{n},\frac{1}{n}]^{n}$. In particular, there exists a number
$w>0$ and set $L\subset\mathcal{L}_{n}$ with $\dim_{P}(L)\leq n-1+\eta$ so
that for each unit vector $v\in S^{n-1}$ with final coordinate $v_{n}\geq
1/2,$ there is a line $\ell\in L$ with $\operatorname{dir}(\ell)=v$ and
$|\ell\cap K|\geq w$. Use Lemma 2.2 to select a subset $L^{\prime}\subset L$
and a number $\rho_{1}>0$ with
$L^{\prime}\in\operatorname{QStick}(n,t/2,\rho_{1})$ and
$\nu_{n-1}(\operatorname{dir}(L^{\prime}))>0$.

Let $k_{1}$ be an integer to specified below, with
$2^{-k_{1}}<\rho_{1}^{2n/t}$. By the definition of Hausdorff dimension, we can
cover $K$ by a union of cubes
$K\subset\bigcup_{k=k_{1}}^{\infty}\bigcup_{Q\in\mathcal{Q}_{k}}Q$, where
$\mathcal{Q}_{k}$ is a set of $2^{-k}$-cubes with

$\\#\mathcal{Q}_{k}\lesssim(2^{k})^{\dim_{H}K+\varepsilon/2}.$ (2.7)

For each $\ell\in L^{\prime}$, we have

$K\cap\ell\ \subset\
\bigcup_{k=k_{1}}^{\infty}\Big{(}\ell\cap\bigcup_{Q\in\mathcal{Q}_{k}}Q\Big{)}.$

Since $|K\cap\ell|\geq w$, there exists at least one index $k$ so that

$\Big{|}\ell\cap\bigcup_{Q\in\mathcal{Q}_{k}}Q\Big{|}\geq\frac{w}{100k^{2}}.$
(2.8)

For each $k\geq k_{1}$, let $L_{k}$ be the set of lines $\ell\in L^{\prime}$
for which (2.8) holds for that choice of $k$. Then
$L^{\prime}=\bigcup_{k}L_{k}$, so there exists an index $k_{2}\geq k_{1}$ with
$\nu_{n-1}(\operatorname{dir}(L_{k_{2}}))\geq\frac{\nu_{n-1}(\operatorname{dir}(L^{\prime}))}{100k_{2}^{2}}$.

Let $\delta=2^{-k_{2}}$, so in particular $\frac{1}{k_{2}}\geq\frac{\log
2}{\log(1/\delta)}$. Let $\Omega$ be a $\delta$-separated subset of
$\operatorname{dir}(L_{k_{2}})$ of cardinality
$\gtrsim\frac{\nu_{n-1}(\operatorname{dir}L^{\prime})\delta^{1-n}}{\log(1/\delta)^{2}}$.
For each direction $v\in\Omega$, let $T$ be a $\delta$-tube with coaxial line
$\ell\in L_{k_{2}}$ pointing in direction $v$. By (2.8), we can choose $T$ so
that the set

$Y(T)=\bigcup_{\begin{subarray}{c}Q\in\mathcal{Q}_{k_{2}}\\\
Q\cap\ell\neq\emptyset\end{subarray}}Q$

satisfies $Y(T)\subset T$ and $|Y(T)|\gtrsim
w\log(1/\delta)^{-2}\delta^{n-1}$.

We claim that if $k_{1}$ is chosen sufficiently large, then
$(\mathbb{T},Y)_{\delta}$ satisfies Items (a), (b), and (c) from Definition
2.1. Item (a) is immediate, since $\Omega$ is $\delta$-separated. Item (c)
follows from the inequality

$\sum_{T\in\mathbb{T}}|Y(T)|\gtrsim\frac{(\\#\Omega)w\delta^{n-1}}{\log(1/\delta)^{2}}\gtrsim\frac{w\nu_{n-1}(\operatorname{dir}L^{\prime})}{\log(1/\delta)^{4}}.$

This quantity is larger than $\delta^{s}$ provided we select $k_{1}$
sufficiently small large, depending on
$\nu_{n-1}(\operatorname{dir}L^{\prime}),$ $w$, and $n$.

Our final task is to show that $(\mathbb{T},Y)_{\delta}$ satisfies Item (b).
Let $\rho\in[\delta,1]$. Our goal is to find a set of $\rho$-tubes
$\tilde{\mathbb{T}}$ that covers $\mathbb{T}$, so that at most $\delta^{-t}$
of them are pairwise parallel. We will divide our analysis into cases.

Case 1: $\rho\in[\delta,\rho_{1}).$ let $\tilde{\mathbb{T}}$ be a collection
of $\rho$-tubes, so that the set of coaxial lines $\tilde{L}$ has the
following properties.

  1. (i)

Each $\tilde{\ell}\in\tilde{L}$ is contained in $N_{\rho}(L^{\prime})$.

  2. (ii)

The $\rho/2$-neighborhoods of the lines in $\tilde{L}$ cover $L^{\prime}$.

  3. (iii)

The $\rho/6$ neighborhoods of the lines in $\tilde{L}$ are disjoint.

Such a collection of tubes can be chosen greedily, as in the proof of the
Vitali covering lemma.

Each tube $T\in\mathbb{T}$ has a coaxial line $\ell\in L_{1}\subset
L^{\prime}$. Hence there is a tube $\tilde{T}\in\tilde{\mathbb{T}}$ so that
$d(\ell,\tilde{\ell})\leq\rho/2,$ and thus $\tilde{T}$ covers $T$. We conclude
that $\tilde{\mathbb{T}}$ covers $\mathbb{T}$.

It remains to show that at most $\rho^{-t}$ tubes from $\tilde{\mathbb{T}}$
can be pairwise parallel. Let
$\tilde{T}_{1},\ldots\tilde{T}_{N}\in\tilde{\mathbb{T}}$ be pairwise parallel,
and let $v=\operatorname{dir}(\tilde{T}_{1})$. In particular,
$|v-\operatorname{dir}(\tilde{T}_{j})|\leq\rho$ for each index $j$. By Item
(i) above, this means $\tilde{T}_{j}\in
N_{\rho}(L^{\prime})\cap\operatorname{dir}^{-1}(N_{\rho}(v))$. By item (iii)
above, we conclude that

$\lambda_{n}\big{(}N_{\rho}(L^{\prime})\cap\operatorname{dir}^{-1}(N_{\rho}(v))\big{)}\geq\sum_{i=1}^{N}\lambda_{n}\big{(}N_{\rho/6}(\tilde{\ell}_{i})\big{)}\gtrsim
N\rho^{2n-2}.$

Thus by Fubini, there exists $v^{\prime}\in N_{\rho}(v)$ with

$|N_{\rho}(L^{\prime})\cap\operatorname{dir}^{-1}(v^{\prime})|\gtrsim
N\rho^{n-1}.$

But since $L^{\prime}\in\operatorname{QStick}(n,t/2,\rho_{1})$ and
$\rho<\rho_{1}$, if $\rho_{1}>0$ is selected sufficiently small (depending on
the implicit constants above, which in turn depend on $n$) then
$N\leq\rho^{-t}\leq\delta^{-t}$.

Case 2: $\rho\in[\rho_{1},1]$. Let $\tilde{\mathbb{T}}$ be a set of
$\rho$-tubes, whose corresponding coaxial lines form a $\rho/2$-net in
$\mathcal{L}_{n}$. Then $\tilde{\mathbb{T}}$ clearly covers $\mathbb{T}$. The
maximum number of pairwise parallel tubes in $\tilde{\mathbb{T}}$ is at most
$\\#\tilde{\mathbb{T}}\sim\rho^{2-2n}$, and this is $\leq\delta^{-t}$
(provided $\rho_{1}$, and hence $\delta$ is selected sufficiently small
compared to $n$), since $\delta\leq 2^{-k_{1}}\leq\rho_{1}^{2n/t}$.

We have shown that $(\mathbb{T},Y)_{\delta}$ satisfies Items (a), (b), and (c)
from Definition 2.1. Thus we have that

$M(s,t,\delta)\leq\Big{|}\bigcup_{T\in\mathbb{T}}Y(T)\Big{|}\leq\Big{|}\bigcup_{Q\in\mathcal{Q}_{k_{2}}}Q\Big{|}\leq(\\#\mathcal{Q}_{k_{2}})\delta^{n}\leq\delta^{n-\dim
K-\varepsilon/2},$

where the final inequality used (2.7). We conclude that

$\frac{\log M(s,t,\delta)}{\log\delta}\geq n-\dim_{H}K-\varepsilon/2.$ (2.9)

We have just shown that for every $k_{1}$ sufficiently large, there exists
$\delta\in(0,2^{-k_{1}}]$ for which (2.9) holds. In particular,

$N(s,t)\geq n-\dim_{H}K-\varepsilon/2.$ (2.10)

To conclude the proof, we compare (2.6) and (2.10). ∎

Proposition 2.1 says that Theorems 1.1 and 1.1′ will follow from the estimate
$\sigma_{3}=0$.

##  3 Extremal families of tubes and multi-scale structure

In this section, we will study families of tubes that nearly extremize the
quantity $M$ from Definition 2.1.

######  Definition 3.1.

Let $\varepsilon,\delta>0$. We say a pair $(\mathbb{T},Y)_{\delta}$ is
$\varepsilon$-extremal if it satisfies Items (a), (b), and (c) from Definition
2.1 with $s=t=\varepsilon$, and furthermore
$\Big{|}\bigcup_{T\in\mathbb{T}}Y(T)\Big{|}\leq\delta^{\sigma_{n}-\varepsilon}$.

From the definition of $\sigma_{n}$, we immediately conclude that extremal
collections of tubes always exist for small $\delta$. More precisely, we have
the following.

######  Lemma 3.1.

Let $\varepsilon,\delta_{0}>0$. Then there exists a $\varepsilon$-extremal
collection of tubes $(\mathbb{T},Y)_{\delta}$ for some
$\delta\in(0,\delta_{0}]$.

The main result of this section says that extremal collections of tubes must
be coarsely self-similar at every scale. To state this precisely, we need the
following definition.

######  Definition 3.2.

Let $(\mathbb{T},Y)_{\delta}$ be a collection of $\delta$ tubes and let
$\tilde{T}$ be a $\rho$ tube that covers each tube in $\mathbb{T}$. The _unit
rescaling of $(\mathbb{T},Y)_{\delta}$ relative to $\tilde{T}$_ is the pair
$(\hat{\mathbb{T}},\hat{Y})_{\delta/\rho}$ defined as follows. The coaxial
lines of the tubes $\hat{T}\in\hat{\mathbb{T}}$ are the images of the coaxial
lines of the tubes $T\in\mathbb{T}$ under the map
$\phi\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$, defined as follows. If
$\tilde{\ell}$ is the coaxial line of $\tilde{T}$, then $\phi$ is the
composition of a rigid transformation that sends the point
$\tilde{\ell}\cap\\{x_{n}=0\\}$ to the origin and sends the line
$\tilde{\ell}$ to the $x_{n}$-axis, with the anisotropic dilation

$(x_{1},\ldots,x_{n})\mapsto\big{(}cx_{1}/\rho,\
cx_{2}/\rho,\ldots,cx_{n-1}/\rho,\ cx_{n}\big{)}.$ (3.1)

Each shading $\hat{Y}(\hat{T})$ is the union of $\delta/\rho$ cubes that
intersect $\phi(Y(T))$. The constant $c=c(n)\sim 1$ is chosen so that the
coaxial lines of the tubes in $\hat{T}$ are in $\mathcal{L}_{n}$, and
$\hat{Y}(\hat{T})\subset\hat{T}$. See Figure 1

Figure 1: The unit rescaling of $(\mathbb{T},Y)_{\delta}$. For clarity, only a
few cubes in $Y(T)$ have been drawn.

We can now state the main result of this section. It says that extremal
collections of tubes look self-similar at all scales.

######  Proposition 3.2.

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 $p\in\mathbb{R}^{n}$,
$\\#\tilde{\mathbb{T}}(p)\leq\rho^{-\sigma_{n}-\varepsilon}$

  4. (iv)

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

Before proving Proposition 3.2, we will analyze the unit rescaling of extremal
collections of tubes that are covered by a single $\rho$-tube. A precise
formulation is given below. In the arguments that follow, we will make
frequent use of the fact that if $(\mathbb{T},Y)_{\delta}$ is
$\varepsilon$-extremal, then
$\delta^{1-n+\varepsilon}\lesssim\\#\mathbb{T}\lesssim\delta^{1-n-\varepsilon},$
and if $(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ is a sub-collection with
$\sum_{T\in\mathbb{T}^{\prime}}|Y^{\prime}(T)|\geq\delta^{\varepsilon^{\prime}}$
for some $\varepsilon^{\prime}\geq\varepsilon$, then
$(\mathbb{T}^{\prime},Y^{\prime})$ is $\varepsilon^{\prime}$-extremal.

######  Lemma 3.3.

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 tubes that are covered by a $\rho$-tube
$\tilde{T}$, with $\rho\in[\delta^{1-\varepsilon},1]$.

Suppose that $(\mathbb{T},Y)_{\delta}$ satisfies Items (a) and (b) from
Definition 2.1 with $\eta$ in place of $s$, and that$(\mathbb{T},Y)_{\delta}$
satisfies the following analogue of Item (c):
$\sum_{T\in\mathbb{T}}|Y(T)|\geq\delta^{\eta}\rho^{n-1}$.

Then there is a refinement $(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ so that
the unit rescaling of $(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ relative to
$\tilde{T}$ satisfies Items (a), (b), and (c) from Definition 2.1 with
$\varepsilon$ in place of $s$ and $t$. Furthermore,

$\\#\mathbb{T}^{\prime}(p)\leq(\rho/\delta)^{\sigma+\varepsilon}\quad\textrm{for
all}\ p\in\mathbb{R}^{n}.$ (3.2)

###### Proof.

To begin, observe that the transformation $\phi$ from Definition 3.2 has the
following properties.

  * •

If $\ell_{1},\ell_{2}$ are lines with $d(\tilde{\ell},\ell_{i})\leq\rho$ for
$i=1,2$. Then

$\begin{split}\rho^{-1}d(\ell_{1},\ell_{2})\lesssim d(\phi(\ell_{1}),&\
\phi(\ell_{2}))\leq\rho^{-1}d(\ell_{1},\ell_{2}),\\\
\rho^{-1}\angle\big{(}\operatorname{dir}(\ell_{1}),\
\operatorname{dir}(\ell_{2})\big{)}\lesssim\angle\big{(}\operatorname{dir}(\phi(\ell_{1})),\
&\operatorname{dir}(\phi(\ell_{2}))\big{)}\leq\rho^{-1}\angle\big{(}\operatorname{dir}(\ell_{1}),\
\operatorname{dir}(\ell_{2})\big{)}.\end{split}$ (3.3)

  * •

If $\tau\in[\delta,\rho]$ and $T^{\dagger}$ is a $\tau$-tube covered by
$2\tilde{T}$, then $\phi(T^{\dagger})$ is contained in the $\tau/\rho$ tube
with coaxial line $\phi(\ell^{\dagger})$.

  * •

For every measurable $X\subset\mathbb{R}^{n}$ we have

$|\phi(X)|\sim\rho^{1-n}|X|.$ (3.4)

By (3.3), there is a refinement of $(\mathbb{T},Y)_{\delta}$ whose unit
rescaling relative to $\tilde{T}$ consists of essentially distinct tubes. Let
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ be a refinement with this
property, which is also constant multiplicity, i.e. there is a number $\mu$ so
that $\\#\mathbb{T}^{\prime}(p)=\mu$ for all $p\in E_{\mathbb{T}^{\prime}}$.
Let $(\hat{\mathbb{T}},\hat{Y})_{\delta/\rho}$ be the unit rescaling of
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ relative to $\tilde{T}$. In
particular, $(\hat{\mathbb{T}},\hat{Y})_{\delta/\rho}$ satisfies Property (a)
from Definition 2.1.

By (3.4), we have

$\Big{|}\bigcup_{T\in\mathbb{T}}Y(T)\Big{|}\sim\rho^{n-1}\Big{|}\bigcup_{\hat{T}\in\hat{\mathbb{T}}}\hat{Y}(\hat{T})\Big{|}.$
(3.5)

Our next task is to obtain a lower bound on the RHS of (3.5). Let $s,t>0$ be
chosen so that

$N(s,t)\leq\sigma_{n}+\varepsilon/4.$ (3.6)

Decreasing $s$ and $t$ if necessary, we may suppose that
$s,t\leq\varepsilon/2$. We will show that
$(\hat{\mathbb{T}},\hat{Y})_{\delta/\rho}$ satisfies Properties (b) and (c)
from Definition 2.1 for this value of $s$ and $t$, and all sufficiently small
$\delta$.

