# On the dimension of exceptional parameters for nonlinear projections, and
the discretized Elekes-Rónyai theorem

Orit E. Raz, Joshua Zahl

###### Abstract

We consider four related problems. (1) Obtaining dimension estimates for the
set of exceptional vantage points for the pinned Falconer distance problem.
(2) Nonlinear projection theorems, in the spirit of Kaufman, Bourgain, and
Shmerkin. (3) The parallelizability of planar $d$-webs. (4) The Elekes-Rónyai
theorem on expanding polynomials.

Given a Borel set $A$ in the plane, we study the set of exceptional vantage
points, for which the pinned distance $\Delta_{p}(A)$ has small dimension,
that is, close to $(\dim A)/2$. We show that if this set has positive
dimension, then it must have very special structure. This result follows from
a more general single-scale nonlinear projection theorem, which says that if
$\phi_{1},\phi_{2},\phi_{3}$ are three smooth functions whose associated 3-web
has non-vanishing Blaschke curvature, and if $A$ is a
$(\delta,\alpha)_{2}$-set in the sense of Katz and Tao, then at least one of
the images $\phi_{i}(A)$ must have measure much larger than $|A|^{1/2}$, where
$|A|$ stands for the measure of $A$. We prove analagous results for $d$ smooth
functions $\phi_{1},\ldots,\phi_{d}$, whose associated $d$-web is not
parallelizable.

We use similar tools to characterize when bivariate real analytic functions
are “dimension expanding” when applied to a Cartesian product: if $P$ is a
bivariate real analytic function, then $P$ is either locally of the form
$h(a(x)+b(y))$, or $P(A,B)$ has dimension at least $\alpha+c$ whenever $A$ and
$B$ are Borel sets with Hausdorff dimension $\alpha$. Again, this follows from
a single-scale estimate, which is an analogue of the Elekes-Rónyai theorem in
the setting of the Katz-Tao discretized ring conjecture.

##  1 Introduction

We consider four related problems. (1) Obtaining dimension estimates for the
set of exceptional vantage points for the pinned Falconer distance problem.
(2) Nonlinear projection theorems, in the spirit of Kaufman, Bourgain, and
Shmerkin. (3) The parallelizability of planar $d$-webs. (4) The Elekes-Rónyai
theorem on expanding polynomials. We will briefly discuss each of these items.

We will begin with projection theory, and specifically the study of
exceptional parameters for nonlinear analogues of the linear projection
function $\pi_{e}\colon\mathbb{R}^{2}\to\mathbb{R}$, which is the orthogonal
projection onto $\operatorname{span}(e)$. A direction $e\in S^{1}$ is called
exceptional for a planar set $A$ if the dimension of the projection
$\pi_{e}(A)$ is smaller than the dimension of the projection in a “typical”
direction. Results of Kaufman and Bourgain quantify the size of the set of
exceptional directions. There are several results concerning nonlinear
generalizations of the projection function $\pi_{e}$. An important example of
a nonlinear projection is the pinned distance function $d_{p}(q)=|p-q|$. The
theme of many of these results is that under suitable constraints, nonlinear
projections are as well-behaved as linear projections, with respect to the
size of the set of exceptional parameters. We discover a new and unexpected
phenomenon, which says that often nonlinear projections are much better
behaved than linear ones.

Our results can also be expressed in the discretized setting of
$(\delta,\alpha)_{2}$ sets, which were introduced by Katz and Tao [25] as a
single-scale model for planar Borel sets that have Hausdorff dimension
$\alpha$. In brief, a $(\delta,\alpha)_{2}$ set is a union of about
$\delta^{-\alpha}$ interior-disjoint squares of side-length $\delta$, which
satisfy a Frostman-type non-concentration condition. It is straightforward to
construct an example of a $(\delta,\alpha)_{2}$ set $A$ that has projection of
size about $|A|^{1/2}$ in many different directions, or equivalently, a set
that has small image under may different linear functions $\pi_{e}$. We prove
that morally speaking, this phenomenon can only occur for linear functions.

The fibers of each function $\pi_{e}$ are parallel lines; more generally, we
say a set of smooth functions $\phi_{1},\ldots,\phi_{d}$ determine a _p_
arallelizable (planar) $d$-web if after a change of coordinates, the fibers of
each function $\phi_{i}$ are parallel lines. We prove that if a set of smooth
functions $\phi_{1},\ldots,\phi_{d}$ do not determine a parallelizable planar
$d$-web (in a certain quantitative sense), and if $A$ is a
$(\delta,\alpha)_{2}$ set, then at least one of the images $\phi_{i}(A)$ must
be much larger than $|A|^{1/2}$.

Finally, we shall discuss the connection with the Elekes-Rónyai theorem on
expanding polynomials. If $\phi_{1},\phi_{2},\phi_{3}$ are smooth functions,
then after a change of coordinates we can locally write $\phi_{1}(x,y)=x$,
$\phi_{2}(x,y)=y$, and $\phi_{3}(x,y)=P(x,y)$ for some smooth function $P$.
Suppose that $A$ is a $(\delta,\alpha)_{2}$ set, and that $\phi_{1}(A)$ and
$\phi_{2}(A)$ each have size about $|A|^{1/2}$. If we write
$A_{1}=\phi_{1}(A)$ and $A_{2}=\phi_{2}(A)$, then $A\subset A_{1}\times
A_{2}$, and the sets $A$ and $A_{1}\times A_{2}$ have comparable size.
Following the discussion above, we would like to show that $|P(A)|$ is much
larger than $|A|^{1/2}$, or almost equivalently, that $P(A_{1}\times A_{2})$
is much larger than $|A|^{1/2}$. If $P$ was a polynomial and if $A_{1}$ and
$A_{2}$ were discrete sets of cardinality $n$, then Elekes and Rónyai [14]
proved that either $P(A_{1}\times A_{2})$ is much larger than $n$, or $P$ has
a very specific form (for example $P(x,y)=x+y$). We will reduce the results
discussed above about non-parallelizable $d$-webs to a discretized analogue of
the Elekes-Rónyai theorem for smooth functions.

###  1.1 Exceptional directions for nonlinear projections

In [23], Kaufman proved that if $A\subset\mathbb{R}^{2}$ is a Borel set and if
$\pi_{e}\colon\mathbb{R}^{2}\to\mathbb{R}$ is the orthogonal projection onto
$\operatorname{span}(e)$, then $\pi_{e}(A)$ can only have small dimension for
a small exceptional set of directions $e\in\mathbb{S}^{1}$.

######  Theorem 1.1 (Kaufman).

Let $A\subset\mathbb{R}^{2}$ be a Borel set and let $s\leq\dim A$. Then

$\dim\\{e\in\mathbb{S}^{1}:\dim\pi_{e}(A)<s\\}\leq s.$ (1.1)

Here and throughout the paper, $\dim$ refers to Hausdorff dimension. Very
recently, Orponen and Shmerkin [33] obtained a quantitatively small (but
qualitatively very significant) improvement to the RHS of (1.1) whenever
$0<s<\dim A$ and $s<1$. In a slightly different direction, Bourgain [4]
obtained a large improvement over (1.1) when $s$ is close to $(\dim A)/2$. As
a special case of his result, we have the following.

######  Theorem 1.2 (Bourgain).

Let $A\subset\mathbb{R}^{2}$ be a Borel set. Then

$\dim\\{e\in\mathbb{S}^{1}\colon\dim\pi_{e}(A)\leq(\dim A)/2\\}=0.$ (1.2)

An example by Kaufman and Mattila [24] shows that (1.2) cannot hold if $(\dim
A)/2$ is replaced by $(\dim A)/2+c$ for any $c>0$.

Estimates similar to those in Theorem 1.1 hold for a broad class of
generalized projections (see [34] and the references therein), and recently
Shmerkin [43] proved an analogue of Theorem 1.2 where the set of orthogonal
projections $\\{\pi_{e}\\}_{e\in S^{1}}$ is replaced by a family
$\\{F_{\lambda}\\}_{\lambda\in\Lambda}$ of nonsingular $C^{2}$ functions
$F_{\lambda}\colon\mathbb{R}^{2}\to\mathbb{R}$, referred to as _nonlinear
projections_ , that satisfy a natural transversality condition.