Property (c) is straightforward: by (3.4) we have

$\sum_{\hat{T}\in\hat{\mathbb{T}}}|\hat{Y}(\hat{T})|\gtrsim\rho^{1-n}\sum_{T\in\mathbb{T}}|Y(T)|\geq\delta^{\eta}\geq(\rho/\delta)^{\eta/\varepsilon}.$
(3.7)

If $\eta,\delta_{0}>0$ are selected sufficiently small, then the LHS of (3.7)
is at most $(\rho/\delta)^{s}$.

For property (b), let $\tau\in(\delta/\rho,1)$. Let $\mathbb{T}^{*}$ be a
collection of $\tau\rho$-tubes that cover $\mathbb{T}$, at most
$\delta^{-\eta}$ of which are pairwise parallel (such a collection exists by
hypothesis, since $\delta\leq\tau\rho\leq 1$). We can suppose that each
$T^{*}\in\mathbb{T}^{*}$ covers at least one tube from $\mathbb{T}$, and thus
by the triangle inequality, each $T^{*}$ is covered by $2\tilde{T}$. Define
$\hat{\mathbb{T}}^{*}$ to be the set of $\tau$ tubes whose coaxial lines are
given by $\\{\phi(\ell^{*})\colon\ell^{*}\in\mathbb{T}^{*}\\}$. By (3.3), if
$T\in\mathbb{T}$ is covered by $T^{*}\in\mathbb{T}^{*}$, then the
corresponding tube $\hat{T}$ is covered by $\hat{T}^{*}$; in particular,
$\hat{\mathbb{T}}^{*}$ covers $\hat{\mathbb{T}}$. By (3.3), if $\delta_{0}$ is
chosen sufficiently small then at most $(\delta/\rho)^{-2\eta/\varepsilon}$
tubes from $\hat{\mathbb{T}}^{*}$ are pairwise parallel. Thus if $\eta>0$ is
chosen sufficiently small, then $(\hat{\mathbb{T}},\hat{Y})_{\delta/\rho}$
satisfies property (b) for the value of $t$ specified in (3.6).

We have shown that $(\hat{\mathbb{T}},\hat{Y})_{\delta/\rho}$ satisfies
Properties (a), (b), and (c) from Definition 2.1 with $\varepsilon$ in place
of $s$ and $t$.

All that remains is to establish (3.2), i.e. to bound
$\mu\leq(\rho/\delta)^{\sigma_{n}+\varepsilon}$. We have also shown that
$(\hat{\mathbb{T}},\hat{Y})_{\delta/\rho}$ satisfies Properties (a), (b), and
(c) from Definition 2.1 for the values of $s$ and $t$ specified by (3.6). Thus
if $\delta/\rho$ is sufficiently small, then

$\Big{|}\bigcup_{\hat{T}\in\hat{\mathbb{T}}}\hat{Y}(\hat{T})\Big{|}\geq(\delta/\rho)^{\sigma_{n}+\varepsilon/2}.$
(3.8)

We can ensure that $\delta/\rho$ is sufficiently small for (3.8) to hold by
selecting $\delta_{0}>0$ appropriately, since the hypothesis
$\rho\geq\delta^{1-\varepsilon}$ forces $\delta/\rho\searrow 0$ as
$\delta\searrow 0$.

On the other hand, since $(\hat{\mathbb{T}},\hat{Y})_{\delta/\rho}$ satisfies
Property (b) from Definition 2.1 with $s\leq\varepsilon/2$, we have

$\mu=\Big{(}\sum_{\hat{T}\in\hat{\mathbb{T}}}|\hat{Y}(\hat{T})|\Big{)}\Big{(}\Big{|}\bigcup_{\hat{T}\in\hat{\mathbb{T}}}\hat{Y}(\hat{T})\Big{|}\Big{)}^{-1}\lesssim(\delta/\rho)^{\varepsilon/2}\Big{(}\Big{|}\bigcup_{\hat{T}\in\hat{\mathbb{T}}}\hat{Y}(\hat{T})\Big{|}\Big{)}^{-1}\leq(\delta/\rho)^{\sigma_{n}+\varepsilon}.\qed$

###### Proof of Proposition 3.2.

Step 1: Constructing $\tilde{\mathbb{T}}$.
To begin, we will construct a set of $\rho$-tubes that covers a refinement of
$\mathbb{T}$, with the property that each $T\in\mathbb{T}$ is contained in
exactly one $\rho$-tube, and $T$ is close to the coaxial line of this
$\rho$-tube. Here are the details. Let $\tilde{\mathbb{T}}_{0}$ be a set of
$\rho/(2n)$ tubes that covers $\mathbb{T}$, so that the coaxial line of each
$\tilde{T}\in\tilde{\mathbb{T}}_{0}$ coincides with the coaxial line of a tube
from $\mathbb{T}$, and at most $O(\delta^{-\eta})$ tubes from
$\tilde{\mathbb{T}}_{0}$ are pairwise parallel. $\tilde{\mathbb{T}}_{0}$ can
be constructed by first selecting a cover of $\mathbb{T}$ as described in Item
(b) of Definition 2.1, and then replacing each tube $\tilde{T}$ in this cover
by $O(1)$ tubes $\tilde{T}^{\prime}$ with
$d(\tilde{T},\tilde{T}^{\prime})\lesssim\rho$.

Next, let $\tilde{\mathbb{T}}_{1}\subset\tilde{\mathbb{T}}_{0}$ be chosen so
that the tubes in $2n\tilde{\mathbb{T}}_{1}$ are essentially distinct; each
$T\in\mathbb{T}$ is covered by at most one tube in $2n\tilde{\mathbb{T}}_{1}$;
and

$\sum_{\tilde{T}\in\tilde{\mathbb{T}}_{1}}\sum_{T\in\mathbb{T}[\tilde{T}]}|Y(T)|\sim\sum_{T\in\mathbb{T}}|Y(T)|.$

Define $\tilde{\mathbb{T}}_{2}=2n\tilde{\mathbb{T}}_{1}$. Then
$\tilde{\mathbb{T}}_{2}$ is a set of essentially distinct $\rho$-tubes, and
hence they satisfies Property (a) from Definition 2.1. Our final set
$\tilde{\mathbb{T}}$ will be a subset of $\tilde{\mathbb{T}}_{2}$, and thus it
will continue to satisfy Property (a) from Definition 2.1.

Next, we will show that $\tilde{\mathbb{T}}_{2}$ satisfies Property (b) from
Definition 2.1 with $t=\eta/\varepsilon$. To do this, let $\tau\in[\rho,1].$
Then there is a set $\mathbb{T}^{\dagger}$ of $\tau$ tubes that covers
$\mathbb{T}$, at most $\delta^{-\eta}\leq\rho^{-\eta/\varepsilon}$ of which
are pairwise parallel. But since each $\tilde{T}\in\tilde{\mathbb{T}}_{2}$
shares a coaxial line with some $T\in\mathbb{T}$, and $\tau\geq\rho$,
$\mathbb{T}^{\dagger}$ also covers $\tilde{\mathbb{T}}_{2}.$ Our final set
$\tilde{\mathbb{T}}$ will be a subset of $\tilde{\mathbb{T}}_{2}$, and thus it
will continue to satisfy Property (b) from Definition 2.1 with this same value
of $t$.

Define $\mathbb{T}_{2}\subset\mathbb{T}$ to be the set of tubes that are
covered by some tube from $\tilde{\mathbb{T}}_{2}$, and let $Y_{2}(T)=Y(T)$
for each $T\in\mathbb{T}_{2}$. Then $(\mathbb{T}_{2},Y_{2})$ is a refinement
of $(\mathbb{T},Y)$. Furthermore, each $T\in\mathbb{T}_{2}$ is covered by
exactly one tube from $\tilde{\mathbb{T}}_{2}$; and if $\tilde{T}$ covers $T$,
then $T$ is close to the coaxial line of $\tilde{T}$, in the sense that
$\frac{1}{2n}\tilde{T}$ also covers $T$.

Step 2: Fine scale structure. In this step we will analyze the structure of
the sets $\mathbb{T}_{2}[\tilde{T}]$. By dyadic pigeonholing, we can select a
set $\tilde{\mathbb{T}}_{3}\subset\tilde{\mathbb{T}}_{2}$ and a refinement
$(\mathbb{T}_{3},Y_{3})$ of $(\mathbb{T}_{2},Y_{2})$ so that for each
$\tilde{T}\in\tilde{\mathbb{T}}_{3}$, we have

$\sum_{T\in\mathbb{T}_{3}[\tilde{T}]}|Y_{3}(T)|=(\\#\tilde{\mathbb{T}}_{3})^{-1}\sum_{T\in\mathbb{T}_{3}}|Y_{3}(T)|.$
(3.9)

Since $O(\delta^{-\eta})$ tubes from $\tilde{\mathbb{T}}_{3}$ are pairwise
parallel, we have $\\#\tilde{\mathbb{T}}_{3}\lesssim\delta^{-\eta}\rho^{1-n}$
and thus
$\sum_{T\in\mathbb{T}_{3}[\tilde{T}]}|Y_{3}(T)|\gtrapprox_{\delta}\delta^{2\eta}\rho^{n-1}$.

If $\eta$ and $\delta_{0}$ are chosen sufficiently small, then we can apply
Lemma 3.3 to each set $(\mathbb{T}_{3}[\tilde{T}],Y_{3})_{\delta}$ with
$\varepsilon^{2}/3$ in place of $\varepsilon$. Let
$(\mathbb{T}_{4},Y_{4})_{\delta}$ be the union of the refined collections
$\\{(\mathbb{T}_{3}[\tilde{T}],Y_{3})_{\delta}\colon\tilde{T}\in\tilde{\mathbb{T}}_{3}\\}$
that are the output of Lemma 3.3. Let
$\tilde{\mathbb{T}}_{4}=\tilde{\mathbb{T}}_{3}$. Then for each
$\tilde{T}\in\tilde{\mathbb{T}}_{4}$, we have

$\\#\mathbb{T}_{4}[\tilde{T}](p)\leq(\rho/\delta)^{\sigma_{n}+\varepsilon^{2}/3},$
(3.10)

so in particular $(\mathbb{T}_{4},Y_{4})_{\delta}$ and
$\tilde{\mathbb{T}}_{4}$ satisfy Item (iv) from the statement of Proposition
3.2.

Furthermore, for each $\tilde{T}\in\tilde{\mathbb{T}}_{4}$, the unit rescaling
of $(\mathbb{T}_{4}[\tilde{T}],Y_{4})_{\delta}$ relative to $\tilde{T}$
satisfies Items (a), (b), and (c) from Definition 2.1 with
$s=t=\varepsilon^{2}$. Items (a) and (b) will continue to hold if
$\mathbb{T}_{4}$ is replaced by a refinement.

Step 3: Coarse scale structure.
Thus far, we have analyzed the fine-scale behavior of the “thin” tubes
$\mathbb{T}_{4}[\tilde{T}]$ for each $\tilde{T}\in\tilde{\mathbb{T}}_{4}$. Our
next task is to analyze the coarse-scale behavior of the “fat” tubes
$\tilde{\mathbb{T}}_{4}$.

First, we will construct a shading on the fat tubes. Observe that for each
$\tilde{T}\in\tilde{\mathbb{T}}_{4}$ and each $\rho$-cube $Q$, we have that

$Q\ \cap\\!\bigcup_{T\in\mathbb{T}_{4}[\tilde{T}]}Y_{4}(T)\neq\emptyset\ \
\textrm{implies}\ \ Q\subset\tilde{T}.$ (3.11)

Indeed, as noted at the end of Step 1, if $T\in\mathbb{T}_{4}[\tilde{T}]$,
then $T$ is covered by $\frac{1}{2n}\tilde{T}$, and thus $Q$ intersects
$\frac{1}{2n}\tilde{T}$. But this implies $Q\subset\tilde{T}$.

Each $\tilde{T}\in\tilde{\mathbb{T}}_{4}$ contains $\sim\rho^{-1}$
$\rho$-cubes, and for each such cube we have

$\sum_{T\in\mathbb{T}_{4}[\tilde{T}]}|Y_{4}(T)\cap
Q|\lesssim\rho\sum_{T\in\mathbb{T}_{4}[\tilde{T}]}|T|.$

The directions $\\{\operatorname{dir}(T)\colon
T\in\mathbb{T}_{4}[\tilde{T}]\\}$ are contained in a disk of radius $\rho$ in
$S^{n-1}$. Since at most $\delta^{-\eta}$ tubes from $\mathbb{T}_{4}$ can be
pairwise parallel, we have

$\\#\mathbb{T}_{4}[\tilde{T}]\lesssim\delta^{-\eta}(\rho/\delta)^{n-1},$
(3.12)

and thus

$\sum_{T\in\mathbb{T}_{4}[\tilde{T}]}|Y_{4}(T)\cap
Q|\lesssim\delta^{-\eta}\rho^{n}.$ (3.13)

Let $c_{1}\sim 1$ be a constant to be chosen below. We say a $\rho$-cube is
_heavy_ for the tube $\tilde{T}$ if

$\sum_{T\in\mathbb{T}_{4}[\tilde{T}]}|Y_{4}(T)\cap Q|\geq
c_{1}\rho\sum_{T\in\mathbb{T}_{4}[\tilde{T}]}|Y_{4}(T)|.$

Denote the set of heavy cubes by $\mathcal{Q}_{0}(\tilde{T})$. If the constant
$c_{1}$ is chosen appropriately, then for each
$\tilde{T}\in\tilde{\mathbb{T}}_{4}$ we have

$\sum_{Q\in\mathcal{Q}_{0}(\tilde{T})}\sum_{T\in\mathbb{T}_{4}[\tilde{T}]}|Y_{4}(T)\cap
Q|\geq\frac{1}{2}\sum_{T\in\mathbb{T}_{4}[\tilde{T}]}|Y_{4}(T)|.$

By dyadic pigeonholing, there exists a weight $w$ so that if we define

$\mathcal{Q}_{1}(\tilde{T})=\Big{\\{}Q\in\mathcal{Q}_{0}(\tilde{T})\colon
w\leq\sum_{T\in\mathbb{T}_{4}[\tilde{T}]}|Y_{4}(T)\cap Q|<2w\Big{\\}},$ (3.14)

then

$\sum_{\tilde{T}\in\tilde{\mathbb{T}}_{4}}\sum_{Q\in\mathcal{Q}_{1}(\tilde{T})}\sum_{T\in\mathbb{T}_{4}[\tilde{T}]}|Y_{4}(T)\cap
Q|\gtrapprox_{\delta}\sum_{\tilde{T}\in\tilde{\mathbb{T}}_{4}}\sum_{T\in\mathbb{T}_{4}[\tilde{T}]}|Y_{4}(T)|=\sum_{T\in\mathbb{T}_{4}}|Y_{4}(T)|.$
(3.15)

Comparing (3.13) and (3.15), we have

$\sum_{\tilde{T}\in\tilde{\mathbb{T}}_{4}}\\#\mathcal{Q}_{1}(\tilde{T})\gtrapprox_{\delta}\delta^{2\eta}\rho^{-n}.$
(3.16)

For each $\tilde{T}\in\tilde{\mathbb{T}}_{4}$, define
$\tilde{Y}_{4}(\tilde{T})=\bigcup_{Q\in\mathcal{Q}_{1}(\tilde{T})}Q$. By
(3.11), $Y_{4}$ is a shading of $\mathbb{T}_{4}$. (3.16) is precisely the
statement that

$\sum_{\tilde{T}\in\tilde{\mathbb{T}}_{4}}|\tilde{Y}_{4}(\tilde{T})|\gtrapprox_{\delta}\delta^{2\eta}.$
(3.17)

Note that $(\tilde{\mathbb{T}}_{4},\tilde{Y}_{4})_{\rho}$ satisfies Items (a),
(b), and (c) from Definition 2.1 with $t=\eta/\varepsilon$ and
$s=3\eta/\varepsilon$.

Let $(\mathbb{T}_{5},Y_{5})_{\delta}$ be a refinement of
$(\mathbb{T}_{4},Y_{4})_{\delta}$ and let
$(\tilde{\mathbb{T}}_{5},\tilde{Y}_{5})_{\rho}$ be a refinement of
$(\tilde{\mathbb{T}}_{4},\tilde{Y}_{4})_{\rho}$, so that the following holds

  * •

There is a number $\mu_{\operatorname{fine}}$ so that for each
$\tilde{T}\in\tilde{\mathbb{T}}_{5}$ and each $p\in
E_{\mathbb{T}_{5}[\tilde{T}]}$, we have
$\\#\mathbb{T}_{5}[\tilde{T}](p)=\mu_{\operatorname{fine}}$; we do this by
refining the shading $Y_{4}$ to ensure that this property holds.

  * •

There is a number $w>0$ so that for each $\tilde{T}\in\tilde{\mathbb{T}}_{5}$
and each $\rho$-cube $Q\subset Y_{5}(\tilde{T})$, we have $|Q\cap
E_{\mathbb{T}_{5}[\tilde{T}]}|=w$; we do this by refining the shading $Y_{4}$,
while preserving the previous property (each $\delta$-cube is either preserved
or deleted from all shadings in $\mathbb{T}_{5}[\tilde{T}]$)

  * •

There is a number $\mu_{\operatorname{coarse}}$ so that for each $p\in
E_{\tilde{\mathbb{T}}_{5}}$ we have
$\\#\tilde{\mathbb{T}}_{5}(p)=\mu_{\operatorname{coarse}}$; we do this by
refining the shading $\tilde{Y}_{4}$, while preserving the previous
properties.

  * •

$(\tilde{\mathbb{T}}_{5},\tilde{Y}_{5})_{\rho}$ covers
$(\mathbb{T}_{5},Y_{5})_{\delta}$.

Since $(\tilde{\mathbb{T}}_{5},\tilde{Y}_{5})_{\rho}$ satisfies Items (a),
(b), and (c) from Definition 2.1 with $t=\eta/\varepsilon$ and
$s=4\eta/\varepsilon$, if $\eta$ and $\delta_{0}$ are chosen sufficiently
small (which forces $\rho$ to be sufficiently small), then
$|E_{\tilde{\mathbb{T}}_{5}}|\geq\rho^{\sigma_{n}+\varepsilon/2}$, and thus

$\mu_{\operatorname{coarse}}\lesssim\rho^{-\sigma_{n}-\varepsilon/2-\eta/\varepsilon}.$
(3.18)

Thus if $\eta$ is chosen sufficiently small, then Item (iii) from the
statement of Proposition 3.2 holds.

At this point, $(\tilde{\mathbb{T}}_{5},\tilde{Y}_{5})_{\rho}$ is a balanced
cover of $(\mathbb{T}_{5},Y_{5})_{\delta}$, and this pair satisfies Items (iv)
and (iii). In addition, $(\tilde{\mathbb{T}}_{5},\tilde{Y}_{5})_{\rho}$
satisfies Items (a), (b), and (c) from Definition 2.1 with
$t=\eta/\varepsilon$ and $s=4\eta/\varepsilon$, and for each
$\tilde{T}\in\tilde{\mathbb{T}}_{5}$, the unit rescaling of
$(\mathbb{T}_{5}[\tilde{T}],Y_{5})_{\delta}$ relative to $\tilde{T}$ satisfies
Items (a) and (b) from Definition 2.1 with $t=\varepsilon^{2}.$

Next we will analyze the multiplicities $\mu_{\operatorname{fine}}$ and
$\mu_{\operatorname{coarse}}$ defined above. (3.18) gives an upper bound on
$\mu_{\operatorname{coarse}}$, while by (3.10) we have
$\mu_{\operatorname{fine}}\leq(\rho/\delta)^{\sigma_{n}+\varepsilon^{2}/3}\leq\rho^{-\varepsilon/3}(\rho/\delta)^{\sigma_{n}}$.
On the other hand,

$\int\sum_{T\in\mathbb{T}_{5}}\chi_{Y_{5}(T)}(p)dp\gtrapprox_{\delta}\delta^{\eta}.$
(3.19)

Since the original collection of tubes $(\mathbb{T},Y)_{\delta}$ is
$\eta$-extremal, the integrand in (5.20) is supported on a set of size at most
$\delta^{\sigma_{n}-\eta}$, and it is pointwise bounded by
$\mu_{\operatorname{fine}}\mu_{\operatorname{coarse}}$. We conclude that

$\mu_{\operatorname{fine}}\gtrapprox_{\delta}\delta^{2\eta}\rho^{\varepsilon/2+\eta/\varepsilon}(\rho/\delta)^{\sigma_{n}},\quad\mu_{\operatorname{coarse}}\gtrapprox_{\delta}\delta^{2\eta}\rho^{-\sigma_{n}+\varepsilon/3}.$
(3.20)

Selecting $\eta>0$ sufficiently small, we conclude that
$|E_{\tilde{\mathbb{T}}_{5}}|\leq\rho^{\sigma_{n}-\varepsilon}$, and for each
$\tilde{T}\in\tilde{\mathbb{T}}_{5}$, the unit-rescaling
$(\hat{\mathbb{T}},\hat{Y})_{\delta/\rho}$ of each set
$(\mathbb{T}_{5}[\tilde{T}],Y_{5})_{\delta}$ satisfies
$|E_{\hat{T}}|\leq(\delta/\rho)^{\sigma_{n}-\varepsilon}$.

We are almost done, except it is possible that the unit-rescaling described
above might fail to satisfy

$\sum_{\hat{T}\in\hat{\mathbb{T}}}|\hat{Y}(\hat{T})|\geq(\rho/\delta)^{\varepsilon}$
(3.21)

for some choices of $\tilde{T}\in\tilde{\mathbb{T}}_{5}$. To fix this problem,
we define $\tilde{\mathbb{T}}_{6}\subset\tilde{\mathbb{T}}_{5}$ to be the set
of tubes for which (3.21) holds. We define $\tilde{Y}_{6}=\tilde{Y}_{5}$,
$Y_{6}=Y_{5}$, and
$\mathbb{T}_{6}=\bigcup_{\tilde{T}\in\tilde{\mathbb{T}}_{6}}\mathbb{T}_{5}[\tilde{T}]$.
Then $(\mathbb{T}_{6},Y_{6})$ is a refinement of $(\mathbb{T},Y)$ which,
together with $(\tilde{\mathbb{T}}_{6},\tilde{Y}_{6})_{\rho}$, satisfies the
conclusions of Proposition 3.2 ∎

######  Remark 3.1.

The conclusions of Lemma 3.1 and Proposition 3.2 are the only consequences of
stickiness that we will use to prove Theorem 1.1. In particular, if Lemma 3.1
and Proposition 3.2 hold for some other class of Kakeya sets, then it should
be possible to prove the analogue of Theorem 1.1 in that setting as well.

We conclude this section with a few final observations about extremal
collections of tubes. First, if $(\mathbb{T},Y)_{\delta}$ is an extremal
collection of tubes, then the directions of the tubes passing through a
typical point cannot focus too tightly. This phenomena is sometimes called
“robust transversality.” We will record a precise version below.

######  Lemma 3.4.

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. Then after
replacing $(\mathbb{T},Y)_{\delta}$ by a refinement, for each
$p\in\mathbb{R}^{n}$ and each $v\in S^{n-1}$, we have

$\\#\\{T\in\mathbb{T}(p)\colon\angle(v,\operatorname{dir}(T)<\delta^{\varepsilon}\\}\leq\delta^{-\sigma_{n}+\varepsilon\sigma_{n}/2}.$
(3.22)

###### Proof.

Apply Proposition 3.2 with $\varepsilon\sigma_{n}/2$ in place of
$\varepsilon$, and $\rho=\delta^{\varepsilon}$. Since the resulting collection
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ of tubes are essentially distinct, the
$\delta$-tubes contributing to (3.22) must be covered by $O(1)$ $\rho$-tubes
from $\tilde{\mathbb{T}}$. (3.22) now follows from Item (i) from Proposition
3.2. ∎

The next lemma says that extremal collections of tubes remain extremal after a
mild re-scaling. The proof is straightforward (though somewhat tedious), and
is omitted.

######  Lemma 3.5.

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. Let
$Q\subset[-1,1]^{n}$ be an axis-parallel rectangular prism, and suppose
$\sum_{T\in\mathbb{T}}|Y(T)\cap Q|\geq\delta^{\eta}$. Let $\phi$ be a
translation composed with a dilation of the form
$(x_{1},\ldots,x_{n})\mapsto(r_{1}x_{1},\ldots,r_{n}x_{n})$, where
$\delta^{\eta}\leq r_{i}\leq\delta^{-\eta}$, and suppose
$\phi(Q)\subset[-1,1]^{n}$.

Then the image of $(\mathbb{T},Y\cap Q)_{\delta}$ under $\phi$ is
$\varepsilon$-extremal. More precisely, there exists an $\varepsilon$-extremal
collection of tubes $(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ for some
$\delta\leq\rho\leq\delta^{1-\eta}$ so that each
$\tilde{T}\in\tilde{\mathbb{T}}$ has a coaxial line of the form $\phi(\ell)$,
where $\ell$ is the coaxial line of a tube $T\in\mathbb{T}$. Furthermore, for
each such pair $T,\tilde{T}$, we have that $\tilde{Y}(\tilde{T})$ is the set
of $\rho$-cubes that intersect $\phi(Y(T)\cap Q).$

##  4 Planiness and graininess

For the remainder of the paper we will specialize to the case $n=3$, and we
will define $\sigma=\sigma_{3}$. Our goal is to prove that $\sigma=0$. In this
section we will establish the existence of extremal collections of tubes
$(\mathbb{T},Y)_{\delta}$ with additional structural properties. The first of
these is the existence of a Lipschitz plane map.

######  Definition 4.1.

Let $(\mathbb{T},Y)_{\delta}$ be a set of $\delta$-tubes. A _plane map_ for
$(\mathbb{T},Y)_{\delta}$ is a function $V\colon E_{\mathbb{T}}\to S^{2}$ so
that

$|\operatorname{dir}(T)\cdot V(p)|\leq\delta\quad\textrm{for all}\ p\in
E_{\mathbb{T}},\ T\in\mathbb{T}(p).$

Next, we will show that each $\\{z=z_{0}\\}$ slice of the Kakeya set is
contained in a union of long thin rectangles, which we will call global
grains. These global grains will be arranged like an Ahlfors-David regular
set. The specific property we need is the following.

######  Definition 4.2.

For $n\geq 1,\ \alpha\in(0,n]$, $C\geq 1$ and $\delta>0$, We say a set
$E\subset\mathbb{R}^{n}$ is a $(\delta,\alpha,C)_{n}$-ADset if for all
$x\in\mathbb{R}^{n}$, all $\rho\geq\delta$, and all $r\geq\rho$, we have

$\mathcal{E}_{\rho}(E\cap B(x,r))\leq C(r/\rho)^{\alpha}.$ (4.1)

######  Remark 4.1.

Note that if $E$ is a $(\delta,\alpha,C)_{n}$-ADset, then so is every subset
of $E$. This observation will be used extentensively in the arguments that
follows. Definition 4.2 only imposes upper bounds on the size of
$\mathcal{E}_{\rho}(E\cap B(x,r))$, while Ahlfors regularity usually requires
a matching lower bound. However, if $E$ is a $(\delta,\alpha,C)_{n}$-ADset
that has bounded diameter and $\mathcal{E}_{\delta}(E)$ has size roughly
$\delta^{-\alpha}$, then a large subset of $E$ will satisfy a lower bound
analogue of (4.1).

With these definitions, we can now state the main result of this section.

######  Proposition 4.1.

For all $\varepsilon,\delta_{0}>0$, there exists $\delta\in(0,\delta_{0}]$ and
a $\varepsilon$-extremal set of tubes $(\mathbb{T},Y)_{\delta}$ with the
following two properties.

  1. 1.

$E_{\mathbb{T}}$ is a union of global grains with Lipschitz slope function.
There is a 1-Lipschitz function $f\colon[-1,1]\to\mathbb{R}$ so that
$\big{(}E_{\mathbb{T}}\cap\\{z=z_{0}\\}\big{)}\cdot(1,f(z_{0}),0)$ is a
$(\delta,1-\sigma,\delta^{-\varepsilon})_{1}$-ADset for each $z_{0}\in[-1,1]$.

  2. 2.

$E_{\mathbb{T}}$ is a union of local grains with Lipschitz plane map
$(\mathbb{T},Y)_{\delta}$ has a 1-Lipschitz plane map $V$. For all
$\rho\in[\delta,1]$ and all $p\in\mathbb{R}^{3}$,
$V(p)\cdot\big{(}B(p,\rho^{1/2})\cap E_{\mathbb{T}}\big{)}$ is a
$(\rho,1-\sigma,\delta^{-\varepsilon})_{1}$-ADset.

######  Remark 4.2.

Item 1 implies that each slice $E_{\mathbb{T}}\cap\\{z=z_{0}\\}$ is contained
in a union of at most $\delta^{\sigma-1-\epsilon}$ rectangles of dimensions
roughly $1\times\delta$, and thus by Fubini,
$|E_{\mathbb{T}}|\lesssim\delta^{\sigma-\epsilon}$. On the other hand, this
containment is nearly sharp, since
$|E_{\mathbb{T}}|\geq\delta^{\sigma+\varepsilon}$.

Item 2 implies that each ball $B(p,\rho^{1/2})\cap N_{\rho}(E_{\mathbb{T}})$
is a union of at most $\rho^{\sigma/2-1/2}$ parallel rectangular slabs of
dimensions roughly $\rho^{1/2}\times\rho^{1/2}\times\rho$. Since
$E_{\mathbb{T}}$ can be covered by
$\delta^{-O(\varepsilon)}\rho^{\sigma/2-3/2}$ balls of radius $\rho^{1/2}$,
this implies that
$|N_{\rho}(E_{\mathbb{T}})|\lesssim\delta^{-O(\varepsilon)}\rho^{\sigma}$. On
the other hand, this containment is nearly sharp, since
$|N_{\rho}(E_{\mathbb{T}})|\gtrsim\delta^{-O(\varepsilon)}\rho^{\sigma}$.

Our main tool for proving Proposition 4.1 will be the multilinear Kakeya
theorem of Bennett-Carbery-Tao [1]. The version stated below can be found in
[6].

######  Theorem 4.2 (Multilinear Kakeya).

Let $\mathbb{T}$ be a set of $\delta$-tubes in $\mathbb{R}^{3}$. Then

$\int\Big{(}\sum_{T_{1},T_{2},T_{3}\in\mathbb{T}}\chi_{T_{1}}\chi_{T_{2}}\chi_{T_{3}}|v_{1}\wedge
v_{2}\wedge v_{3}|\Big{)}^{1/2}\lesssim\big{(}\delta^{2}\\#\mathbb{T})^{3/2},$
(4.2)

where in the above expression, $v_{i}=\operatorname{dir}(T_{i})$ is a unit
vector pointing in the direction of the tube $T_{i}$, and $|v_{1}\wedge
v_{2}\wedge v_{3}|$ is the area of the parallelepiped spanned by
$v_{1},v_{2}$, and $v_{3}$.

###  4.1 Finding a plane map

In this section, we will show that every extremal set of tubes has a
refinement with a plane map. First, we will show that every extremal set of
tubes has a refinement that “weakly” agrees with a plane map.