A particularly interesting class of nonlinear projections are the pinned
distance functions $d_{p}(q)=|p-q|$. In [32], Orponen proved a packing
dimension version of the Falconer distance conjecture in the plane for
Ahlfors-regular sets by showing that the nonlinear projection $d_{p}(q)$ could
be analyzed using multi-scale averages of suitably chosen linear projections,
and thus the problem of computing the size of the pinned distance set
$\Delta_{p}(E)=\\{d_{p}(q)\colon q\in E\\}$ was closely related to the problem
of computing the size of certain orthogonal projections of $E$ at various
scales and locations. This powerful new idea has been extended in subsequent
works [26, 41, 43]. In particular, Shmerkin [43] proved an analogue of Theorem
1.2 for pinned distances.

######  Theorem 1.3 (Shmerkin).

Let $A\subset\mathbb{R}^{2}$ be a Borel set. Then

$\dim\\{p\in\mathbb{R}^{2}\colon\dim\Delta_{p}(A)\leq(\dim A)/2\\}=0.$ (1.3)

The following example due to Elekes [13] shows that the quantity $(\dim A)/2$
on the LHS of (1.3) cannot be replaced by $(\dim A)/2+c$ for any $c>0$.

######  Example 1.

Let $0<\alpha<2$ and let $c>0$. Let $X,Y,S\subset[0,1]$ be Borel sets (see
e.g. [24]) so that $\dim(X\times Y)=\alpha$, $\dim(S)>0$, and
$\dim(X-sY)<\alpha/2+c$ for each $s\in S$. For each $(a,b)\in X\times Y$,
define $q_{a,b}=(a/2,\sqrt{1+b-a^{2}/4})$, and define $A=\\{q_{a,b}\colon a\in
X,\ b\in Y\\}$. We have $\dim A=\alpha$. Define $K=S\times\\{0\\}$. Then for
each $(x,y)=q_{a,b}\in A$ and $s\in S$, we have
$|(s,0)-q_{a,b}|^{2}=(s^{2}+1)+(b-sa)$. Thus for each $p=(s,0)\in K$, we have
$\dim(\Delta_{p}(A))=\dim(B-sA)<\alpha/2+c$. We conclude that

$\\{p\in\mathbb{R}^{2}\colon\dim\Delta_{p}(A)\leq(\dim A)/2+c\\}\supset K,$
(1.4)

so in particular the set on the LHS of (1.4) has positive dimension.

In Example 1, the set $K$ is contained in a line. It is possible to construct
slightly more complicated multi-scale examples where the set (1.4) is not
contained in one line, but can be contained in a union of countably many
lines. The next theorem says that if $c>0$ is small, then this is essentially
the only way that the set (1.4) can have positive dimension. To make this
statement precise, we will need the following definition.

######  Definition 1.4.

We say a Borel set $K\subset\mathbb{R}^{2}$ is _curved_ if there exist
$\beta>0$ and $C>0$, and a Borel probability measure $\mu$ supported on $K$ so
that for all $r>0$,

$\mu^{3}(\\{(p_{1},p_{2},p_{3})\in K^{3}\colon|T(p_{1},p_{2},p_{3})|<r\\})\leq
Cr^{\beta},$ (1.5)

where $|T(p_{1},p_{2},p_{3})|$ is the area of the triangle spanned by
$p_{1},p_{2},p_{3}$. If $K$ is not curved, we say it is _flat_. In particular,
if $\dim(K)=0$ then $K$ is flat.

For example, if $K\subset\mathbb{R}^{2}$ is contained in a countable union of
lines, then it is flat. If $K$ has positive dimension and is contained in a
smooth plane curve that has nonvanishing curvature (or more generally, whose
curvature never vanishes to infinite order), then $K$ is curved. If
$K=K_{1}\times K_{2}$, where $K_{1},K_{2}\subset\mathbb{R}$ are Borel sets of
positive dimension, then $K$ is curved. If $K^{\prime}\subset K$ has positive
dimension and $K^{\prime}$ is curved, then $K$ is curved.

######  Theorem 1.5.

For each $0<\alpha<2$ there exists $c(\alpha)>0$ so that the following holds.
Let $A\subset\mathbb{R}^{2}$ be a Borel set, with $\dim(A)=\alpha.$ Then the
set $\\{p\in\mathbb{R}^{2}\colon\dim\Delta_{p}(A)\leq\dim(A)/2+c\\}$ is flat.

######  Corollary 1.6.

For each $0<\alpha<2$ there exists $c(\alpha)>0$ so that the following holds.
Let $A\subset\mathbb{R}^{2}$ be a Borel set, with $\dim(A)=\alpha.$ Let
$\gamma\subset\mathbb{R}^{2}$ be a smooth curve whose curvature never vanishes
to infinite order. Then

$\dim\\{p\in\gamma\colon\dim\Delta_{p}(A)\leq\dim(A)/2+c\\}=0.$

Theorem 1.5 is an example of a general phenomenon, which says that at a single
scale, it is impossible for a set to have small image simultaneously under
three nonlinear projections, provided these projections satisfy a certain non-
degeneracy condition. This condition will be described below.

###  1.2 A single-scale estimate: the role of Blaschke curvature

######  Definition 1.7.

Let $U\subset\mathbb{R}^{2}$ be a connected open set, and let
$\phi_{1},\phi_{2},\phi_{3}\colon U\to\mathbb{R}$ be smooth functions whose
gradients are non-vanishing and pairwise linearly independent at each point of
$U$. Blaschke [2] defined the curvature form

$2\frac{\partial}{\partial\phi_{1}}\frac{\partial}{\partial\phi_{2}}\log\frac{\partial\phi_{3}/\partial\phi_{1}}{\partial\phi_{3}/\partial\phi_{2}}d\phi_{1}\wedge
d\phi_{2}.$ (1.6)

This curvature form is invariant under change of coordinates, and it captures
important information about the 3-web given by the level-sets of the functions
$\phi_{1},\phi_{2},\phi_{3}$ on $U$. Indeed, Blaschke [2, Section 9] (see also
[21] for the exact formulation given above) showed that the curvature form
(1.6) depends only on the 3-web (i.e. the triple of foliations defined by the
level sets $\\{\phi_{i}=c\\}_{c\in\mathbb{R}}$), and not on the choice of
$\phi_{1},\phi_{2},\phi_{3}$.

If each of $\phi_{1},\phi_{2},\phi_{3}$ is linear, then the curvature form
(1.6) vanishes identically. Conversely, if the curvature form (1.6) vanishes
identically on $U$, then around each point $p\in U$ there is a local change of
coordinates so that

$\phi_{1}(x,y)=x;\quad\phi_{2}(x,y)=y;\quad\phi_{3}(x,y)=x+y.$ (1.7)

This observation is important in the context of projection theory for the
following reason: If $\phi_{1},\phi_{2},\phi_{3}$ are of the form (1.7), and
if $E=A\times A$, where $A\subset\mathbb{R}$ is an arithmetic progression,
then $\\#\phi_{i}(E)\leq 2(\\#E)^{1/2}$ for each index $i=1,2,3$. Thus if the
curvature form (1.6) vanishes identically, then for each integer $N\geq 1$ we
can construct a set $E\subset U$ with $\\#E=N$ so that $\\#\phi_{i}(E)\leq
2N^{1/2}$ for $i=1,2,3$. Similarly, it is possible to construct discretized
sets $E\subset U$ satisfying a Frostman type non-concentration condition so
that $\phi_{1}(E),\phi_{2}(E)$, and $\phi_{3}(E)$ each have size roughly
$|E|^{1/2}$. In this section we will describe several converses to this
statement, which are manifestations of the following principle: If the
curvature form (1.6) does not vanish identically, then one of
$\phi_{1}(E),\phi_{2}(E)$, or $\phi_{3}(E)$ must be substantially larger than
$|E|^{1/2}$.

Returning to the pinned Falconer distance problem, let $p_{1},p_{2},p_{3}$ be
points in $\mathbb{R}^{2}$ and let $\phi_{i}(q)=|p_{i}-q|^{2}$. As we will
show in Section 5, the curvature form (1.6) vanishes identically if and only
if $p_{1},p_{2},p_{3}$ are collinear. If $p_{1},p_{2},p_{3}$ are not
collinear, then the gradients of $\phi_{i}$ and $\phi_{j}$ are linearly
dependent precisely on $\ell_{i,j}$, the line spanned by $p_{i}$ and $p_{j}$.
We can now state the following single-scale version of Theorem 1.5.