######  Lemma 4.3.

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 $\eta$-extremal set of $\delta$-tubes. Then
there is a refinement $(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ and a
function $V\colon E_{\mathbb{T}}\to S^{2}$ with the following property: for
each $p\in E_{\mathbb{T}^{\prime}}$, we have

$|V(p)\cdot\operatorname{dir}(T)|\leq\delta^{\sigma/2}\quad\textrm{for all}\
T\in\mathbb{T}^{\prime}(p).$ (4.3)

###### Proof.

Without loss of generality, we can suppose that $\varepsilon\leq\sigma/4$. We
will select $\eta$ sufficiently small to ensure that
$\eta\leq\min(\varepsilon/40,\ \varepsilon\sigma/24)$. Apply Lemma 3.4 to
$(\mathbb{T},Y)_{\delta}$ with $\varepsilon/4$ in place of $\varepsilon$ to
obtain a refinement $(\mathbb{T}_{1},Y_{1})_{\delta}$ so that for each
$p\in\mathbb{R}^{3}$ and each unit vector $v\in S^{2}$, we have

$\\#\\{T\in\mathbb{T}_{1}(p)\colon\angle(v,\operatorname{dir}(T))\leq\delta^{\varepsilon/4}\\}\leq\delta^{-\sigma+(\varepsilon/4)\sigma/2}\leq\delta^{-\sigma+3\eta}.$
(4.4)

Let $(\mathbb{T}_{2},Y_{2})$ be a constant multiplicity refinement of
$(\mathbb{T}_{1},Y_{1})$, with multiplicity
$\delta^{-\sigma+2\eta}\lessapprox\mu\lesssim\delta^{-\sigma-2\eta}$. By
Theorem 4.2 we have

$\int\Big{(}\sum_{T_{1},T_{2},T_{3}\in\mathbb{T}_{2}}\chi_{Y_{2}(T_{1})}\chi_{Y_{2}(T_{1})}\chi_{Y_{2}(T_{3})}|v_{1}\wedge
v_{2}\wedge
v_{3}|\Big{)}^{1/2}\lesssim\big{(}\delta^{2}\\#\mathbb{T}_{2})^{3/2}\lesssim\delta^{-(3/2)\eta}.$
(4.5)

We say a point $p\in E_{\mathbb{T}_{2}}$ is _broad_ if

$\\#\\{(T_{1},T_{2},T_{3})\in(\mathbb{T}_{2}(p))^{3}\colon|v_{1}\wedge
v_{2}\wedge v_{3}|\geq\delta^{\sigma-10\eta}\\}\geq\delta^{-3\sigma+3\eta},$

otherwise we say it is narrow (in the above expression and the expressions to
follow, we set $v_{i}=\operatorname{dir}(T_{i})$). Both the set of broad and
the set of narrow points are unions of $\delta$-cubes. Observe that each broad
point contributes at least $\delta^{-\sigma-(7/2)\eta}$ to the integrand on
the LHS of (4.5), and thus by (4.5) the measure of the set of broad points
$\lesssim\delta^{\sigma+2\eta}$. On the other hand, $E_{\mathbb{T}_{2}}$ has
measure $\gtrapprox\delta^{\sigma+\eta}$.

Define $Y_{3}(T)=\\{p\in Y_{2}(T)\colon p\ \textrm{narrow}\\}$, and define
$\mathbb{T}_{3}=\mathbb{T}_{2}$. Then it is still true that
$\\#\mathbb{T}_{3}(p)=\mu$ for each $p\in E_{\mathbb{T}_{3}}$, and
$|E_{\mathbb{T}_{3}}|\geq\frac{1}{2}|E_{\mathbb{T}_{2}}|$.

For each $p\in E_{\mathbb{T}_{3}}$, we have

$\\#\\{(T_{1},T_{2},T_{3})\in\mathbb{T}_{3}(p)^{3}\colon|v_{1}\wedge
v_{2}\wedge
v_{3}|<\delta^{\sigma-10\eta}\\}\geq\mu^{3}-\delta^{-3\sigma+3\eta}\geq\frac{1}{2}\mu^{3}.$
(4.6)

On the other hand, by (4.4), for each $i,j\in\\{1,2,3\\}$ with $i\neq j$ we
have

$\\#\\{(T_{1},T_{2},T_{3})\in\mathbb{T}_{3}(p)^{3}\colon|v_{i}\wedge
v_{j}|<\delta^{\varepsilon/4}\\}\leq\mu^{2}\delta^{-\sigma+3\eta}\leq\frac{1}{100}\mu^{3}.$
(4.7)

Comparing (4.6) and (4.7), we conclude that there are at least $\mu^{3}/4$
triples $(T_{1},T_{2},T_{3})$ with $|v_{1}\wedge v_{2}\wedge
v_{3}|<\delta^{\sigma-2\eta}\leq\delta^{\sigma-\varepsilon/4}$ and
$|v_{i}\wedge v_{j}|\geq\delta^{\varepsilon/4}$ for each $i\neq j$. By
pigeonholing, there exists $T_{1},T_{2}$ so that there are at least $\mu/4$
choices of $T_{3}$ so that $(T_{1},T_{2},T_{3})$ is a triple of the above
form. Define

$V(p)=\frac{\operatorname{dir}(T_{1})\times\operatorname{dir}(T_{2})}{|\operatorname{dir}(T_{1})\times\operatorname{dir}(T_{2})|}.$

Then for each $T_{3}$ of the type described above, we have

$|V(p)\cdot
v(T_{3})|=\frac{|\operatorname{dir}(T_{1})\wedge\operatorname{dir}(T_{2})\wedge\operatorname{dir}(T_{3})|}{|\operatorname{dir}(T_{1})\times\operatorname{dir}(T_{2})|}\leq\frac{\delta^{\sigma-10\eta}}{\delta^{\varepsilon/4}}\leq\delta^{\sigma/2}.$
(4.8)

Finally, let $\mathbb{T}^{\prime}=\mathbb{T}_{3}$, and define the shading

$Y^{\prime}(T)=\\{p\in Y_{3}(T)\colon|V(p)\cdot
v(T)|\leq\delta^{\sigma/2}\\}.$ (4.9)

$(\mathbb{T}^{\prime},Y)_{\delta}$ is a refinement of
$(\mathbb{T},Y)_{\delta}$ that satisfies (4.3). ∎

The next lemma shows that by moving to a coarser scale, we can ensure that a
suitable thickening of the tubes from Lemma 4.3 strongly agree with their
plane map.

######  Lemma 4.4.

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 $\eta$-extremal set of $\delta$-tubes, and let
$\rho\in[\delta^{1-\varepsilon},\delta^{\varepsilon}]$. Suppose there exists a
function $V\colon E_{\mathbb{T}}\to S^{2}$ so that for each $p\in
E_{\mathbb{T}}$, we have

$|V(p)\cdot v(T)|\leq\rho/2\quad\textrm{for all}\ T\in\mathbb{T}^{\prime}(p).$
(4.10)

Then there exists a refinement $(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ of
$(\mathbb{T},Y)_{\delta}$ and a $\varepsilon$-extremal set of $\rho$-tubes
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ with plane map $\tilde{V}$ so that
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ covers
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$. Furthermore, $\tilde{V}$ is
constant on each $\rho$-cube, and $V$ and $\tilde{V}$ are consistent on each
$\rho$-cube, in the sense that there exists $p\in Q$ with $\tilde{V}(Q)=V(p)$.

###### Proof.

Let $\varepsilon_{1}>0$ be a small constant to be chosen below. We will select
$\eta$ very small compared to $\varepsilon_{1}$, and $\varepsilon_{1}$ small
compared to $\varepsilon$. Apply Proposition 3.2 to $(\mathbb{T},Y)_{\delta}$,
with $\varepsilon_{1}$ in place of $\varepsilon$ and with $\rho$ as stated in
the lemma. We obtain a refinement $(\mathbb{T}_{1},Y_{1})_{\delta}$ of
$(\mathbb{T},Y)_{\delta}$ and a $\varepsilon_{1}$-extremal collection of
$\rho$-tubes $(\tilde{\mathbb{T}}_{1},\tilde{Y}_{1})_{\rho}$ that covers
$(\mathbb{T}_{1},Y_{1})_{\delta}$.

After refining $(\tilde{\mathbb{T}}_{1},\tilde{Y}_{1})_{\rho}$ (which induces
a refinement on $(\mathbb{T}_{1},Y_{1})_{\delta}$), we may suppose that for
each $\rho$-cube $Q\subset E_{\tilde{\mathbb{T}}_{1}}$, there is a point
$p_{Q}\in Q$ with $\\#\mathbb{T}_{1}(p_{Q})\gtrapprox\delta^{-\sigma+2\eta}$.
We say the tube $\tilde{T}\in\tilde{\mathbb{T}}_{1}$ is _associated_ to the
$\rho$-cube $Q$ if $p_{Q}\in\bigcup_{T\in\mathbb{T}_{1}[\tilde{T}]}Y_{1}(T)$;
this necessarily implies $Q\subset\tilde{Y}_{1}(\tilde{T})$. By Item (iv) from
Proposition 3.2, each $\rho$-cube $Q\subset E_{\tilde{T}_{1}}$ satisfies

$\\#\\{\tilde{T}\in\tilde{T}_{1}\colon\tilde{T}\ \textrm{associated to}\
Q\\}\gtrapprox\frac{\delta^{-\sigma+2\eta}}{(\rho/\delta)^{\sigma+\varepsilon_{1}}}\gtrapprox\delta^{2\eta+\varepsilon_{1}}\rho^{-\sigma}.$

On the other hand, by Item (iii) from Proposition 3.2, the number of tubes in
$\tilde{\mathbb{T}}_{1}$ whose shading contains $Q$ is at most
$\rho^{-\sigma-\varepsilon_{1}}$. Thus if we define $\tilde{Y}_{2}(\tilde{T})$
to be the union of $\rho$-cubes associated to $\tilde{T}$ and define
$\tilde{\mathbb{T}}_{2}=\tilde{\mathbb{T}}_{1}$ then

$\sum_{\tilde{T}\in\tilde{\mathbb{T}}_{2}}|\tilde{Y}_{2}(\tilde{T})|\gtrapprox\delta^{2\eta+2\varepsilon_{1}}\sum_{\tilde{T}\in\tilde{\mathbb{T}}_{1}}|\tilde{Y}_{1}(T)|.$
(4.11)

Selecting $\varepsilon_{1}$ sufficiently small, we can ensure that the LHS of
(4.11) is at least $\rho^{\varepsilon}$.

Next, we shall define the function $\tilde{V}(p)\colon
E_{\tilde{\mathbb{T}}_{2}}\to S^{2}$ as follows. For each $\rho$-cube
$Q\subset E_{\tilde{\mathbb{T}}_{2}}$, we define $\tilde{V}(p)=V(p_{Q})$ for
all $p\in Q$. Clearly $\tilde{V}$ is constant on $\rho$-cubes, and satisfies
the consistency condition with $V$ stated in Lemma 4.4. It remains to show
that $\tilde{V}$ is a plane map for
$(\tilde{\mathbb{T}}_{2},\tilde{Y}_{2})_{\rho}.$ Let $p\in
E_{\tilde{\mathbb{T}}_{2}}$ be contained in a $\rho$-cube $Q$, and let
$\tilde{T}\in\tilde{\mathbb{T}}_{2}(p)$. Then there exists a tube
$T\in\mathbb{T}_{2}(T)\cap\mathbb{T}(p)$. We have

$|\tilde{V}(p)\cdot\operatorname{dir}(\tilde{T})|=|V(p_{Q})\cdot\operatorname{dir}(\tilde{T})|\leq|V(p_{Q})\cdot\operatorname{dir}(T)|+|V(p_{Q})\cdot(\operatorname{dir}(\tilde{T})-\operatorname{dir}(T)|\leq\rho/2+|\operatorname{dir}(\tilde{T})-\operatorname{dir}(T)|\leq\rho,$

where the final inequality used (4.10) and the observation that
$|\operatorname{dir}(\tilde{T})-\operatorname{dir}(T)|\leq\rho/2$ (since
$\tilde{T}$ covers $T$). ∎

###  4.2 Lipschitz regularity of the plane map

In this section we will show that if an extremal collection of tubes has a
plane map, then after restricting to a sub-collection, this plane map must be
Lipschitz with controlled Lipschitz norm.

######  Lemma 4.5.

For all $\varepsilon>0,$ there exists $\eta>0,\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let
$(\mathbb{T},Y)_{\delta}$ be a $\eta$-extremal collection of tubes with plane
map $V$, and let $\rho\in[\delta^{1-\varepsilon},\delta^{\varepsilon}]$. Then
there exists a $\varepsilon$-extremal sub-collection
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$, so that if $p,q\in
E_{\mathbb{T}^{\prime}}$ are contained in a common $\rho$-cube, then
$|V(p)-V(q)|\leq\rho$.

###### Proof.

Let $\varepsilon_{1},\varepsilon_{2}>0$ be small constants to be chosen below.
We will select $\eta$ very small compared to $\varepsilon_{1}$,
$\varepsilon_{1}$ very small compared to $\varepsilon_{2}$, and
$\varepsilon_{2}$ very small compared to $\varepsilon$.

Let $(\mathbb{T}_{1},Y_{1})_{\delta}$ be the refinement obtained by applying
Lemma 3.4 to $(\mathbb{T},Y)_{\delta}$ with $\varepsilon_{1}$ in place of
$\varepsilon$. Next, apply Proposition 3.2 to
$(\mathbb{T}_{1},Y_{2})_{\delta}$ with $\varepsilon_{2}$ in place of
$\varepsilon$, and $\rho$ as specified in the lemma. We obtain a refinement
$(\mathbb{T}_{2},Y_{2})_{\delta}$ of $(\mathbb{T}_{1},Y_{1})$ and a
$\varepsilon_{2}$-extremal set of $\rho$-tubes
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$.

After refining $(\mathbb{T}_{2},Y_{2})_{\delta}$ and
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$, we may suppose that
$(\mathbb{T}_{2},Y_{2})_{\delta}$ has constant multiplicity
$\mu\gtrapprox\delta^{-\sigma+2\eta}$, and
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ is still a balanced cover of
$(\mathbb{T}_{2},Y_{2})_{\delta}$. If $\eta>0$ is selected sufficiently small
(depending on $\varepsilon_{1}$ and $\sigma$), then for each $p\in
E_{\mathbb{T}_{2}}$, there are at least $\mu^{2}/2$ pairs
$T_{1},T_{2}\in\mathbb{T}_{2}(p)$ with
$\angle(\operatorname{dir}(T_{1}),\operatorname{dir}(T_{2}))\geq\delta^{\varepsilon_{1}}$.
Thus by Item (iv) from Proposition 3.2, there are at least
$(\mu^{2}/2)/\big{(}(\rho/\delta)^{\sigma+\varepsilon_{2}}\big{)}^{2}\gtrapprox\delta^{4\eta+2\varepsilon_{2}}\rho^{-2\sigma}$
pairs $\tilde{T}_{1},\tilde{T}_{2}\in\tilde{\mathbb{T}}$ with
$p\in\bigcup_{T\in\mathbb{T}_{2}[\tilde{T}]}Y_{3}(T)$,
$p\in\bigcup_{T\in\mathbb{T}_{2}[\tilde{T}^{\prime}]}Y_{3}(T)$, and
$\angle\big{(}\operatorname{dir}(\tilde{T}_{1}),\operatorname{dir}(\tilde{T}_{2})\big{)}\geq\delta^{\varepsilon_{1}}-2\rho$;
we will choose $\varepsilon_{1}$ sufficiently small so that
$\delta^{\varepsilon_{1}}-2\rho\geq\delta^{\varepsilon_{1}}/2$. In particular,
for each $\rho$-cube $Q\subset E_{\tilde{\mathbb{T}}}$, we have

$\begin{split}\Big{|}\Big{\\{}(p,\tilde{T},\tilde{T}^{\prime})\in(Q\cap
E_{\mathbb{T}_{3}})\times\tilde{\mathbb{T}}^{2}\colon
p\in\bigcup_{T\in\mathbb{T}_{2}[\tilde{T}]}Y_{3}(T)\cap\bigcup_{T\in\mathbb{T}_{2}[\tilde{T}^{\prime}]}Y_{3}(T),\
&\angle\big{(}\operatorname{dir}(\tilde{T}_{1}),\operatorname{dir}(\tilde{T}_{2})\big{)}\geq\delta^{\varepsilon_{1}}/2\Big{\\}}\Big{|}\\\
&\gtrapprox|Q\cap
E_{\mathbb{T}_{3}}|\delta^{4\eta+2\varepsilon_{2}}\rho^{-2\sigma}\\\
&\gtrapprox\delta^{4\eta+4\varepsilon_{2}}|Q\cap
E_{\mathbb{T}_{3}}|\big{(}\\#\tilde{\mathbb{T}}(Q)\big{)}^{2},\end{split}$
(4.12)

where $|\cdot|$ on the LHS of (4.12) denotes the product of Lebesgue measure
in $\mathbb{R}^{3}$ and counting measure, and the final inequality used Item
(iii) from Proposition 3.2. Thus by pigeonholing, we can select two tubes from
$\tilde{\mathbb{T}}$, which we will denote by $\tilde{T}_{Q}$ and
$\tilde{T}^{\prime}_{Q},$ with
$\angle\big{(}\operatorname{dir}(\tilde{T}_{1}),\operatorname{dir}(\tilde{T}_{2})\big{)}\geq\delta^{\varepsilon_{1}}/2,$
so that

$\Big{|}Q\
\cap\Big{(}\bigcup_{T\in\mathbb{T}_{2}[\tilde{T}_{Q}]}Y_{3}(T)\Big{)}\cap\Big{(}\bigcup_{T\in\mathbb{T}_{2}[\tilde{T}_{Q}^{\prime}]}Y_{3}(T)\Big{)}\Big{|}\gtrapprox\delta^{4\eta+4\varepsilon_{2}}|Q\cap
E_{\mathbb{T}_{3}}|.$ (4.13)

\begin{overpic}[scale={0.3}]{figures/figLipInCube} \put(15.0,57.0){$p$}
\put(10.0,52.0){$Q$} \put(70.0,35.0){$\tilde{T}_{Q}$}
\put(67.0,68.0){$\tilde{T}_{Q}^{\prime}$} \put(55.0,24.0){$\tilde{T}_{p}$}
\put(55.0,78.0){$\tilde{T}_{p}^{\prime}$} \end{overpic} Figure 2: $V(p)$ is
almost orthogonal to $\operatorname{dir}(\tilde{T}_{Q})$ and
$\operatorname{dir}(\tilde{T}_{Q}^{\prime})$

Define $F_{Q}$ to be the set on the LHS of (4.13). Let
$\mathbb{T}_{3}=\mathbb{T}_{2}$, and for each $T\in\mathbb{T}_{3}$ define

$Y_{3}(T)=Y_{2}(T)\cap\bigcup_{Q}F_{Q}.$

Then
$\sum_{T\in\mathbb{T}_{3}}|Y_{3}(T)|\gtrapprox\delta^{4\eta+4\varepsilon_{2}}$.
Let $Q$ be a $\rho$-cube, and let $p\in E_{\mathbb{T}_{3}}\cap Q$. Then there
exists $T_{p}\in\mathbb{T}_{3}[\tilde{T}_{Q}]$ and
$T_{p}^{\prime}\in\mathbb{T}_{3}[\tilde{T}_{Q}^{\prime}]$ so that $p\in
Y_{3}(T_{p})\cap Y_{3}(T_{p}^{\prime})$, and hence

$|V(p)\cdot\operatorname{dir}(\tilde{T}_{Q})|\leq|V(p)\cdot\operatorname{dir}(T_{p})|+|V(p)\cdot(\operatorname{dir}(\tilde{T}_{Q})-\operatorname{dir}(T_{p}))|\leq\delta+\rho,$

and similarly for $|V(p)\cdot\operatorname{dir}(\tilde{T}^{\prime}_{Q})|$. See
Figure 2. We conclude that for all $p\in Q$, we have

$\angle\Big{(}V(p),\
\frac{\operatorname{dir}(\tilde{T}_{Q})\times\operatorname{dir}(\tilde{T}^{\prime}_{Q})}{|\operatorname{dir}(\tilde{T}_{Q})\times\operatorname{dir}(\tilde{T}^{\prime}_{Q})|}\Big{)}\lesssim\delta^{-\varepsilon_{1}}\rho.$
(4.14)

(4.14) says that $(\mathbb{T}_{3},Y_{3})$ almost satisfies the conclusion of
Lemma 4.5, except the RHS of (4.14) is $\delta^{-\varepsilon_{1}}\rho$ rather
than $\rho$. To fix this, we will pigeonhole to select a set
$E_{4}\subset\mathbb{T}_{T_{4}}$ that is a union of $\delta$-cubes so that
(i): for each $\rho$-cube $Q$, $\angle(V(p),V(q))\leq\rho$ for all $p,q\in Q$,
and (ii) if we define $Y_{4}(T)=Y_{3}(T)\cap E_{4}$ and
$\mathbb{T}_{4}=\mathbb{T}_{3}$, then
$\sum_{T\in\mathbb{T}_{4}}|Y_{4}(T)|\gtrsim\delta^{2\varepsilon_{1}}\sum_{T\in\mathbb{T}_{3}}|Y_{3}(T)|$.
If we choose $\varepsilon_{1},\varepsilon_{2},\eta$ sufficiently small, then
$(\mathbb{T}_{4},Y_{4})_{\delta}$ will be $\varepsilon$-extremal. ∎

Applying Lemma 4.5 iteratively $O(1)$ times to various translates of the
standard grid, we obtain the following

######  Corollary 4.6.

For all $\varepsilon>0$, there exists $\eta>0,\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let
$(\mathbb{T},Y)_{\delta}$ be a $\eta$-extremal collection of tubes with plane
map $V$, and let $\rho\in[\delta^{1-\varepsilon},\delta^{\varepsilon}]$. Then
there exists a $\varepsilon$-extremal sub-collection
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$, so that if $p,q\in
E_{\mathbb{T}^{\prime}}$ satisfy $|p-q|\leq\rho$, then $|V(p)-V(q)|\leq
2\rho$.

Corollary 4.6 says that after restricting to a sub-collection, the plane map
$V$ will be Lipschitz at a specific scale $\rho$. The next lemma iterates this
result at many scales.

######  Lemma 4.7.

For all $\varepsilon>0$, there exists $\eta>0,\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let
$(\mathbb{T},Y)_{\delta}$ be a $\eta$-extremal collection of tubes with plane
map $V$ that is constant on $\delta$-cubes. Then there exists a
$\varepsilon$-extremal sub-collection
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ so that $V$ is
$\delta^{-\varepsilon}$-Lipschitz on $E_{\mathbb{T}^{\prime}}$.

###### Proof.

After a refinement, we can suppose that if $p,q\in E_{\mathbb{T}}$ satisfy
$|p-q|\leq\delta$, then $p$ and $q$ are contained in a common $\delta$-cube,
and hence $V(p)=V(q)$. Let $N=2/\varepsilon$. If $\eta,\delta_{0}$ are
selected sufficiently small, then we can apply Corollary 4.6 iteratively to
$(\mathbb{T},Y)_{\delta}$ for $\rho=\rho_{j}=\delta^{j/N},$ $j=1,\ldots,N-1$,
and the resulting sub-collection $(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$
will be $\varepsilon$-extremal. We claim that $V$ is
$2\delta^{-1/N}$-Lipschitz on $(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$. To
see this, let $p,q\in E_{\mathbb{T}^{\prime}}$. If $|p-q|\leq\delta$ then
$V(p)=V(q)$ and we are done. If $|p-q|>\delta^{1/N},$ then $|V(p)-V(q)|\leq
2$, and again we are done. Otherwise, there is an index $1\leq k\leq N-1$ so
that $\delta^{(k+1)/N}<|p-q|\leq\delta^{k/N}$, and

$|V(p)-V(q)|\leq\delta^{k/N}\leq\delta^{-1/N}|p-q|.\qed$

Combining the results of this section with Lemma 3.1, we have the following

######  Lemma 4.8.

For all $\varepsilon,\delta_{0}>0$, there exists $\delta\in(0,\delta_{0}]$ and
a $\varepsilon$-extremal set of tubes $(\mathbb{T},Y)_{\delta}$ that has a
$\delta^{-\varepsilon}$-Lipschitz plane map.

###  4.3 Grains

In this section, we will show that if an extremal set of tubes has a Lipschitz
plane map, then after restricting to a sub-collection, the tubes arrange
themselves into parallel slabs, which are called grains. In what follows, $V$
will be a plane map for a collection of tubes. If $Q$ is a cube with center
$p$, we define $V(Q)=V(p_{Q})$ (this is a unit vector in $\mathbb{R}^{3}$),
and we define the projection $\pi_{Q}(p)=p\cdot V(Q)$.

Our goal is to show that the set of grains inside a cube arrange themselves
into a AD-regular set, in the sense of Definition 4.2. As a first step, the
next lemma controls how many grains can occur inside a cube at a given scale.

######  Lemma 4.9.

For all $\varepsilon>0,$ there exists $\eta>0,\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let
$(\mathbb{T},Y)_{\delta}$ be a $\eta$-extremal collection of tubes with
$\delta^{-\eta}$-Lipschitz plane map $V$. Let
$\rho\in[\delta^{1-\varepsilon},\delta^{\varepsilon}]$ and let
$\tau\in[\rho^{1-\varepsilon},\rho^{1/2}]$.

Then there exists a $\varepsilon$-extremal sub-collection
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$, so that for each $\tau$-cube $Q$,
we have

$\mathcal{E}_{\rho}\Big{(}V(Q)\cdot\big{(}E_{\mathbb{T}^{\prime}}\cap
Q\big{)}\Big{)}\leq\rho^{-\varepsilon}(\tau/\rho)^{1-\sigma}.$ (4.15)

###### Proof.

Let $\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}>0$ be small constants to
be chosen below. We will select $\eta$ very small compared to
$\varepsilon_{1}$, $\varepsilon_{1}$ very small compared to $\varepsilon_{2}$,
$\varepsilon_{2}$ very small compared to $\varepsilon_{3}$, and
$\varepsilon_{3}$ very small compared to $\varepsilon$.

Applying Proposition 3.2 to $(\mathbb{T},Y)_{\delta}$ with $\varepsilon_{1}/2$
in place of $\varepsilon$, and then again to the output
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ with $\tau$ in place of $\rho$ and
$\varepsilon_{2}$ in place of $\varepsilon$, we obtain a refinement
$(\mathbb{T}_{1},Y_{1})_{\delta}$ of $(\mathbb{T},Y)_{\delta}$ that is covered
by a $\varepsilon_{1}$-extremal set $(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$,
which in turn is covered by a $\varepsilon_{2}$-extremal set
$(\accentset{\approx}{\mathbb{T}},\accentset{\approx}{Y})_{\tau}$. Note that
$V$ is also a plane map for $(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ and
$(\accentset{\approx}{\mathbb{T}},\accentset{\approx}{Y})_{\tau}$.

After refining $(\mathbb{T}_{1},Y_{1})_{\delta}$ and
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$, we can ensure that
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ is a balanced cover of
$(\mathbb{T}_{1},Y_{1})_{\delta}$. In the arguments below, we will find a sub-
collection of $(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ with certain favorable
properties. Since $(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ is a balanced cover
of $(\mathbb{T}_{1},Y_{1})_{\delta}$, this sub-collection of
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ will induce a sub-collection of
$(\mathbb{T}_{1},Y_{1})_{\delta}$ of the same relative density.

Apply Lemma 3.4 to $(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$, with
$\varepsilon_{3}$ in place of $\varepsilon$. After further refinement, we can
find a collection $(\tilde{\mathbb{T}}_{1},\tilde{Y}_{1})_{\rho}$ with the
following property (_P_): For each $\tilde{T}\in\tilde{\mathbb{T}}_{1}$ and
each $\tau$-cube $Q$ that intersects $\tilde{Y}_{1}(\tilde{T})$, we have
$|Q\cap\tilde{Y}_{1}(\tilde{T})|\geq\rho^{2+2\varepsilon_{1}}\tau$ (for
comparison, the maximal size of an intersection of the form $\tilde{T}\cap Q$
is bounded by $100\rho^{2}\tau$).

Let $(\tilde{\mathbb{T}}_{2},\tilde{Y}_{2})_{\rho}$ be a constant multiplicity
refinement of $(\tilde{\mathbb{T}}_{1},\tilde{Y}_{1})_{\rho}$, and let
$(\tilde{\mathbb{T}}_{3},\tilde{Y}_{3})_{\rho}$ be a refinement of
$(\tilde{\mathbb{T}}_{2},\tilde{Y}_{2})_{\rho}$ that again has property (P).
The reason for the above refinements is as follows: If
$\tilde{T}\in\tilde{\mathbb{T}}_{3}$ and if $Q$ is a $\tau$-cube that
intersects $\tilde{Y}_{3}(\tilde{T})$, then
$|Q\cap\tilde{Y}_{3}(\tilde{T})|\geq\rho^{2+2\varepsilon_{1}}\tau$. Thus we
can choose a set of points $p_{1},\ldots p_{k}\in
Q\cap\tilde{Y}_{3}(\tilde{T})$ that are $\geq\rho^{1-\varepsilon_{3}}$
separated, with
$k\gtrsim\rho^{2\varepsilon_{1}+\varepsilon_{3}}\rho^{-1}\tau$. For each such
point $p_{i}$, we have
$\\#\tilde{\mathbb{T}}_{2}(p_{i})=\mu\gtrapprox\rho^{-\sigma+\varepsilon_{1}}$.
But since we refined $(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ using Lemma 3.4,
if $\varepsilon_{1}$ is sufficiently small compared to $\varepsilon_{3}$, then
there is a tube $\tilde{T}_{i}\in\tilde{\mathbb{T}}_{1}(p_{i})$ with
$\angle(\operatorname{dir}(\tilde{T}),\operatorname{dir}(\tilde{T}_{i}))\geq\rho^{\varepsilon_{3}}$.

We claim that the union of the sets
$\\{Q\cap\tilde{Y}(\tilde{T}_{i})\\}_{i=1}^{k}$ has large volume. Since the
points $p_{i}$ are $\rho^{1-\varepsilon_{3}}$ separated and the tubes
$\tilde{T}_{i}$ make angle $\geq\rho^{\varepsilon_{3}}$ with $\tilde{T}$, we
can label the points $p_{1},\ldots,p_{k}$ so that
$\operatorname{dist}\big{(}\tilde{T}_{i}\cap\tilde{T},\
\tilde{T}_{j}\cap\tilde{T}\big{)}\gtrsim\rho^{1-\varepsilon_{3}}|i-j|$. Then
if distinct tubes $\tilde{T}_{i}$ and $\tilde{T}_{j}$ intersect inside $Q$, we
must have
$\angle(v(\tilde{T}_{i}),v(\tilde{T}_{j}))\gtrsim\rho^{1-\varepsilon_{3}}|i-j|/\tau$.
But this implies that for distinct $\tilde{T}_{i}$ and $\tilde{T}_{j}$ we have

$|Q\cap\tilde{T}_{i}\cap\tilde{T}_{j}|\lesssim\frac{\rho^{3}}{\angle(v(\tilde{T}_{i}),v(\tilde{T}_{j}))+\rho}\lesssim\frac{\rho^{2+\varepsilon_{3}}\tau}{|i-j|}.$

Thus we have the following Cordoba-type $L^{2}$ estimate.

$\Big{|}Q\cap\bigcup_{i=1}^{k}\tilde{Y}_{1}(\tilde{T}_{i})\Big{|}\geq\frac{\Big{(}\sum_{i=1}^{k}|Q\cap\tilde{Y}_{1}(\tilde{T}_{i})|\Big{)}^{2}}{\big{\|}\sum_{i=1}^{k}\chi_{\tilde{T}_{i}}\big{\|}_{L^{2}(Q)}^{2}}\gtrapprox\frac{k^{2}\big{(}\rho^{2\varepsilon_{1}+2}\tau\big{)}^{2}}{(k\rho^{2+\varepsilon_{3}}\tau)}\geq\rho^{6\varepsilon_{1}+\varepsilon_{3}}\tau^{2}\rho.$
(4.16)

\begin{overpic}[scale={0.4}]{figures/figFullGrain}
\put(26.0,35.0){{\tiny$\tilde{T}_{1}$}}
\put(33.0,35.0){{\tiny$\tilde{T}_{2}$}}
\put(65.0,33.0){{\tiny$\tilde{T}_{k}$}} \par\end{overpic} Figure 3: The union
of the sets $Q\cap\tilde{Y}_{1}(T_{i}),\ i=1,\ldots,k$ fill out most of the
slab (blue) inside $Q$

(4.16) says that $\bigcup_{i=1}^{k}\tilde{Y}_{1}(\tilde{T}_{i})$ has large
volume inside of $Q$. On the other hand, we claim that this set is contained
inside a rectangular prism of dimensions roughly
$\tau\times\tau\times\delta^{-\eta}\rho$. Indeed, since the plane map $V$ is
$\delta^{-\eta}$-Lipschitz and $Q$ has diameter $\leq 2\rho$, if $q\in Q\cap
T$ then we have

$Q\cap\bigcup_{i=1}^{k}\tilde{T}_{i}\ \subset\ Q\cap
N_{4\delta^{-\eta}\rho}(q+V(Q)^{\perp}).$

This is illustrated in Figure 3. Thus

$V(q)\cdot\Big{(}Q\cap\bigcup_{i=1}^{k}\tilde{T}_{i}\Big{)}\subset[V(Q)\cdot
q-4\delta^{-\eta}\rho,\ V(Q)\cdot q+4\delta^{-\eta}\rho].$ (4.17)

Note that $V(Q)\cdot(Q\cap T)$ is also contained in this same interval.