######  Theorem 1.8.

For each $0<\alpha<2$, there exists $c=c(\alpha)>0$ such that the following
holds for all $\delta>0$ sufficiently small. Let
$p_{1},p_{2},p_{3}\subset[0,1]^{2}$ be three points that span a triangle with
area at least $\delta^{c}$. Let $X\subset[0,1]^{2}$ be a union of
$\delta$-squares with $|X|\geq\delta^{2-\alpha+c}$, and suppose that for all
balls $B$ of diameter $r\geq\delta$, $X$ satisfies the Frostman-type non-
concentration condition

$\begin{split}|X\cap B|&\leq r^{\alpha}\delta^{2-\alpha-c}.\end{split}$ (1.8)

Then for at least one index $i,$ we have

$|\Delta_{p_{i}}(X)|\geq\delta^{1-\alpha/2-c}.$ (1.9)

######  Remark 1.9.

Theorem 1.8 fails if we remove the condition that the three points span a
triangle with large area. Indeed, both Example 1 and the “railroad tracks”
example from [25, Figure 1] shows that if the three points $p_{1},p_{2},p_{3}$
are collinear (or almost collinear), then (1.9) can fail for all three points,
or indeed, any number of points111more precisely, for any integer $N$, we can
construct an example where an estimate of the form (1.9) fails for $N$
points..

######  Remark 1.10.

Theorem 1.8 is a discretized version of [15, Theorem 32] (see also [42]),
which bounds the number of “triple points” determined by three families of
circles. See [13, 36] for further discussion of this problem.

The next result is a version of Theorem 1.8 for arbitrary 3-webs.

######  Theorem 1.11.

For each $0<\alpha<2,$ there exists $c=c(\alpha)>0$ so that the following
holds. Let $K\subset U\subset\mathbb{R}^{2}$, where $K$ is compact and $U$ is
open and connected. Let $\phi_{1},\phi_{2},\phi_{3}\colon U\to\mathbb{R}$ be
smooth functions whose gradients are pairwise linearly independent at each
point of $U$, and suppose the Blaschke curvature form (1.6) is non-vanishing
on $U$. Then the following is true for all $\delta>0$ sufficiently small.

Let $X\subset K$ be a union of $\delta$-squares, with
$|X|\geq\delta^{2-\alpha+c}$, and suppose that for all balls $B$ of diameter
$r\geq\delta$, $X$ satisfies the non-concentration condition

$\begin{split}|X\cap B|&\leq r^{\alpha}\delta^{2-\alpha-c}.\end{split}$ (1.10)

Then for at least one index $i$ we have

$|\phi_{i}(X)|\geq\delta^{1-\alpha/2-c}.$ (1.11)

Theorem 1.11 is a discretized projection theorem in the spirit of [4, 43]. In
[4], Bourgain proved that if $X\subset[0,1]^{2}$ is a set satisfying (1.10),
and if $\Theta\subset S^{1}$ is a large (and non-concentrated) set of
directions, then there must exist a direction $e\in\Theta$ so that the
associated orthogonal projection $\pi_{e}(q)$ satisfies (1.11). The set
$\Theta$ must be large, since if we define
$X=[0,1]^{2}\cap(\delta^{\alpha}\mathbb{Z})^{2}$, then whenever $s$ and $t$
are integers with $|s|$ and $|t|$ small, the projection $(x,y)\cdot(s,t)$ will
have small cardinality and thus fail to satisfy (1.11). Thus it is new, and
rather surprising, that Theorem 1.11 only requires three projections.

###  1.3 A single-scale estimate when Blaschke curvature vanishes: the role of
convexity

Let us now consider a quadruple of functions $\phi_{1},\ldots,\phi_{4}$ where
each triple has vanishing Blaschke curvature. To simplify our discussion, we
will work in coordinates where $\phi_{1}(x,y)=x,$ $\phi_{2}(x,y)=y$,
$\phi_{3}(x,y)=x+y$, and $\phi_{4}(x,y)=u(x)+v(y)$. In this setting, we can
still expect expansion provided that at least one of the functions $u$ or $v$
is strictly convex.

######  Theorem 1.12.

For each $0<\alpha<2,$ there exists $c=c(\alpha)>0$ so that the following
holds. Let $u,v\colon[0,1]\to\mathbb{R}$ be smooth. Suppose
$u^{\prime},v^{\prime}$, and $u^{\prime\prime}$ are nonzero on $[0,1]$. Let
$\phi_{1}(x,y)=x,$ $\phi_{2}(x,y)=y,$ $\phi_{3}(x,y)=x+y$, and
$\phi_{4}(x,y)=u(x)+v(y)$. Then the following is true for all $\delta>0$
sufficiently small.

Let $X\subset[0,1]^{2}$ be a union of $\delta$-squares, with
$|X|\geq\delta^{2-\alpha+c}$, and suppose that for all balls $B$ of diameter
$r\geq\delta$, $X$ satisfies the non-concentration condition

$\begin{split}|X\cap B|&\leq r^{\alpha}\delta^{2-\alpha-c}.\end{split}$ (1.12)

Then for at least one index $i$ we have

$|\phi_{i}(X)|\geq\delta^{1-\alpha/2-c}.$ (1.13)

Theorem 1.12 can be viewed as a continuous analogue of [22, Theorem 1.2], and
indeed our proof is inspired by certain ideas from [22].

Finally, we remark that Theorem 1.12 allows us to prove similar expansion
results for $d$-tuples of functions $\phi_{1},\ldots,\phi_{d}$ for $d>4$. In
brief, suppose that $\phi_{1},\ldots,\phi_{d}$ are smooth functions, and that
the Blaschke curvature of each triple $\phi_{i},\phi_{j},\phi_{k}$ is
identically 0. Then after a change of coordinates we can write
$\phi_{1}(x,y)=x,$ $\phi_{2}(x,y)=y,$ $\phi_{3}(x,y)=x+y$, and
$\phi_{i}(x,y)=u_{i}(x)+v_{i}(y)$ for each $i\geq 4$. We can then apply
Theorem 1.12 for each $i\geq 4$. If $u_{i}^{\prime\prime}$ and
$v_{i}^{\prime\prime}$ are identically 0 for each index $i\geq 4$, then the
$d$-web $(\phi_{1},\ldots,\phi_{d})$ is parallelizable. Thus Theorem 1.12 is a
quantitative version of the principle that if $d$ functions
$\phi_{1},\ldots,\phi_{d}$ determine a non-parallelizable $d$-web and if $X$
is a $(\delta,\alpha)_{2}$ set, then $|\phi_{i}(X)|$ must be substantially
larger than $|X|^{1/2}$ for at least one index $i$.

###  1.4 The discretized Elekes-Rónyai theorem

Theorems 1.5 and 1.11 are closely related to the Elekes-Rónyai theorem, which
we will now briefly describe. In [16], Erdős and Szemerédi proved that if
$A\subset\mathbb{R}$ is a finite set, then either the sum set
$A+A=\\{a+a^{\prime}\colon a,a^{\prime}\in A\\}$ or the product set
$A.A=\\{aa^{\prime}\colon a,a^{\prime}\in A\\}$ must have cardinality much
larger than that of $A$.

######  Theorem 1.13 (Erdős-Szemerédi).

There exists $c>0$ so that for all finite sets $A\subset\mathbb{R}$, we have

$\\#(A+A)+\\#(A.A)\gtrsim(\\#A)^{1+c}.$ (1.14)

The Erdős-Szemerédi Theorem quantifies the principle that a subset of
$\mathbb{R}$ cannot be approximately closed under both addition and
multiplication. In [14], Elekes and Rónyai developed this idea in a slightly
different direction, and proved the following:

######  Theorem 1.14 (Elekes-Rónyai).

Let $P$ be a bivariate real polynomial. Then either $P$ is one of the special
forms $P(x,y)=h(a(x)+b(y))$ or $P(x,y)=h(a(x)b(y))$, where $h,a,b$ are
univariate real polynomials, or that for all finite sets of real numbers $A,B$
of cardinality $N$, we have

$\\#P(A,B)=\omega(N).$ (1.15)

If $P(x,y)=h(a(x)+b(y))$ or $P(x,y)=h(a(x)b(y))$, where $h,a,b$ are univariate
real polynomials, then we call $P$ a (polynomial) special form. If $P$ is not
a (polynomial) special form then we call it an expanding polynomial. Theorem
1.14 has since been generalized in several directions. See [15, 37, 38, 44,
30, 39, 1], and the references therein.

Similar questions can be asked for metric entropy. In this direction, Katz and
Tao proposed the discretized ring conjecture [25], which was solved by
Bourgain [3]. The discretized ring conjecture (now a theorem) is similar to
the Erdős-Szemerédi theorem, except cardinality has been replaced by metric
entropy. It was an important ingredient in the proof of Theorem 1.2

######  Theorem 1.15 (Bourgain).

For each $0<\alpha<1$, there is a number $c=c(\alpha)>0$ and $s=s(\alpha)>0$
so that the following holds for all $\delta>0$ sufficiently small. Let
$A\subset[1,2]$ be a union of $\delta$-intervals, with
$|A|=\delta^{1-\alpha}$. Suppose that $A$ satisfies the following non-
concentration condition for each interval $J$

$|A\cap J|\leq|J|^{\alpha}\delta^{1-\alpha-s}.$ (1.16)

Then

$|A+A|+|A.A|\geq\delta^{1-\alpha-c}.$

The non-concentration condition (1.16) arises naturally when discretizing
fractal sets; see [25] or Section 9 for details. In particular, in [4]
Bourgain used a variant of Theorem 1.15 to prove the following “dimension
expansion” result for subsets of $\mathbb{R}$:

######  Theorem 1.16 (Bourgain).

For each $0<\alpha<1$ there is a number $c=c(\alpha)>0$ so that the following
holds. Let $A\subset\mathbb{R}$ be a Borel set with $\dim A=\alpha$. Then
there exists $\lambda\in A$ so that

$\dim(A+\lambda A)\geq\alpha+c.$

Theorem 1.16 implies the Erdős-Volkmann ring conjecture [17] (proved by Edgar
and Miller [12] slightly earlier, and also addressed directly by Bourgain in
[3]), which asserts that there does not exist a proper Borel subring of the
reals with positive Hausdorff dimension.

Theorem 1.15 has seen a number of extensions and generalizations. In [6],
Bourgain and Gamburd proved a variant of 1.15 with a less restrictive version
of the non-concentration condition (1.16), while in [18] Guth, Katz, and the
second author found a simple new proof of Theorem 1.15 that yields an explicit
lower bound on the exponent $\varepsilon$. Theorem 1.15 has also been
generalized to $SU(2)$ [6] as well as other settings [5, 19, 20, 40].

We prove a discretized version of the Elekes-Rónyai theorem for analytic
functions, in the spirit of Theorem 1.15. Before stating our result, we will
define what it means for an analytic function to be a special form (cf. [35,
Lemma 10]).

######  Definition 1.17.

Let $P(x,y)$ be analytic on a connected open set $U\subset\mathbb{R}^{2}$. We
say $P$ is an (analytic) special form if on each connected region of
$U\backslash\big{(}\big{\\{}\frac{d}{dx}P=0\big{\\}}\cup\big{\\{}\frac{d}{dy}=0\big{\\}}\big{)}$,
there are univariate real analytic functions $h$, $a$, and $b$ so that
$P(x,y)=h(a(x)+b(y))$.

Note that every polynomial special form is also an analytic special form. We
are now ready to state our main result on expanding functions.

######  Theorem 1.18.

Let $0<\kappa\leq\alpha<1$, let $U\subset\mathbb{R}^{2}$ be a connected open
set that contains $[0,1]^{2}$, and let $P\colon U\to\mathbb{R}$ be analytic
(resp. polynomial). Then either $P$ is an analytic (resp. polynomial) special
form, or there exists $\varepsilon=\varepsilon(\alpha,\kappa)>0$ and
$\eta=\eta(\alpha,\kappa,P)>0$ so that the following is true for all
$\delta>0$ sufficiently small.

Let $A,B\subset[0,1]$ be unions of $\delta$ intervals, and suppose that for
all intervals $J$, $A$ and $B$ satisfy the non-concentration conditions

$\begin{split}|A\cap J|&\leq|J|^{\kappa}\delta^{1-\alpha-\eta},\\\ |B\cap
J|&\leq|J|^{\kappa}\delta^{1-\alpha-\eta}.\end{split}$ (1.17)

Let $E\subset A\times B$ be a union of $\delta$-squares, with
$|E|\geq\delta^{2-2\alpha+\eta}$. Then we have the growth estimate

$|P(E)|\geq\delta^{1-\alpha-\varepsilon}.$ (1.18)

######  Remark 1.19.

Note that the growth exponent $\varepsilon$ in Theorem 1.18 is independent of
$P$, and in particular independent of the degree of $P$ when $P$ is a
polynomial. This is interesting for the following reasons. First, if $P$ is a
polynomial then it might be possible (though likely difficult) to prove an
analogue of Theorem 1.18 using Bourgain’s discretized sum-product theorem
(Theorem 1.15) and tools from Ruzsa calculus. For example, these ideas were
used in [7] to prove expansion over $\mathbb{F}_{p}$ for a certain explicitly
specified polynomial. However, the entropy growth exponent $\varepsilon$
arising from such a strategy would necessarily depend on the polynomial $P$
(the degree of $P$, the number of monomial terms, etc).

Second, there is a general intuition in additive combinatorics that states
that one should expect a quantity such as $P(A\times B)$ to be large when
there are few solutions to $P(x,y)=P(x^{\prime},y^{\prime})$, and conversely.
Concretely, if $A$ and $B$ have cardinality $\delta^{-\alpha}$, then the
statements “there are at about $\delta^{-3\alpha+\varepsilon}$
$\delta$-separated solutions to $P(x,y)=P(x^{\prime},y^{\prime})$” and
“$P(A,B)$ has measure about $\delta^{1-\alpha-\varepsilon}$” are often morally
equivalent. In the present setting, however, this intuition is not entirely
correct—there exists a sequence of polynomials $P_{1},P_{2},\ldots$ of
increasing degree so that there are $\delta^{-3\alpha+o(1)}$ solutions to
$P_{j}(x,y)=P_{j}(x^{\prime},y^{\prime})$, and yet the measure of
$P_{j}(A\times B)$ remains small.

We will also prove the following “dimension expansion” version of Theorem
1.18, which is an analogue of Theorem 1.16.

######  Theorem 1.20.

For every $0<\alpha<1$, there exists $c=c(\alpha)>0$ so that the following
holds. Let $U\subset\mathbb{R}^{2}$ be a connected open set that contains
$[0,1]^{2}$ and let $P\colon U\to\mathbb{R}$ be analytic (resp. polynomial).
Then either $P$ is an analytic (resp. polynomial) special form, or for every
pair of Borel sets $A,B\subset[0,1]$ of dimension at least $\alpha$, we have

$\dim P(A\times B)\geq\alpha+c.$

###  1.5 Proof ideas

If $P(x,y)$ is a smooth function, we will define an auxiliary function
$K_{P}(x,y)$ that measures the extent to which $P$ “looks like” a special form
(in the sense of Definition 1.17) near the point $(x,y)$. $K_{P}(x,y)$ is
closely related to the Blaschke curvature of the 3-web defined by the
functions $\phi_{1}(x,y)=x,$ $\phi_{2}(x,y)=y$, and $\phi_{3}(x,y)=P(x,y)$ at
the point $(x,y)$. We will show that if $P$ is analytic and $K_{P}$ vanishes
identically, then $P$ is an analytic special form.

At the other extreme, if $K_{P}$ is bounded away from 0, then we will prove
that the set of solutions to $P(a,b)=P(a^{\prime},b^{\prime})$, where
$a,a^{\prime}\in A$ and $b,b^{\prime}\in B$ has small $\delta$-covering
number. We will call this an “energy dispersion” estimate, since it is an
upper bound on the $L^{2}$ norm of the multiplicity function
$m(z)=\mathcal{E}_{\delta}(P^{-1}(z)\cap(A\times B))$, where
$\mathcal{E}_{\delta}(\cdot)$ denotes the $\delta$-covering number. This
energy dispersion estimate will be stated precisely in Proposition 3.4, and it
is one of the main technical results of this paper. It is closely related to
the $L^{2}$ flattening lemma that was used by Bourgain and Gamburd [5, 6] to
establish expansion in the Cayley graph of $SL_{2}(\mathbb{F}_{p})$ and the
existence of a spectral gap in certain free subgroups of $SU(2)$.

Theorem 1.11 follows from Proposition 3.4 and a formula that relates the
auxiliary function $K_{P}$ to Blaschke curvature. Theorem 1.8 and Theorem 1.5
are proved by combining Proposition 3.4 with tools from real algebraic
geometry, while Theorem 1.18 is proved by combining Proposition 3.4 with tools
from semi-analytic geometry.