Combining (4.16) and (4.17), we conclude that if $t\in V(Q)\cdot\big{(}Q\cap
E_{\tilde{\mathbb{T}}_{2}}\big{)}$, then

$\Big{|}Q\cap
E_{\tilde{\mathbb{T}}_{1}}\cap\pi_{Q}^{-1}([t-4\delta^{-\eta}\rho,\
t+4\delta^{-\eta}\rho])\Big{|}\gtrapprox\rho^{6\varepsilon_{1}+\varepsilon_{3}}\tau^{2}\rho.$
(4.18)

Informally, (4.18) says that each $\rho$-separated point $t\in
V(Q)\cdot\big{(}Q\cap E_{\tilde{\mathbb{T}}_{2}}\big{)}$ corresponds to a
$\tau\times\tau\times\rho$ grain (i.e. rectangular prism whose short axis is
parallel to $V(Q)$) that has nearly full intersection with $Q\cap
E_{\tilde{\mathbb{T}}_{1}}$.

If $|t-t^{\prime}|>8\delta^{-\eta}\rho$ then the sets
$\pi_{Q}^{-1}([t-4\delta^{-\eta}\rho,\ t+4\delta^{-\eta}\rho]$ and
$\pi_{Q}^{-1}([t^{\prime}-4\delta^{-\eta}\rho,\
t^{\prime}+4\delta^{-\eta}\rho]$ are disjoint, and hence (4.18) implies that

$\mathcal{E}_{8\delta^{-\eta}\rho}\Big{(}V(Q)\cdot\big{(}Q\cap
E_{\tilde{\mathbb{T}}_{2}}\big{)}\Big{)}\lessapprox\frac{|Q\cap
E_{\tilde{\mathbb{T}}_{1}}|}{\rho^{6\varepsilon_{1}+\varepsilon_{3}}\tau^{2}\rho}\leq\frac{\rho^{\sigma-\varepsilon_{1}}\tau^{3-\sigma-\varepsilon_{2}}}{\rho^{6\varepsilon_{1}+\varepsilon_{3}}\tau^{2}\rho}\lesssim\rho^{-7\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}}(\tau/\rho)^{1-\sigma},$

and thus

$\mathcal{E}_{\rho}\Big{(}V(Q)\cdot\big{(}Q\cap
E_{\tilde{\mathbb{T}}_{2}}\big{)}\Big{)}\leq
8\delta^{-\eta}\mathcal{E}_{8\delta^{-\eta}\rho}\Big{(}V(Q)\cdot\big{(}Q\cap
E_{\tilde{\mathbb{T}}_{2}}\big{)}\Big{)}\lessapprox\delta^{-\eta}\rho^{-7\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}}(\tau/\rho)^{1-\sigma}.$
(4.19)

To conclude the proof, select
$\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\eta$ sufficiently small
(depending on $\varepsilon$) so that the LHS of (4.19) is at most
$\rho^{-\varepsilon}(\tau/\rho)^{1-\sigma}$ (this is possible since
$\rho\leq\delta^{\varepsilon}$). Let $(\mathbb{T}_{2},Y_{2})_{\delta}$ be the
refinement of $(\mathbb{T}_{1},Y_{1})_{\delta}$ induced by the refinement
$(\tilde{\mathbb{T}}_{2},\tilde{Y}_{2})_{\rho}$. Since
$E_{\mathbb{T}_{2}}\subset E_{\tilde{\mathbb{T}}_{2}}$, (4.19) implies that
$(\mathbb{T}_{2},Y_{2})_{\delta}$ satisfies (4.15). ∎

Lemma 4.9 controls the size of
$V(Q_{\tau})\cdot\big{(}E_{\mathbb{T}^{\prime}}\cap Q_{\tau})$ at scale
$\rho$, where $Q_{\tau}$ is a cube of side-length
$\tau\in[\rho^{1-\varepsilon},\rho^{1/2}]$; note that this set is contained in
an interval of length $\sim\tau$. The next lemma establishes a similar bound
for the (potentially larger) quantity
$J\cap\big{(}V(Q_{\rho^{1/2}})\cdot\big{(}E_{\mathbb{T}^{\prime}}\cap
Q_{\rho^{1/2}})\big{)},$ where $Q_{\rho^{1/2}}$ is a cube of side-length
$\rho^{1/2}$, and $J$ is an interval of length $\tau$.

######  Lemma 4.10.

For all $\varepsilon>0,$ there exists $\eta>0,\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let
$(\mathbb{T},Y)_{\delta}$ be a $\eta$-extremal collection of tubes with
$\delta^{-\eta}$-Lipschitz plane map $V$. Let
$\rho\in[\delta^{1-\varepsilon},\delta^{\varepsilon}]$ and let
$\tau\in[\rho^{1-\varepsilon},\rho^{1/2}]$.

Then there exists a $\varepsilon$-extremal sub-collection
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$, so that the following holds. Let
$Q$ be a $\rho^{1/2}$-cube and let $F=V(Q)\cdot(E_{\mathbb{T}^{\prime}}\cap
Q)$. Then for each interval $J$ of length $\tau$, we have

$\mathcal{E}_{\rho}(J\cap F)\leq\rho^{-\varepsilon}(\tau/\rho)^{1-\sigma}.$
(4.20)

###### Proof.

For notational convenience, we will suppose that $\rho^{1/2}/\tau$ is an
integer, and thus every $\tau$-cube is contained in a single $\rho^{1/2}$
cube. The case where $\rho^{1/2}/\tau$ is not an integer can be handled
similarly. Let $\varepsilon_{1},\ldots,\varepsilon_{4}>0$ be small constants
to be chosen below. We will select $\eta$ very small compared to
$\varepsilon_{1}$, and $\varepsilon_{i}$ very small compared to
$\varepsilon_{i+1}$. We will select $\varepsilon_{4}$ small compared to
$\varepsilon$.

Apply Proposition 3.2 to $(\mathbb{T},Y)_{\delta}$ with $\varepsilon_{1}$ in
place of $\varepsilon$, and $\rho$ as above. We obtain a refinement of
$(\mathbb{T},Y)_{\delta}$, and a $\varepsilon_{1}$-extremal collection of
$\rho$-tubes, $(\tilde{\mathbb{T}}_{1},\tilde{Y}_{1})_{\rho}$. In what
follows, we will describe various sub-collections of
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$. Since
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ is a balanced cover of
$(\mathbb{T},Y)_{\delta}$, each sub-collection of
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ will induce an analogous sub-
collection of $(\mathbb{T},Y)_{\delta}$, of the same relative density.

Applying Lemma 4.9 to $(\tilde{\mathbb{T}}_{1},\tilde{Y}_{1})_{\rho}$ with
$\varepsilon_{2}$ in place of $\varepsilon$ and $\tau$ as above, and then
again to the output with $\varepsilon_{3}$ in place of $\varepsilon$ and
$\rho^{1/2}$ in place of $\tau$, we obtain a $\varepsilon_{3}$-extremal sub-
collection $(\tilde{\mathbb{T}}_{2},\tilde{Y}_{2})_{\rho}$ of
$(\tilde{\mathbb{T}}_{1},\tilde{Y}_{1})_{\rho}$ with the following two
properties.

  * •

For each $\rho^{1/2}$-cube $Q$, we have

$\mathcal{E}_{\rho}\big{(}V(Q)\cdot(E_{\tilde{\mathbb{T}}_{2}}\cap
Q)\big{)}\leq\rho^{-\varepsilon_{3}}\rho^{\frac{\sigma-1}{2}}.$ (4.21)

  * •

For each $\rho^{1/2}$-cube $Q$, and each $\tau$-cube $W$, we have

$\mathcal{E}_{\rho}\big{(}V(Q)\cdot(E_{\tilde{\mathbb{T}}_{2}}\cap
W)\big{)}\lesssim\delta^{-\eta}\rho^{-\varepsilon_{2}}(\tau/\rho)^{\sigma-1}.$
(4.22)

Note that in (4.22) we have the unit vector $V(Q)$ rather than $V(W)$. This
requires justification. By Lemma 4.9 we get the estimate
$\mathcal{E}_{\rho}\big{(}V(W)\cdot(E_{\tilde{\mathbb{T}}_{2}}\cap
W)\big{)}\lesssim\rho^{-\varepsilon_{2}}(\tau/\rho)^{\sigma-1}$. To obtain
(4.22), we bound

$\begin{split}\mathcal{E}_{\rho}\big{(}V(Q)\cdot(E_{\tilde{\mathbb{T}}_{2}}\cap
W)\big{)}&\leq\delta^{-\eta}\mathcal{E}_{\delta^{-\eta}\rho}\big{(}V(Q)\cdot(E_{\tilde{\mathbb{T}}_{2}}\cap
W)\\\
&\lesssim\delta^{-\eta}\mathcal{E}_{\delta^{-\eta}\rho}\big{(}V(W)\cdot(E_{\tilde{\mathbb{T}}_{2}}\cap
W)\\\
&\leq\delta^{-\eta}\mathcal{E}_{\rho}\big{(}V(W)\cdot(E_{\tilde{\mathbb{T}}_{2}}\cap
W)\\\
&\leq\delta^{-\eta}\rho^{-\varepsilon_{2}}(\tau/\rho)^{\sigma-1}.\end{split}$

The key step is the second inequality, which used the fact that
$\angle(V(Q),V(W))\lesssim\rho^{1/2}$,
$\operatorname{diam}(W)\lesssim\rho^{1/2}$, and $V$ is
$\delta^{-\eta}$-Lipschitz.

Apply Proposition 3.2 to $(\tilde{\mathbb{T}}_{2},\tilde{Y}_{2})_{\rho}$ with
$\varepsilon_{4}$ in place of $\varepsilon$ and $\rho^{1/2}$ in place of
$\rho$. We obtain a refinement $(\tilde{\mathbb{T}}_{3},\tilde{Y}_{3})_{\rho}$
of $(\tilde{\mathbb{T}}_{2},\tilde{Y}_{2})_{\rho}$, which is contained in a
union of $\lesssim\rho^{-\varepsilon_{4}}\rho^{\frac{\sigma-3}{2}}$
$\rho^{1/2}$-cubes.

After a further refinement, we can suppose that each $\rho^{1/2}$-cube $Q$
that intersects $E_{\tilde{\mathbb{T}}_{3}}$ satisfies
$|E_{\tilde{\mathbb{T}}_{3}}\cap
Q|\gtrapprox\rho^{2\varepsilon_{4}}\rho^{\frac{\sigma+3}{2}}.$

The two properties described above continue to hold for
$(\tilde{\mathbb{T}}_{3},\tilde{Y}_{3})_{\rho}$. Therefore, if $Q$ is a
$\rho^{1/2}$-cube that intersects $E_{\tilde{\mathbb{T}}_{3}}$, then by
(4.21), $E_{\tilde{\mathbb{T}}_{3}}\cap Q$ is contained in a union of at most
$\rho^{-\varepsilon_{3}}\rho^{\frac{\sigma-1}{2}}$ “grains,” which are sets of
the form $Q\cap N_{\rho}(\pi_{Q}^{-1}(t))$, where $t\in\mathbb{R}$. The union
of these grains has volume
$\lesssim\rho^{-\varepsilon_{3}}\rho^{\frac{\sigma+3}{2}}$, while on the other
hand $E_{\tilde{\mathbb{T}}_{3}}\cap Q$ has volume
$\gtrapprox\rho^{2\varepsilon_{4}}\rho^{\frac{\sigma+3}{2}}$. This means that
after further refining $(\tilde{\mathbb{T}}_{3},\tilde{Y}_{3})_{\rho}$ to
obtain a pair $(\tilde{\mathbb{T}}_{4},\tilde{Y}_{4})_{\rho}$, we can suppose
each for each $\rho^{1/2}$-cube $Q$ and each $t\in
V(p_{Q})\cdot(E_{\tilde{\mathbb{T}}_{4}}\cap Q)$, we have

$|E_{\tilde{\mathbb{T}}_{4}}\cap Q\ \cap\
\pi_{Q}^{-1}(N_{\rho}(t))|\gtrapprox\rho^{2\varepsilon_{4}+\varepsilon_{3}}\rho^{2}.$
(4.23)

Next, let $Q$ be one of the $\rho^{1/2}$ cubes described above. By Fubini,
there must be a point $q\in Q$ so that the line $L=q+\mathbb{R}V(p_{Q})$
satisfies

$|L\cap E_{\tilde{\mathbb{T}}_{4}}\cap
Q|\gtrapprox\rho^{2\varepsilon_{4}}\rho^{\frac{\sigma+3}{2}}\gtrsim\rho^{\varepsilon_{2}+2\varepsilon_{4}}|E_{\tilde{\mathbb{T}}_{4}}\cap
Q|.$ (4.24)

For each cube $Q$, let $G_{Q}$ be the union of grains that intersect $L\cap
E_{\tilde{\mathbb{T}}_{4}}\cap Q$, i.e.

$G_{Q}=Q\cap\Big{[}\pi_{Q}^{-1}\Big{(}N_{\rho}\big{(}\pi(L\cap
E_{\tilde{\mathbb{T}}_{4}}\cap Q)\big{)}\Big{)}\Big{]}.$

Then by (4.23) and (4.24), we have

$|E_{\tilde{\mathbb{T}}_{4}}\cap
G_{Q}|\gtrapprox\rho^{\varepsilon_{2}+\varepsilon_{3}+4\varepsilon_{4}}|E_{\tilde{\mathbb{T}}_{4}}\cap
Q|.$ (4.25)

Let $\tilde{\mathbb{T}}_{5}=\tilde{\mathbb{T}}_{4}$, and define the shading
$\tilde{Y}_{5}$ by
$\tilde{Y}_{5}(\tilde{T})=\tilde{Y}_{4}(\tilde{T})\cap\bigcup_{Q}G_{Q}$. By
(4.25) and the bound $\varepsilon_{2}+\varepsilon_{3}+4\varepsilon_{4}\leq
5\varepsilon_{4}$, we have
$\sum_{\tilde{T}\in\tilde{\mathbb{T}}_{5}}|\tilde{Y}_{5}(\tilde{T})|\gtrapprox\rho^{5\varepsilon_{4}}\sum_{\tilde{T}\in\tilde{\mathbb{T}}_{4}}|\tilde{Y}_{4}(\tilde{T})|$.

Unwinding definitions, we see that if $Q$ is a $\rho^{1/2}$-cube and if
$J\subset\mathbb{R}$ is an interval, then

$\mathcal{E}_{\rho}\Big{(}J\cap\big{(}V(p_{Q})\cdot(E_{\tilde{\mathbb{T}}_{5}}\cap
Q)\big{)}\Big{)}\sim\mathcal{E}_{\rho}\Big{(}V(p_{Q})\cdot(L_{J}\cap
E_{\tilde{\mathbb{T}}_{3}})\Big{)},$

where $L_{J}=L\cap\pi_{Q}^{-1}(J)$, and $L$ is the line from (4.24). In
particular, if $J$ is an interval of length $\tau$, then $L_{J}$ is contained
in a union of $O(1)$ $\tau$-cubes $W$, and hence

$\mathcal{E}_{\rho}\Big{(}J\cap\big{(}V(p_{Q})\cdot(E_{\tilde{\mathbb{T}}_{5}}\cap
Q)\big{)}\Big{)}\lesssim\sum_{W}\mathcal{E}_{\rho}\big{(}V(p)\cdot(E_{\tilde{\mathbb{T}}_{5}}\cap
W)\big{)},$

where the sum is taken over the $O(1)$ $\tau$-cubes $W$ described above. Since
the contribution from each of these $\tau$-cubes is
$\leq\rho^{-\varepsilon_{2}}(\tau/\rho)^{1-\sigma},$ we conclude that

$\mathcal{E}_{\rho}\Big{(}J\cap\big{(}V(p_{Q})\cdot(E_{\tilde{\mathbb{T}}_{5}}\cap
Q)\big{)}\Big{)}\lesssim\rho^{-\varepsilon_{2}}(\tau/\rho)^{1-\sigma}.$ (4.26)

We claim that the sub-collection $(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$
of $(\mathbb{T},Y)_{\delta}$ induced by
$(\tilde{\mathbb{T}}_{5},\tilde{Y}_{5})_{\rho}$ satisfies the conclusions of
Lemma 4.10. First, if $\varepsilon_{1},\ldots,\varepsilon_{4}$ are chosen
appropriately, then
$\sum_{T\in\mathbb{T}^{\prime}}|Y^{\prime}(T)|\geq\delta^{\varepsilon}$, so
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ is $\varepsilon$-extremal. Next,
let $Q$ be a $\rho^{1/2}$ cube, let $p\in Q$, and let $J$ be an interval of
length $\tau$. Then since $E_{\mathbb{T}^{\prime}}\subset
E_{\tilde{\mathbb{T}}_{5}}$, we have

$\mathcal{E}_{\rho}\Big{(}J\cap\big{(}V(p)\cdot(E_{\mathbb{T}^{\prime}}\cap
Q)\big{)}\Big{)}\lesssim\mathcal{E}_{\rho}\Big{(}J\cap\big{(}V(p)\cdot(E_{\tilde{\mathbb{T}}_{5}}\cap
Q)\big{)}\Big{)}.$ (4.27)

(4.20) now follows from (4.26). ∎

For a fixed $\rho$, the next lemma records what happens when we iterate Lemma
4.10 for many different scales $\tau$.