Proposition 3.4 is proved using ideas from additive combinatorics. We will
give a brief sketch of the proof, which is inspired by the proof of the
Elekes-Rónyai theorem in [38]. The goal is to prove an estimate of the form
$\mathcal{E}_{\delta}(\\{P(a,b)=P(a^{\prime},b^{\prime})\\})\leq\delta^{-3\alpha+\varepsilon}$
for some $\varepsilon=\varepsilon(\alpha)>0$. Here and for the rest of this
proof sketch, we will always suppose that $a,a^{\prime}\in A$ and
$b,b^{\prime}\in B$. Suppose instead that
$\mathcal{E}_{\delta}(\\{P(a,b)=P(a^{\prime},b^{\prime})\\})\geq\delta^{-3\alpha+\varepsilon}$;
if $\varepsilon>0$ is sufficiently small we will obtain a contradiction.

For the purpose of this sketch, we will suppose that for each $x,x^{\prime}$
and $y$, there is a unique $g(x,x^{\prime},y)$ so that
$P(x,y)=P(x^{\prime},g(x,x^{\prime},y)).$ In particular, if
$P(a,b)=P(a^{\prime},b^{\prime}),$ then $g(a,a^{\prime},b)\in B$. This means
there are many triples $(a,a^{\prime},b)\in A\times A\times B$ for which
$g(a,a^{\prime},b)\in B$. Concretely, we have an estimate of the form

$|\\{(a,a^{\prime},b)\colon g(a,a^{\prime},b)\in
B\\}|\geq\delta^{3-3\alpha+O(\varepsilon)}.$ (1.19)

Next, we will write $g(a,a^{\prime},b)=g_{b}(a,a^{\prime})$; we will think of
$\\{g_{b}\\}_{b\in B}$ as a family of (nonlinear) projections from
$\mathbb{R}^{2}\to\mathbb{R}$. The bound (1.19) implies that for many $b\in
B$, there is a set $E_{b}\subset A\times A$ with
$|E_{b}|\geq\delta^{O(\varepsilon)}|A\times A|$ so that $g_{b}(E_{b})\subset
B$, and in particular,
$|g_{b}(E_{b})|\leq\delta^{-O(\varepsilon)}|E_{b}|^{1/2}$. This is precisely
the setting where Shmerkin’s projection theorem (discussed in Section 1.1) can
be applied. Shmerkin’s projection theorem says that if the family of
projection maps $\\{g_{b}\\}$ satisfies a certain non-degeneracy condition
(which is indeed satisfied if the auxiliary function $K_{P}$ is bounded away
from 0), then it is impossible for $|g_{b}(E_{b})|$ to be small for many
different projection maps $g_{b}$. This is a contradiction, which completes
the proof.

###  1.6 Structure of the paper

In Section 2 we will introduce Shmerkin’s nonlinear discretized projection
theorem, and we will recast this theorem into a (slightly technical) statement
about energy dispersion; this will be Lemma 2.5. In Section 3 we will
introduce the auxiliary function $K_{P},$ which helps quantify whether $P$
(locally) looks like a special form, and we show that it has two key
properties. First, if $K_{P}$ vanishes identically, then $P$ is a special
form. Second, when $K_{P}$ is large, then there can be few solutions to
$P(x,y)=P(x^{\prime},y^{\prime})$; this will be Proposition 3.4. Proposition
3.4 is used to prove most of the results in the paper. The structure of the
paper is as follows.

Lem 2.5Prop 3.4Thm 1.12Thm 1.18Thm 1.11Thm 1.8Thm 1.5Thm 1.20§3§6§8§4§9§5§7

###  1.7 Notation

In what follows, $\delta$ will denote a small positive number. We will be
interested in the asymptotic behaviour various quantities as $\delta\searrow
0$. If $f$ and $g$ are functions, we write $f\lesssim g$ (or $f=O(g)$ or
$g=\Omega(f)$) if there is a constant $C$ (independent of $\delta$) so that
$f\leq Cg$. Such a constant will be called an “implicit constant in the
$\lesssim$ notation.” If $f\lesssim g$ and $g\lesssim f$, we write $f\sim g$.
While our main theorems involve functions in the plane, some of our
intermediate results will be slightly more general and will involve functions
with domain $\mathbb{R}^{d}$. All implicit constants in the $\lesssim$
notation are allowed to depend on $d$. We will also write $f=O_{\alpha}(g)$ if
there is a constant $C$ (which may depend on $\alpha$) so that $f\leq Cg$.

If $X\subset\mathbb{R}^{d}$, we will use $\mathcal{E}_{\delta}(X)$ to denote
the $\delta$-covering number of $X$; $\\#X$ to denote the cardinality of $X$;
and $|X|$ to denote the Lebesgue measure of $X$. To improve clarity, we
sometimes use $\lambda_{d}(X)$ in place of $|X|$ to emphasize the ambient
dimension. If $t>0$ we use $N_{t}(X)$ to denote the $t$-neighborhood of $X$.

Our arguments will frequently involve sets of the form $I_{1}\times
I_{2}\times\ldots\times I_{d},$ where $I_{1},\ldots,I_{d}$ are closed
intervals. We will refer to such sets as rectangles.

###  1.8 Thanks

The authors would like to thank Pablo Shmerkin for numerous helpful comments
and suggestions. The authors would like to thank Michael Christ for
stimulating discussions, and for alerting them to Blaschke curvature, and its
connection to polynomial and analytic special forms.

##  2 Shmerkin’s nonlinear discretized projection theorem and its consequences

In this section we will discuss Shmerkin’s nonlinear generalization [43] of
Bourgain’s discretized projection theorem [4]. Shmerkin’s nonlinear projection
theorem concerns smooth functions of the form $G\colon I\to\mathbb{R}$, where
$I\subset[0,1]^{3}$ is a rectangle. We will call these “projection functions.”

In what follows, it will be helpful to introduce some additional notation. If
$G\colon I\to\mathbb{R}$ is a projection function, define
$G_{(z)}(x,y)=G(x,y,z)$. In particular,

$\nabla G_{(z)}(x,y)=\big{(}\partial_{x}G(x,y,z),\
\partial_{y}G(x,y,z)\big{)},$

so $\nabla G_{(z)}(x,y)$ is a vector in $\mathbb{R}^{2}$. For such a function
$G$, and for $(x,y,z)\in I$, define the map

$\theta_{(x,y)}(z)=\angle\operatorname{dir}(\nabla G_{(z)}(x,y)).$ (2.1)

With these definitions, we can now state Shmerkin’s result from [43].

######  Theorem 2.1 (Shmerkin).

For every $\eta>0$ and $C>0$, there exists $\tau=\tau(\eta)>0$ and
$\delta_{0}=\delta_{0}(\eta,C)>0$ such that the following holds for all
$0<\delta<\delta_{0}$. Let $I=I_{1}\times I_{2}\times I_{3}\subset[0,1]^{3}$
be a rectangle, let $X\subset I_{1}\times I_{2}$ be a union of
$\delta$-squares, and let $Z\subset I_{3}$ be a union of $\delta$-intervals.

Let $G\colon I\to\mathbb{R}$ be a projection function that satisfies

$\sup_{z\in Z}\|G_{(z)}\|_{C^{2}(I_{1}\times
I_{2})}<C,\quad\quad\inf_{(x,y,z)\in X\times Z}|\nabla G_{(z)}(x,y)|>C^{-1}.$
(2.2)

Suppose that $X$ satisfies the non-concentration estimate

$|X\cap Q|\leq\delta^{\eta}|X|,$ (2.3)

whenever $Q\subset[0,1]^{2}$ is a square of side-length $|X|^{1/2}$. Suppose
furthermore that for each $(x,y)\in X$ and each arc $J\subset S^{1}$ of length
$|J|\geq\delta$, we have the “transversality” estimate

$|\\{z\in Z\colon\theta_{(x,y)}(z)\in J\\}|\leq\delta^{-\tau}|J|^{\eta}|Z|.$
(2.4)

Then there is a small bad set $Z_{\operatorname{bad}}\subset Z$ with
$|Z_{\operatorname{bad}}|\leq\delta^{\tau}|Z|$ so that for all $z\in
Z\backslash Z_{\operatorname{bad}}$, the following holds: Let
$X^{\prime}\subset X$ with $|X^{\prime}|\geq\delta^{\tau}|X|$. Then

$\mathcal{E}_{\delta}(\\{G(x,y,z)\colon(x,y)\in
X^{\prime}\\})\geq\delta^{-\tau}\mathcal{E}_{\delta}(X)^{1/2}.$ (2.5)

Our task for the remainder of this section is to reformulate Theorem 2.1 as an
energy dispersion estimate.

###  2.1 Weakening the hypotheses of Theorem 2.1

Our first task is to state a version of Theorem 2.1 where the bounds (2.2) are
slightly less restrictive. We will also recast several of the hypotheses and
conclusions to align more closely with the setup of Theorem 1.18.

######  Lemma 2.2.

For every $0<\alpha<1$ and $0<\kappa\leq\alpha$, there exists
$\sigma=\sigma(\alpha,\kappa)>0$ and $\delta_{0}=\delta_{0}(\alpha,\kappa)$
such that the following holds for all $0<\delta<\delta_{0}$.

Let $I=I_{1}\times I_{2}\times I_{3}\subset[0,1]^{3}$ and let $G\colon
I\to\mathbb{R}$ be a projection function that satisfies the non-degeneracy
conditions

$\displaystyle\|G\|_{C^{2}(I)}$ $\displaystyle\leq\delta^{-\sigma},$ (2.6)
$\displaystyle\inf_{I}|\nabla G_{(z)}(x,y)|$
$\displaystyle\geq\delta^{\sigma},$ (2.7)
$\displaystyle\inf_{I}|\partial_{z}\theta_{(x,y)}(z)|$
$\displaystyle\geq\delta^{\sigma}.$ (2.8)

Let $A_{1},A_{2},A_{3}$ be sets, with $A_{i}\subset I_{i},$ and let
$A=A_{1}\times A_{2}\times A_{3}$. Suppose that for each index $i$ and each
interval $J$ of length at least $\delta$, we have the non-concentration
condition

$\mathcal{E}_{\delta}(A_{i}\cap
J)\leq\delta^{-\sigma}|J|^{\kappa}\delta^{-\alpha}.$ (2.9)

Let $R\subset\mathbb{R}$ be a set with
$\mathcal{E}_{\delta}(R)\leq\delta^{-\alpha-\sigma}$. Then

$\mathcal{E}_{\delta}\big{(}\\{(a_{1},a_{2},a_{3})\in A\colon
G(a_{1},a_{2},a_{3})\in R\\}\big{)}\leq\delta^{-3\alpha+\sigma}.$ (2.10)

###### Proof.

Our proof consists of three main steps. While there are a few technical
details, each of these steps follow standard arguments. First, note that our
projection function $G$ satisfies the conditions (2.6) and (2.7). These are
different from the requirement (2.2) that is needed when applying Theorem 2.1.
We will apply a re-scaling argument to create a new function $\tilde{G}$ that
satisfies (2.2).

Second, our projection function $G$ satisfies the condition (2.8), and our
sets $A_{1},A_{2},A_{3}$ satisfy the non-concentration condition (2.9). These
are different from the corresponding requirements (2.4) and (2.3). We will
verify that our new function $\tilde{G}$ and appropriately constructed sets
$X$ and $Z$ satisfy (2.3) and (2.4).

Third, we will show that our desired bound (2.10) follows by applying Theorem
2.1 to our newly constructed function $\tilde{G}$. In brief, if the bound
(2.10) failed then there must exist a large subset $A_{3}^{\prime}\subset
A_{3}$ so that $G(A_{1},A_{2},z)$ has small $\delta$-covering number for each
$z\in A_{3}^{\prime}$. We will show that this contradicts Theorem 2.1.

Constructing $\tilde{G}$: A re-scaling argument. First we will find a slightly
smaller sub-rectangle of $I$ that still captures most of the triples from
(2.10). Let $I^{\prime}=I_{1}^{\prime}\times I_{2}^{\prime}\times
I_{3}^{\prime}\subset I$ be a rectangle with
$|I_{1}^{\prime}|,|I_{2}^{\prime}|\leq\delta^{2\sigma}$ and
$|I_{3}^{\prime}|\leq\delta^{4\sigma}$ that maximizes the quantity

$\mathcal{E}_{\delta}\big{(}\\{(a_{1},a_{2},a_{3})\in I^{\prime}\cap A\colon
G(a_{1},a_{2},a_{3})\in R\\}\big{)}.$

Define $A_{i}^{\prime}=A_{i}\cap I_{i}^{\prime}$. and define
$A^{\prime}=A_{1}^{\prime}\times A_{2}^{\prime}\times A_{3}^{\prime}$. Since
$I\subset[0,1]^{3}$, by pigeonholing, we have

$\begin{split}\mathcal{E}_{\delta}&\big{(}\\{(a_{1},a_{2},a_{3})\in A\colon
G(a_{1},a_{2},a_{3})\in R\\}\big{)}\\\
&\leq\delta^{-8\sigma}\mathcal{E}_{\delta}\big{(}\\{(a_{1},a_{2},a_{3})\in
A^{\prime}\colon G(a_{1},a_{2},a_{3})\in R\\}\big{)}.\end{split}$ (2.11)

Let

$m=\inf_{I^{\prime}}|\nabla G_{(z)}(x,y)|.$

By (2.7) we have $m\geq\delta^{\sigma}$. By (2.6) we have

$\sup_{I^{\prime}}|\nabla G_{(z)}(x,y)|\leq
m+\delta^{-\sigma}\operatorname{diam}(I^{\prime})\leq 4m.$ (2.12)

Let $x_{0}$ be the left endpoint of $I_{1}^{\prime}$, let $y_{0}$ be the left
endpoint of $I_{2}^{\prime}$, and let $z_{0}$ be the left endpoint of
$I_{3}^{\prime}$. Define
$\tilde{I}_{1}=\delta^{-2\sigma}(I_{1}^{\prime}-x_{0})$,
$\tilde{I}_{2}=\delta^{-2\sigma}(I_{2}^{\prime}-y_{0})$, and
$\tilde{I}_{3}=I_{3}^{\prime}-z_{0}$. Thus $\tilde{I}\subset[0,1]^{3}$ is a
rectangle with bottom corner $(0,0,0)$. For $(x,y,z)\in\tilde{I}$, define

$\tilde{G}(x,y,z)=m^{-1}\delta^{-2\sigma}(G(\delta^{2\sigma}x+x_{0},\delta^{2\sigma}y+y_{0},z+z_{0})-G(x_{0},y_{0},z_{0})).$

We have $\tilde{G}\colon\tilde{I}\to\mathbb{R}$, with $\tilde{G}(0,0,0)=0$.
Indeed, if $(x,y,z)\in\tilde{I}$, then $(\delta^{2\sigma}x+x_{0}\in
I_{1}^{\prime};\delta^{2\sigma}y+y_{0}\in I_{2}^{\prime};$ and $z+z_{0}\in
I_{3}^{\prime}$, and thus
$G(\delta^{2\sigma}x+x_{0},\delta^{2\sigma}y+y_{0},z+z_{0})-G(x_{0},y_{0},z_{0})$
is well-defined.

We have

$\begin{split}\partial_{x}\tilde{G}(x,y,z)&=m^{-1}G_{1}(\delta^{2\sigma}x+x_{0},\delta^{2\sigma}y+y_{0},z+z_{0}),\\\
\partial_{y}\tilde{G}(x,y,z)&=m^{-1}G_{2}(\delta^{2\sigma}x+x_{0},\delta^{2\sigma}y+y_{0},z+z_{0}),\end{split}$
(2.13)

where for clarity we write $G_{1}(\cdot,\cdot,\cdot)$ (resp.
$G_{2}(\cdot,\cdot,\cdot)$ ) to denote the partial derivative of $G$ with
respect to its first (resp. second) variable; this notation will not be used
anywhere else in the paper. (2.13) and the definition of $m$ implies that

$\inf_{\tilde{I}}|\nabla\tilde{G}_{(z)}(x,y)|\geq 1.$ (2.14)

On the other hand, by (2.12) we have

$\sup_{\tilde{I}}|\nabla\tilde{G}_{(z)}(x,y)|\leq 6.$

This implies $|\tilde{G}(x,y,z)-\tilde{G}(0,0,z)|\leq 10$ for all
$(x,y,z)\in\tilde{I}$. By (2.6), we have
$|\tilde{G}(0,0,z)|\leq\delta^{-\sigma}m^{-1}|z|\leq\delta^{-2\sigma}|I_{3}^{\prime}|\leq
1,$ for all $z\in\tilde{I}_{3}$. Thus