######  Lemma 4.11.

For all $\varepsilon>0,$ there exists $\eta>0,\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let
$(\mathbb{T},Y)_{\delta}$ be a $\eta$-extremal collection of tubes with
$\delta^{-\eta}$-Lipschitz plane map $V$. Let
$\rho\in[\delta^{1-\varepsilon},\delta^{\varepsilon}]$.

Then there exists a $\varepsilon$-extremal sub-collection
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$, so that the following holds. Let
$Q$ be a $\rho^{1/2}$-cube. Then $V(Q)\cdot(E_{\mathbb{T}^{\prime}}\cap Q)$ is
a $(\rho,1-\sigma;\delta^{-\varepsilon})_{1}$-ADset.

###### Proof.

Let $N\geq 1/\varepsilon$ be an integer. For each $0\leq j<N$, iteratively
apply Lemma 4.10 to $(\mathbb{T},Y)$ with $\varepsilon/2$ in place of
$\varepsilon$, $\rho$ as above, and $\tau=\tau_{j}=\rho^{(j+N)/2N}$. We will
select $\eta>0$ sufficiently small so that the resulting refinement
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ is $\varepsilon$-extremal. Let
$F=V(Q)\cdot(E_{\mathbb{T}^{\prime}}\cap Q)$.

Let $Q$ be a $\rho^{1/2}$-cube and let $J\subset\mathbb{R}$ be an interval of
length at least $\rho$. Shrinking $J$ if needed, we may suppose $J\subset
V(Q)\cdot Q$, so in particular $|J|\leq 2\rho^{1/2}$. Thus there is an integer
$0\leq j<N$ so that $\rho^{(j+N+1)/2N}<|J|\leq 2\rho^{(j+N)/2N}$. Let
$\tilde{J}$ be an interval of length $\tau_{j}=\rho^{(j+N)/2N}$ that contains
$J$ (if $|J|>\rho^{1/2}$ then a slight modification is needed; we need to
cover $J$ by a union of two intervals of length $\rho^{1/2}$). By Lemma 4.10,
we have

$\mathcal{E}_{\rho}(J\cap F)\leq\mathcal{E}_{\rho}(\tilde{J}\cap
F)\leq\delta^{-\varepsilon_{1}}(\rho^{(j+N)/2N}/\rho)^{1-\sigma}\leq\delta^{-\varepsilon_{1}}\rho^{-1/2N}(|J|/\rho)^{1-\sigma}.\qed$

The next lemma records what happens when we iterate Lemma 4.11 for many
different scales $\rho$.

######  Lemma 4.12.

For all $\varepsilon>0$, there exists $\delta_{0},\eta>0$ so that the
following holds. Let $\delta\in(0,\delta_{0}]$ and let
$(\mathbb{T},Y)_{\delta}$ be an $\eta$-extremal collection of tubes with a
$\delta^{-\eta}$-Lipschitz plane map $V$.

Then there exists a $\varepsilon$-extremal sub-collection
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$, so that

$V(p)\cdot\big{(}E_{\mathbb{T}^{\prime}}\cap B(p,\rho^{1/2}\big{)}\ \textrm{is
a}\ (\rho,1-\sigma;\delta^{-\varepsilon})_{1}\textrm{-ADset}\quad\textrm{for
all}\ \rho\in[\delta,1],\ p\in\mathbb{R}^{3}.$ (4.28)

###### Proof.

Let $N\geq 2/\varepsilon$ be an integer. For each $1\leq j<N$, iteratively
apply Lemma 4.11 to $(\mathbb{T},Y)$ with $\varepsilon/3$ in place of
$\varepsilon$, $C=N$, and $\rho=\rho_{j}=\delta^{j/N}$. We will select
$\eta>0$ sufficiently small so that the resulting refinement
$(\mathbb{T}^{\prime},Y^{\prime})_{\delta}$ is $\varepsilon$-extremal.

Next, let $\rho\in[\delta,1]$, let $p\in\mathbb{R}^{3}$, and let $J$ be an
interval of length at least $\rho$. Our goal is to show that

$\mathcal{E}_{\rho}\Big{(}J\ \cap\
\Big{[}V(p)\cdot\big{(}E_{\mathbb{T}^{\prime}}\cap
B(p,\rho^{1/2}\big{)}\Big{]}\Big{)}\leq\delta^{-\varepsilon}(|J|/\rho)^{1-\sigma}.$
(4.29)

Case 1: $\rho\geq\delta^{1/N}$. In this case, (4.29) follows from the simple
observation that $E_{\mathbb{T}^{\prime}}\subset B(0,1)$, and
$\mathcal{E}_{\rho}([-1,1])\leq 2\rho^{-1}\leq\delta^{-\varepsilon}$.

Case 2: $\rho\in[\delta^{1-1/N},\delta^{1/N})$. In this case, there is an
integer $1\leq j\leq N-2$ so that $\delta^{(j+1)/N}\leq\rho<\delta^{j/N}$.
$B(p,\rho^{1/2})$ can be covered by a union of $O(1)$ $\rho_{j}^{1/2}$-cubes.
By Lemma 4.11, for each such cube $Q$ we have

$\mathcal{E}_{\rho_{j}}\Big{(}J\ \cap\
\big{(}V(Q)\cdot(E_{\mathbb{T}^{\prime}}\cap
Q)\big{)}\Big{)}\leq\delta^{-\varepsilon_{1}}(|J|/\rho_{j})^{1-\sigma}.$
(4.30)

Recall that $V(Q)=V(p_{Q})$, where $p_{Q}$ is the center of $Q$. Since $Q$
intersects $B(p,\rho^{1/2})$, we have that $|p_{Q}-p|\lesssim\rho_{j}^{1/2}$.
Since $V$ is $\delta^{-\eta}$-Lipschitz, this implies
$\angle(V(Q),V(p))\lesssim\delta^{-\eta}\rho_{j}^{1/2}$. Finally, since $Q$
has diameter $\leq 2\rho_{j}^{1/2}$, we have

$\begin{split}\mathcal{E}_{\delta^{-\eta}\rho_{j}}\Big{(}J\ \cap\
\big{(}V(p)\cdot(E_{\mathbb{T}^{\prime}}\cap
Q)\big{)}\Big{)}&\sim\mathcal{E}_{\delta^{-\eta}\rho_{j}}\Big{(}J\ \cap\
\big{(}V(Q)\cdot(E_{\mathbb{T}^{\prime}}\cap Q)\big{)}\Big{)}\\\
&\leq\delta^{-\varepsilon_{1}}(|J|/\rho_{j})^{1-\sigma},\end{split}$

where the final inequality used (4.30).

Since (4.31) holds for each of the $O(1)$ cubes whose union covers
$B(p,\rho^{1/2})$, we have

$\mathcal{E}_{\delta^{-\eta}\rho_{j}}\Big{(}J\ \cap\
\Big{[}V(p)\cdot\big{(}E_{\mathbb{T}^{\prime}}\cap
B(p,\rho^{1/2}\big{)}\Big{]}\Big{)}\lesssim\delta^{-\varepsilon_{1}}(|J|/\rho_{j})^{1-\sigma}.$
(4.31)

This is almost what we want, except we have bounded the
$\delta^{-\eta}\rho_{j}$-covering number, rather than the $\rho$-covering
number. But since each interval of length $\delta^{-\eta}\rho_{j}$ can be
covered by $\leq\delta^{-1/N-\eta}$ intervals of length $\rho$, (4.31) implies

$\mathcal{E}_{\rho}\Big{(}J\ \cap\
\Big{[}V(p)\cdot\big{(}E_{\mathbb{T}^{\prime}}\cap
B(p,\rho^{1/2}\big{)}\Big{]}\Big{)}\lesssim\delta^{-\varepsilon_{1}-1/N-\eta}(|J|/\rho_{j})^{1-\sigma}.$

(4.29) now follows, provided $\delta_{0}$ and $\eta$ are chosen sufficiently
small.

Case 3: $\rho\in[\delta,\delta^{1-1/N})$. This reduces to Case 2, using the
observation that
$\mathcal{E}_{\rho}(X)\leq\delta^{-1/N}\mathcal{E}_{\delta^{1-1/N}}(X)$.

∎

###  4.4 Proof of Proposition 4.1

We are now ready to prove Proposition 4.1. In brief, we use the results from
Sections 4.1 and 4.2 to find an extremal set of $\delta$-tubes
$(\mathbb{T},Y)_{\delta}$ with a plane map. Next, we use the results from
Section 4.3 to show that this set of tubes arranges into grains of dimensions
$\sqrt{\delta}\times\sqrt{\delta}\times\delta$. We select a tube $T_{0}$ whose
$\sqrt{\delta}$ neighborhood covers about $\delta^{-1}$ tubes from
$(\mathbb{T},Y)_{\delta}$, and consider the unit rescaling (recall Definition
3.2) of this collection of tubes with respect to the $\sqrt{\delta}$
neighborhood $T_{0}$. This is a set of $\sqrt{\delta}$ tubes, which we will
denote by $(\tilde{T},\tilde{Y})_{\sqrt{\delta}}$.

In what follows, we define $\rho=\sqrt{\delta}$. Note that the $\sqrt{\delta}$
neighborhood of $T_{0}$ can be covered by about $\delta^{-1/2}$ balls of
radius $\sqrt{\delta}$, and the restriction of $E_{\mathbb{T}}$ to each of
these balls consists of a union of parallel grains of dimensions
$\sqrt{\delta}\times\sqrt{\delta}\times\delta$. After re-scaling, the
situation is as follows: The unit ball can be covered by $\rho^{-1}$ “slabs”
(rectangular prisms) of dimensions $1\times 1\times\rho$, and the restriction
of $E_{\tilde{\mathbb{T}}}$ to each slab consists of a union of parallel (re-
scaled) grains of dimensions $1\times\rho\times\rho$; these will be called
_global grains_ , and they will satisfy the properties described in Item 1
from the statement of Proposition 4.1.

Finally, we will apply the results from Section 4.3 to show that
$(\tilde{\mathbb{T}},\tilde{Y})_{\rho}$ arranges into grains of dimensions
$\sqrt{\rho}\times\sqrt{\rho}\times\rho$; these will be called _local grains_
, and they will satisfy the properties described in Item 2 from the statement
of Proposition 4.1. We now turn to the details.

###### Proof of Proposition 4.1.

Let $\varepsilon_{1},\ldots,\varepsilon_{4}>0$ be small constants to be chosen
below. We will select $\varepsilon_{i}$ very small compared to
$\varepsilon_{i+1},$ and and $\varepsilon_{4}$ very small compared to
$\varepsilon$.

Step 1: Global grains.
Define $\tau_{0}=\delta_{0}^{\frac{2+\sigma}{\sigma}}$. By Lemma 4.8 followed
by Lemma 4.12, there exists a $\varepsilon_{1}$-extremal collection of tubes
$(\mathbb{T},Y)_{\tau}$ for some $\tau\in(0,\tau_{0}]$ that has a
$\tau^{-\varepsilon_{1}}$-Lipschitz plane map $V$, and satisfies (4.28) with
$\varepsilon_{1}$ in place of $\varepsilon$.

Define $\delta=\tau^{\frac{\sigma}{2+\sigma}}$, so $\delta\in(0,\delta_{0}]$.
Apply Proposition 3.2 to $(\mathbb{T},Y)_{\tau}$, with $\tau$ in place of
$\delta$, $\delta$ in place of $\rho$, and $\varepsilon_{2}$ in place of
$\varepsilon$. We obtain a refinement $(\mathbb{T}_{2},Y_{2})_{\tau}$ of
$(\mathbb{T}_{1},Y_{1})_{\tau}$, and a $\varepsilon_{2}$-extremal collection
of $\delta$-tubes $(\tilde{\mathbb{T}}_{1},\tilde{Y}_{1})_{\delta}$.

Fix a choice of $\delta$-tube $\tilde{T}$; define
$\mathbb{T}^{\dagger}=\mathbb{T}_{2}[\tilde{T}]$, and let $Y^{\dagger}$ be the
restriction of $Y_{2}$ to $\mathbb{T}^{\dagger}$. Since the unit re-scaling of
$(\mathbb{T}^{\dagger},Y^{\dagger})_{\tau}$ relative to $\tilde{T}$ is
$\varepsilon_{2}$-extremal, we can find a sub-collection
$(\mathbb{T}_{1}^{\dagger},Y_{1}^{\dagger})_{\tau}$ of
$(\mathbb{T}^{\dagger},Y^{\dagger})_{\tau}$, with
$\sum|Y_{1}^{\dagger}(T)|\gtrapprox\tau^{\varepsilon_{2}}\sum|Y^{\dagger}(T)|,$
and a tube $T_{0}\in\mathbb{T}_{1}^{\dagger}$ with the following property (P):
for each $\delta$-cube $Q$ that intersects $E_{\mathbb{T}_{1}^{\dagger}}$, we
have $Y_{1}^{\dagger}(T_{0})\cap Q\neq\emptyset$. After enlarging $\tilde{T}$
slightly (we might have to replace $\delta$ by $2\delta$, but this is
harmless), we may suppose that $T_{0}$ and $\tilde{T}$ have the same coaxial
line.

Let $(\hat{\mathbb{T}},\hat{Y})_{\tau/\delta}$ be the unit re-scaling of
$(\mathbb{T}_{1}^{\dagger},Y_{1}^{\dagger})_{\tau}$ relative to $\tilde{T}$.
This re-scaling sends $T_{0}$ to a tube $\hat{T}_{0}$ whose coaxial line is
the $z$-axis. Let

$Z=\\{z\in\delta\mathbb{Z}\colon(0,0,z)\in\hat{Y}(\hat{T}_{0})\\}.$

Then since $T_{0}$ has Property (P), for each $p=(x,y,z)\in
E_{\hat{\mathbb{T}}}$ there is $z_{0}\in Z$ with $|z-z_{0}|\lesssim\delta$.