$|\tilde{G}(x,y,z)|\leq 11\quad\textrm{for all }\ (x,y,z)\in\tilde{I}.$ (2.15)

Next, we can compute

$|\partial_{xx}\tilde{G}(x,y,z)|=|m^{-1}\delta^{2\sigma}(\partial_{xx}G)(\delta^{2\sigma}x+x_{0},\delta^{2\sigma}y+y_{0},z+z_{0})|\leq
1,$

where the final inequality used the bound $m\geq\delta^{\sigma}$ and (2.6). A
similar computation shows that the other second derivatives of $\tilde{G}$
with respect to $x$ and $y$ are bounded by 1, and thus

$\sup_{z\in\tilde{I}_{3}}\|\tilde{G}_{(z)}\|_{C^{2}(\tilde{Q})}\leq 100.$
(2.16)

Finally, if $(x,y,z)\in\tilde{I}$, then

$\tilde{\theta}_{(x,y)}(z)=\theta_{(\delta^{2\sigma}x+x_{0},\delta^{2\sigma}y+y_{0})}(z+z_{0}),$

so for each $(x,y,z)\in\tilde{I}$, (2.8) implies

$|\partial_{z}\tilde{\theta}_{(x,y)}(z)|\geq\delta^{\sigma}.$ (2.17)

Let $\tilde{A}_{1}=N_{\delta}(\delta^{-2\sigma}(A_{1}^{\prime}-x_{0}))$,
$\tilde{A}_{2}=N_{\delta}(\delta^{-2\sigma}(A_{2}^{\prime}-y_{0}))$, and
$\tilde{A}_{3}=N_{\delta}(A_{3}^{\prime}-z_{0})$, and define
$\tilde{A}=\tilde{A}_{1}\times\tilde{A}_{2}\times\tilde{A}_{3}$. We have that
The sets $\tilde{A}_{1}$ and $\tilde{A}_{2}$ satisfy a slightly weaker version
of (2.9)—if $J\subset[0,1]$ is an interval, then

$\begin{split}\mathcal{E}_{\delta}(\tilde{A}_{i}\cap
J)&\leq\delta^{-2\sigma}\mathcal{E}_{\delta}(A_{i}\cap\delta^{\rho}(J+x_{0}))\\\
&\leq|J|^{\kappa}\delta^{-\alpha-3\sigma}.\end{split}$ (2.18)

The set $\tilde{A}_{3}$ is just the $\delta$-thickening of a translate of
$A_{3}^{\prime}$, so (2.18) follows immediately from (2.9).

Finally, let $\tilde{R}=m^{-1}\delta^{-2\sigma}(R-G(x_{0},y_{0},z_{0}))$ and
note that $\mathcal{E}_{\delta}(\tilde{R})\leq\delta^{-\alpha-4\sigma}$. Then

$\mathcal{E}_{\delta}\big{(}\\{(a_{1},a_{2},a_{3})\in A^{\prime}\colon
G(a_{1},a_{2},a_{3})\in
R\\}\big{)}\leq\delta^{4\sigma}\mathcal{E}_{\delta}\big{(}\\{(a_{1},a_{2},a_{3})\in\tilde{A}\colon\tilde{G}(a_{1},a_{2},a_{3})\in\tilde{R}\\}\big{)}.$
(2.19)

Combining (2.11) and (2.19), we see that in order to establish (2.10), it
suffices to prove the bound

$\mathcal{E}_{\delta}\big{(}\\{(a_{1},a_{2},a_{3})\in\tilde{A}\colon\tilde{G}(a_{1},a_{2},a_{3})\in\tilde{R}\\}\big{)}\lesssim\delta^{-3\alpha+5\sigma}.$
(2.20)

We will establish the bound (2.20) by applying Theorem 2.1 to $\tilde{G}$. To
do so, we must verify that $\tilde{G}$, $\tilde{A}_{1}$, $\tilde{A}_{2}$, and
$\tilde{A}_{3}$ satisfy the hypotheses of Theorem 2.1.

Verifying the conditions for Theorem 2.1 Let $\eta=(1-\alpha)\kappa$, and let
$\tau>0$ be the value obtained by applying Theorem 2.1 with this choice of
$\eta$. Let

$\sigma<\min\big{(}(1-\alpha)\kappa/20,\ \tau/11\big{)}.$ (2.21)

Define

$T:=\\{(a_{1},a_{2},a_{3})\in\tilde{A}\colon\tilde{G}(a_{1},a_{2},a_{3})\in\tilde{R}\\}.$

Since $\tilde{A}_{1},\tilde{A}_{2}$ an $\tilde{A}_{3}$ are unions of $\delta$
intervals, to obtain (2.20) it suffices to prove that

$\lambda_{3}(T)\leq\delta^{3-3\alpha+5\sigma}.$ (2.22)

First, we can suppose that for each index $i$,

$\lambda_{1}(\tilde{A}_{i})\geq\delta^{1-\alpha+14\sigma}.$ (2.23)

Indeed, if (2.23) fails for some index $i$ then (2.22) follows from the
trivial bound

$\begin{split}\lambda_{3}(T)&\leq\lambda_{3}(\tilde{A}_{1}\times\tilde{A}_{2}\times\tilde{A}_{3})\\\
&\leq\delta^{2-2\alpha-9\sigma}\min_{1\leq i\leq
3}\lambda_{1}(A_{i}),\end{split}$

where on the last line we used (2.18) with $J=[0,1]$.

Let $X=\tilde{A}_{1}\times\tilde{A}_{2}$ and let $Z=\tilde{A}_{3}$. We will
verify that $\tilde{G}$ and the sets $X$ and $Z$ satisfy the hypotheses of
Theorem 2.1 with parameter $\eta$. To begin, (2.2) follows from (2.14) and
(2.16).

Next we will verify that $X$ satisfies the non-concentration estimate (2.3).
Let $r=\lambda_{2}(X)^{1/2}\leq\delta^{1-\alpha-3\sigma}$. Let $B(x,r)$ be a
ball of radius $r$, and let $J_{1}\times J_{2}$ be a square of side-length
$2r$ that contains $B(x,r)$. Then by (2.18), we have

$\begin{split}\lambda_{2}(X\cap B(x,r))&\leq\lambda_{1}(\tilde{A}_{1}\cap
J_{1})\lambda_{1}(\tilde{A}_{2}\cap J_{2})\\\
&\leq\big{(}(\delta^{-3\sigma}(2\delta^{1-\alpha-3\sigma})^{\kappa}\delta^{1-\alpha}\big{)}^{2}\\\
&\leq
4^{\kappa}\delta^{2\kappa(1-\alpha)-40\sigma}|\tilde{A}_{1}||\tilde{A}_{2}|\\\
&\leq\delta^{2\kappa(1-\alpha)-40\sigma}\lambda_{2}(X),\end{split}$

where on the third line we used (2.23) and on the last line we used (2.21).

Finally, we will verify the transversality estimate (2.4). Note that for each
$(x,y)\in\tilde{I}_{1}\times\tilde{I}_{2}$, the function
$\tilde{\theta}_{(x,y)}(z)$ is monotone and continuous on $\tilde{I}_{3}$.
Thus for each interval $K$, the pre-image

$J:=\theta_{(x,y)}^{-1}(K)$

is necessarily an interval. Moreover, (2.17) implies that for every
$z_{1},z_{2}\in J$, we have

$\frac{|\tilde{\theta}_{(x,y)}(z_{1})-\tilde{\theta}_{(x,y)}(z_{2})|}{|z_{1}-z_{2}|}\geq\delta^{\sigma}.$

Thus $|J|\leq\delta^{-\sigma}|K|$. It follows that for every interval
$K\subset\mathbb{R}$ of length at most $2r$, we have

$\displaystyle|\\{z\in\tilde{A}_{3}\colon\tilde{\theta}_{(x,y)}(z)\in K\\}|$
$\displaystyle=|\tilde{A}_{3}\cap J|$
$\displaystyle\leq\delta^{-\sigma}|J|^{\kappa}\delta^{1-\alpha}$
$\displaystyle\leq\delta^{-\sigma-\kappa\sigma}|K|^{\kappa}\frac{\delta^{1-\alpha}}{|\tilde{A}_{3}|}|\tilde{A}_{3}|$
$\displaystyle\leq\delta^{-(6+\kappa)\sigma}|\tilde{A}_{3}|r^{\eta}r^{\alpha\kappa}2^{\kappa}$
$\displaystyle\leq\delta^{-\tau}|\tilde{A}_{3}|r^{\eta},$

where we used (2.21) and (2.23). Thus (2.4) holds.