Next, we will analyze what (4.28) says about
$(\hat{\mathbb{T}},\hat{Y})_{\tau/\delta}$. For each $z\in Z$, let
$\phi^{-1}(z)$ be the inverse image of $z$ under the unit-rescaling associated
to $\tilde{T}$. Then $\phi^{-1}(z)$ is contained in the coaxial line of
$T_{0}$, and there is a point $p_{z}\in Y_{1}^{\dagger}(T_{0})$ with
$|\phi^{-1}(z)-p_{z}|\lesssim\delta$. By (4.28), we have

$V(p_{z})\cdot\big{(}E_{\mathbb{T}_{1}^{\dagger}}\cap B(p_{z},\delta)\big{)}\
\textrm{is a}\
(\delta^{2},1-\sigma;\tau^{-\varepsilon_{1}})_{1}\textrm{-ADset}.$ (4.32)

But if we consider the image of $B(p_{z},\delta)$ under $\phi$, (4.32) says
that there is a unit vector $\hat{V}(z)\in S^{2}$ so that

$\hat{V}(z)\cdot\Big{(}E_{\hat{\mathbb{T}}}\cap\big{(}\mathbb{R}^{2}\times[z,z+\delta)\big{)}\Big{)}\
\textrm{is a}\
(\delta,1-\sigma;O(\tau^{-\varepsilon_{1}}))_{1}\textrm{-ADset}.$ (4.33)

Since $|V(p_{z})\cdot\operatorname{dir}(L_{0})|\leq\tau$, we have
$|\hat{V}(z)\cdot(0,0,1)|\leq\tau/\delta\leq 1/10$ for all $z\in Z$. In
particular, since $V$ is $\tau^{-\varepsilon_{1}}$-Lipschitz, we have that
that $\hat{V}$ is also $\tau^{-\varepsilon_{1}}$-Lipschitz, and thus for each
interval $I\subset[-1,1]$ of length $\sim\tau^{\varepsilon_{1}}$, we must have
either $|\hat{V}(z)\cdot(1,0,0)|\geq\frac{1}{3}$ or
$|\hat{V}(z)\cdot(0,1,0)|\geq\frac{1}{3}$ for all $z\in I\cap Z$.
Interchanging the $x$ and $y$ axes if necessary, we may replace $Z$ with
$Z\cap I$, and define $\hat{\mathbb{T}}_{1}=\hat{\mathbb{T}}$,
$\hat{Y}_{1}(\hat{T})=\hat{Y}(\hat{T})\cap(\mathbb{R}^{2}\times I)$. If we
choose the interval $I$ appropriately, then
$\sum_{\hat{T}\in\hat{\mathbb{T}}}|\hat{Y}_{1}(\hat{T})|\gtrsim\tau^{\varepsilon_{1}}\sum_{\hat{T}\in\hat{\mathbb{T}}}|\hat{Y}(\hat{T})|$,
and $|\hat{V}(z)\cdot(1,0,0)|\geq\frac{1}{3}$ for all $z\in Z$. In particular,
there are $3\tau^{-2\varepsilon_{1}}$-Lipschitz functions $f,g\colon
Z\to\mathbb{R}$ so that for all $z\in Z$, we have that $\hat{V}(z)$ is
parallel to $(1,f(z),g(z))$ ($f$ is given by dividing the second component of
$\hat{V}$ by the first, and $g$ is defined by dividing the third component of
$\hat{V}$ by the first). Currently $f(z)$ is defined on $Z$, but we will
extend it slightly be defining $f(z+t)=f(z)$, if $z\in Z$ and
$t\in[0,\delta)$. Then by (4.33), for each $p=(x,y,z)\in
E_{\hat{\mathbb{T}}_{1}}$ we have

$(1,f(z),0)\cdot\Big{(}E_{\hat{\mathbb{T}}}\cap\big{(}\mathbb{R}^{2}\times[z,z+\delta)\big{)}\Big{)}\
\textrm{is a}\
(\delta,1-\sigma;O(\tau^{-\varepsilon_{1}}))_{1}\textrm{-ADset}.$ (4.34)

Step 2: Local grains.
Apply Lemma 4.3 to $(\hat{\mathbb{T}}_{1},\hat{Y}_{1})_{\tau/\delta}$ with
$\tau/\delta$ in place of $\delta$ and $\varepsilon_{3}$ in place of
$\varepsilon$; we obtain a refinement
$(\hat{\mathbb{T}}_{2},\hat{Y}_{2})_{\tau/\delta}$ and a map $U\colon
E_{\hat{\mathbb{T}}_{2}}\to S^{2}$ that satisfies

$|U(p)\cdot\operatorname{dir}(\hat{T})|\leq(\tau/\delta)^{\sigma/2}\quad\textrm{for
all}\ p\in E_{\hat{\mathbb{T}}_{2}},\ \hat{T}\in\hat{\mathbb{T}}_{2}(p).$

Next, apply Lemma 4.4 with $\tau/\delta$ in place of $\delta$, $\delta$ in
place of $\rho$ (note that this is a valid choice, since
$\delta=(\tau/\delta)^{\sigma/2}$), and $\varepsilon_{3}$ in place of
$\varepsilon$. We obtain a $\varepsilon_{3}$-extremal collection of
$\delta$-tubes $(\tilde{T},\tilde{Y})_{\delta}$ that covers
$(\hat{\mathbb{T}}_{2},\hat{Y}_{2})_{\tau/\delta}$ and a plane map $W\colon
E_{\tilde{\mathbb{T}}}\to S^{2}$. Each $\delta$-cube $Q\subset
E_{\tilde{\mathbb{T}}}$ intersects $E_{\hat{\mathbb{T}}_{2}}$, and thus by
(4.34), for each $z\in Z$ we have

$(1,f(z),0)\cdot\Big{(}E_{\tilde{\mathbb{T}}}\cap\big{(}\mathbb{R}^{2}\times[z,z+\delta)\big{)}\Big{)}\
\textrm{is a}\
(\delta,1-\sigma;O(\tau^{-\varepsilon_{1}}))_{1}\textrm{-ADset}.$ (4.35)

Indeed, to verify (4.35) let
$\tilde{X}=\big{(}E_{\tilde{\mathbb{T}}}\cap\big{(}\mathbb{R}^{2}\times[z,z+\delta)\big{)}\big{)}$
and let
$\hat{X}=\big{(}E_{\hat{\mathbb{T}}}\cap\big{(}\mathbb{R}^{2}\times[z,z+\delta)\big{)}\big{)}$.
It suffices to show that each $w\in(1,f(z),0)\cdot\tilde{X}$ is contained in
the $O(\delta)$-neighborhood of $(1,f(z),0)\cdot\hat{X}$. For such a $w$,
there exists a $\delta$-cube $Q$ that intersects $\tilde{X}$ with
$w\in(1,f(z),0)\cap Q$. But since $Q$ is a $\delta$-cube, it must be contained
in $\tilde{X}$. Since $Q\cap E_{\hat{\mathbb{T}}}\neq\emptyset,$ there must be
a point $p\in Q\cap\hat{X}$. Thus $|(1,f(z),0)\cdot w-(1,f(z),0)\cdot
p|\lesssim|w-p|\lesssim\delta$.

We will re-write (4.35) as

$(1,f(z_{0}),0)\cdot\Big{(}E_{\tilde{\mathbb{T}}}\cap\\{z=z_{0}\\}\Big{)}\
\textrm{is a}\
(\delta,1-\sigma;O(\tau^{-\varepsilon_{1}}))_{1}\textrm{-ADset},$ (4.36)

for all $z_{0}$ in the domain of $f$. Abusing notation slightly, we can
replace $f$ with its extension to $[-1,1]$; this extension is also
$3\tau^{-2\varepsilon_{1}}$-Lipschitz, and (4.36) continues to hold, since the
set on the LHS of (4.36) is empty if $z_{0}$ is not contained in the original
domain of definition of $f$.

To complete the proof, apply Lemma 4.12 to $(\tilde{T},\tilde{Y})_{\delta}$
with $\varepsilon_{4}$ in place of $\varepsilon$. If
$\varepsilon_{1},\ldots,\varepsilon_{4}$ are chosen sufficiently small, then
the resulting refinement of $(\tilde{T},\tilde{Y})_{\delta}$ almost satisfies
the conclusions of Proposition 4.1, except the function $f$ and the plane map
$W$ are not $1$-Lipschitz. This can be fixed via a mild rescaling, using Lemma
3.5. ∎

##  5 The global grains slope function is $C^{2}$

Our goal in this section is to strength the conclusions of Proposition 4.1 by
increasing the regularity of the global grains slope function. More precisely,
we will prove the following.

######  Proposition 5.1.

For all $\varepsilon,\delta_{0}>0$, there exists $\delta\in(0,\delta_{0}]$ and
a $\varepsilon$-extremal set of tubes $(\mathbb{T},Y)_{\delta}$ with the
following properties:

  1. $1^{\prime}.$

Global grains with $C^{2}$ slope function.
There is a function $f\colon[-1,1]\to\mathbb{R}$ with $\|f\|_{C^{2}}\leq 1$ so
that $\big{(}E_{\mathbb{T}}\cap\\{z=z_{0}\\}\big{)}\cdot(1,f(z_{0}),0)$ is a
$(\delta,1-\sigma,\delta^{-\varepsilon})_{1}$-ADset for each $z_{0}\in[-1,1]$.

  2. 2.

Local grains with Lipschitz plane map
$(\mathbb{T},Y)_{\delta}$ has a 1-Lipschitz plane map $V$. For all
$\rho\in[\delta,1]$ and all $p\in\mathbb{R}^{3}$,
$V(p)\cdot\big{(}B(p,\rho^{1/2})\cap E_{\mathbb{T}}\big{)}$ is a
$(\rho,1-\sigma,\delta^{-\varepsilon})_{1}$-ADset.

Our main tool for proving Proposition 5.1 will be a new result in projection
theory. To state this result at an appropriate level of generality, we
introduce a new type of discretized set.

######  Definition 5.1.

For $n\geq 1,\ \alpha\in(0,n]$, $C\geq 1$ and $\delta>0$, We say a set
$E\subset\mathbb{R}^{n}$ is a $(\delta,\alpha,C)_{n}$-set if $E$ is a union of
$\delta$-cubes, and for all $x\in\mathbb{R}^{n}$ and all $r\geq\delta$ we have

$|E\cap B(x,r)|\leq Cr^{\alpha}|E|.$ (5.1)

We form an analogous definition if $E\subset S^{1}$ is a union of
$\delta$-arcs.

Definition 5.1 is closely related to Katz and Tao’s notion of a
$(\delta,\alpha)_{n}$-set from [18]. However, in Katz and Tao’s definition,
the quantity $|E|$ in the RHS of (5.1) is replaced by $\delta^{n-\alpha}$; the
former could be much larger. Definition 5.1 is also closely related to our
definition of a $(\delta,\alpha,C)_{n}$-ADset from Definition 4.2. In
particular, if $E$ is a union of $\delta$-cubes that is also a
$(\delta,\alpha,C)_{n}$-ADset, and $|E|\geq c\delta^{n-\alpha}$ for some
$c>0$, then $E$ is a $(\delta,\alpha,C/c)_{n}$-set. In practice we will always
consider sets $E\subset\mathbb{R}^{2}$, so we will drop the subscript $2$ from
our notation. The reason for this definition (which is slightly at odds with
the traditional literature) is that several recent results in projection
theory can be more easily stated with this terminology.

######  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
$A_{1},A_{2},A_{3}\subset[0,1]^{2}$ be $(\delta,1,\delta^{-\eta})$-sets. Then
at least one of the following must hold.

  1. (A)

There exist orthogonal lines $\ell$ and $\ell^{\perp}$ whose
$\delta$-neighborhoods cover a significant fraction of $A_{1}\cup A_{2}$ and
$A_{3}$ respectively. In particular,

$\mathcal{E}_{\delta}(N_{\delta}(\ell)\cap
A_{1})\geq\delta^{\varepsilon-1},\quad\mathcal{E}_{\delta}(N_{\delta}(\ell)\cap
A_{2})\geq\delta^{\varepsilon-1},\ \ \textrm{and}\ \
\mathcal{E}_{\delta}(N_{\delta}(\ell^{\perp})\cap
A_{3})\geq\delta^{\varepsilon-1}.$ (5.2)

  2. (B)

Let $G\subset A_{1}\times A_{2}\times A_{3}$ be any set satisfying
$\mathcal{E}_{\delta}(G)\geq\delta^{\eta-3}$. Then there exists
$\rho\geq\delta$ and an interval $I$ of length at least $\delta^{-\eta}\rho$
such that

$\mathcal{E}_{\rho}\big{(}I\cap\\{(a_{1}-a_{2})\cdot
a_{3}\colon(a_{1},a_{2},a_{3})\in
G\\}\big{)}\geq\big{(}|I|/\rho\big{)}^{1-\varepsilon}.$ (5.3)

The proof of Theorem 5.2 uses ideas and results that are different in
character from the rest of the proof of Theorem 1.1, so we will defer the
proof to Section 8.

###  5.1 The slope function is locally linear

In this section we will use Theorem 5.2 to show that if
$(\mathbb{T},Y)_{\delta}$ is an extremal collection of tubes that satisfies
the conclusions of Proposition 4.1, then the restriction of the global grains
slope function to an interval of length $\rho^{1/2}$ looks linear at scale
$\rho$.

To begin, we will study the interplay between the “global grains” and “local
grains” structures from Proposition 4.1. If we restrict $E_{\mathbb{T}}$ to
the $\delta^{1/2}$ neighborhood of a global grain, then we obtain a set
$E\subset E_{\mathbb{T}}$ that has the following two structures: The global
grains property (Item 1 from Proposition 4.1) says that each “horizontal”
(i.e. $z=z_{0}$) slice of $E$ is a union of $1\times\delta$ rectangles
pointing in the direction $(1,f(z_{0}),0)$, and the spacing of these
rectangles forms a ADset in the sense of Definition 4.2. The local grains
property (Item 2 from Proposition 4.1) says that the restriction of $E$ to a
ball of radius $\delta^{1/2}$ is a union of
$\delta^{1/2}\times\delta^{1/2}\times\delta$ grains, whose normal direction is
given by the plane map $V$. We will use these two structures to construct sets
$F$, $G$, and $\mathcal{H}\subset F\times G\times G$ that satisfy the
hypotheses of Theorem 5.2. We will show that Conclusion (B) from Theorem 5.2
would contradict the ADset spacing condition from the global grains structure,
and hence the sets $F$ and $G$ must satisfy Conclusion (A). This in turn
implies that the slope function $f$ is linear at scale $\delta$ over intervals
of length $\delta^{1/2}$. The precise version is stated in Lemma 5.3 below.
The precise hypotheses of this lemma are slightly technical; they are
formulated in this fashion to allow the lemma to be applied at many scales and
locations.

######  Lemma 5.3.

For all $\varepsilon>0$, there exists $\eta,\delta_{0}>0$ so that the
following holds for all $\delta\in(0,\delta_{0}]$. Let $I\subset[-1,1]$ be an
interval of length $\delta^{1/2}$ with left endpoint $z_{0}$, and let
$Z\subset I$. Let $f\colon I\to[0,1]$ be 1-Lipschitz. Let
$E\subset[-1,1]^{2}\times Z$ be a union of $\delta$-cubes contained in the
$\delta^{1/2}$-neighborhood of a line pointing in direction $(-f(z_{0}),1,0)$,
with $|E|\geq\delta^{1+\sigma/2+\eta}$. Let $V\colon E\to S^{2}$ be
1-Lipschitz, with $|V(p)\cdot(0,0,1)|\leq 1/2$ for all $p\in E$. Suppose that
for each $z_{0}\in Z$,

$\big{(}E\cap\\{z=z_{0}\\}\big{)}\cdot\big{(}1,f(z_{0}),0\big{)}\quad\textrm{is
a}\ (\delta,1-\sigma,\delta^{-\eta})_{1}\textrm{-ADset},$ (5.4)

and for each $p\in E$,

$\big{|}\big{(}E\cap B(p,\delta^{1/2})\big{)}\cdot
V(p)\big{|}\leq\delta^{\frac{1+\sigma}{2}-\eta}.$ (5.5)

Then there exists a set $Z_{\operatorname{lin}}\subset Z$ with
$|Z_{\operatorname{lin}}|\geq\delta^{1/2+\varepsilon}$, and a linear function
$L\colon[0,\delta^{1/2}]\to[0,1]$ so that $|f(z)-L(z)|\leq\delta$ for all
$z\in Z_{\operatorname{lin}}$.

###### Proof.

Without loss of generality we may suppose that $I=[0,\delta^{1/2}]$, $f(0)=0$
(and hence $f(Z)\subset[-\delta^{1/2},\delta^{1/2}]$ and
$Z\subset[0,\delta^{1/2}]$), and that $E$ is contained in the
$\delta^{1/2}$-neighborhood of the $y$-axis. If not, then we can apply a
(harmless) translation so that $E$ is contained in the
$\delta^{1/2}$-neighborhood of a line passing through the origin, and then we
can apply a rotation around the $z$-axis so that $f(0)=0$.

Note that the image $E^{\prime}$ of $E$ under this rotation is no longer a
union of $\delta$-cubes (since by definition, $\delta$-cubes are aligned with
the standard grid). However, we can replace $E^{\prime}$ by the union