Applying Theorem 2.1 Apply Theorem 2.1 to $\tilde{G}$,
$X=\tilde{A}_{1}\times\tilde{A}_{2}$, and $Z=\tilde{A}_{3}$. We obtain a set
$(\tilde{A}_{3})_{\operatorname{bad}}\subset\tilde{A}_{3}$ with

$\lambda_{1}((\tilde{A}_{3})_{\operatorname{bad}})\leq\delta^{\tau}\lambda_{1}(\tilde{A}_{3}).$

For each $c\in\tilde{A}_{3}$, define

$X_{c}:=\\{(x,y)\in\tilde{A}_{1}\times\tilde{A}_{2}\colon\tilde{G}(x,y,c)\in\tilde{R}\\}.$

Let

$\tilde{A}_{3}^{\prime}:=\left\\{z\in\tilde{A}_{3}\colon\lambda_{2}(X_{z})\geq\tfrac{\lambda_{3}(T)}{2\lambda_{1}(\tilde{A}_{3})}\right\\}.$

Now, suppose (2.22) fails; we will obtain a contradiction. By Fubini, we have

$\lambda_{3}(T)\leq\lambda_{1}(\tilde{A}_{3}^{\prime})\lambda_{1}(\tilde{A}_{1})\lambda_{1}(\tilde{A}_{2})+\lambda_{1}(\tilde{A}_{3}\setminus\tilde{A}_{3}^{\prime})\cdot\tfrac{\lambda_{3}(T)}{2\lambda_{1}(\tilde{A}_{3})},$

which implies

$\displaystyle|\tilde{A}_{3}^{\prime}|$
$\displaystyle\geq\frac{\lambda_{3}(T)}{2|\tilde{A}_{1}||\tilde{A}_{2}|}$
$\displaystyle=\frac{\lambda_{3}(T)}{2|\tilde{A}_{1}||\tilde{A}_{2}||\tilde{A}_{3}|}|\tilde{A}_{3}|$
$\displaystyle\geq\tfrac{1}{2}\delta^{14\sigma}|\tilde{A}_{3}|,$

where on the final line we used (2.18) (with $J=[0,1]$).

In particular, since $\sigma>0$ satisfies (2.21) (and assuming $\delta>0$ is
sufficiently small depending on $\alpha$ and $\kappa$), we have
$|\tilde{A}_{3}^{\prime}|>\delta^{\tau}|\tilde{A}_{3}|$ and thus

$\tilde{A}_{3}^{\prime}\not\subseteq(\tilde{A}_{3})_{\rm bad}.$ (2.24)

Fix an element $z\in\tilde{A}_{3}^{\prime}\backslash(\tilde{A}_{3})_{\rm bad}$
and define $X^{\prime}=X_{z}$. By the definition of $\tilde{A}_{3}^{\prime}$,
we have

$\lambda_{2}(X_{z})\geq\tfrac{\lambda_{3}(T)}{2|\tilde{A}_{3}|}\geq\tfrac{1}{2}\delta^{2-2\alpha+5\sigma}\geq\tfrac{1}{2}\delta^{11\sigma}\lambda_{2}(X)\geq\delta^{\tau}\lambda_{2}(X),$
(2.25)

and thus by Theorem 2.1 we have

$\mathcal{E}_{\delta}\big{(}\\{\tilde{G}(x,y,z)\colon(x,y)\in
X_{z}\\}\big{)}\geq\delta^{-\tau}\mathcal{E}_{\delta}(X)^{1/2}.$ (2.26)

On the other hand,

$\mathcal{E}_{\delta}\big{(}\\{\tilde{G}(x,y,z)\colon(x,y)\in
X_{z}\\}\big{)}\leq\mathcal{E}_{\delta}(\tilde{R})\leq\delta^{-\alpha-4\sigma}\leq\delta^{-10\sigma}\mathcal{E}_{\delta}(X)^{1/2}\leq\delta^{-\tau}\mathcal{E}_{\delta}(X)^{1/2}.$
(2.27)

This is a contradiction. We conclude that (2.22) holds, which completes the
proof. ∎

###  2.2 Reformulating Theorem 2.1 as an energy dispersion estimate

As the title suggests, in this section we will reformulate Theorem 2.1 as an
energy dispersion estimate. The basic idea is that we can count the number of
solutions to $P(x,y)=P(x^{\prime},y^{\prime})$ with $x,x^{\prime}\in A,\
y,y^{\prime}\in B$ by counting the size of the intersection

$\\{(x,x^{\prime},y,y^{\prime}):P(x,y)-P(x^{\prime},y^{\prime})=0\\}\cap(A^{2}\times
B^{2}).$ (2.28)

We will express the surface $Z(P(x,y)-P(x^{\prime},y^{\prime}))$ as the graph
$y^{\prime}=G(x,x^{\prime},y^{\prime}),$ and then use Lemma 2.2 to estimate
the size (or more accurately, $\delta$-covering number) of the set (2.28). We
now turn to the details.

######  Definition 2.3.

If $F(x,x^{\prime},y,y^{\prime})$ is a function (that is at least twice
differentiable on its domain), define

$\begin{split}H_{F}(x,x^{\prime},y,y^{\prime})&=(\partial_{x}F)(\partial_{y^{\prime}}F)(\partial_{x^{\prime}y}F)-(\partial_{x}F)(\partial_{y}F)(\partial_{x^{\prime}y^{\prime}}F)\\\
&\qquad-(\partial_{x^{\prime}}F)(\partial_{y^{\prime}}F)(\partial_{xy}F)+(\partial_{x^{\prime}}F)(\partial_{y}F)(\partial_{xy^{\prime}}F).\end{split}$
(2.29)

We will be particularly interested in functions $F$ of the form
$F(x,x^{\prime},y,y^{\prime})=P(x,y)-P(x^{\prime},y^{\prime})$. In this case
$H_{F}$ simplifies to

$H_{F}(x,x^{\prime},y,y^{\prime})=\partial_{x}P(x,y)\partial_{y}P(x,y)\partial_{x^{\prime}y^{\prime}}P(x^{\prime},y^{\prime})-\partial_{x^{\prime}}P(x^{\prime},y^{\prime})\partial_{y^{\prime}}P(x^{\prime},y^{\prime})\partial_{xy}P(x,y).$
(2.30)

######  Definition 2.4.

Let $F\colon I_{1}\times I_{2}\times I_{3}\times I_{4}\to\mathbb{R}$, and let
$\pi$ be the projection to the first three coordinates. We say $F$ has full
projection if $\pi(Z(F))=I_{1}\times I_{2}\times I_{3}$.

######  Lemma 2.5.

For every $0<\alpha<1$ and $0<\kappa\leq\alpha,$ there exists
$\varepsilon=\varepsilon(\alpha,\kappa)>0$ and
$\delta_{0}=\delta_{0}(\alpha,\kappa)$ such that the following holds for all
$0<\delta<\delta_{0}$.

Let $I=I_{1}\times I_{2}\times I_{3}\times I_{4}\subset[0,1]^{4}$ be intervals
and let $F\colon I\to\mathbb{R}$ be a smooth function with full projection.
Suppose that

$|H_{F}|,\ |\partial_{x}F|,\ |\partial_{y}F|,\ |\partial_{x^{\prime}}F|,\
|\partial_{y^{\prime}}F|\geq\delta^{\varepsilon}\quad\textrm{on}\ Z(F)\cap I,$
(2.31)

and

$\|F\|_{C^{2}(I)}\leq\delta^{-\varepsilon}.$ (2.32)

Let $A_{i}\subset I_{i}$, $i=1,2,3,4$ be sets. Suppose that for each index
$i=1,\ldots,4$ and each interval $J$ of length at least $\delta$, $A_{i}$
satisfies the non-concentration condition

$\mathcal{E}_{\delta}(A_{i}\cap
J)\leq|J|^{\kappa}\delta^{-\alpha-\varepsilon}.$ (2.33)

Then we have the energy dispersion estimate

$\mathcal{E}_{\delta}\big{(}(A_{1}\times A_{2}\times A_{3}\times A_{4})\cap
Z(F)\big{)}\leq\delta^{-3\alpha+\varepsilon}.$ (2.34)

###### Proof.

By the bound (2.31) for $\partial_{y^{\prime}}F$ and the assumption that $F$
has full projection, we have that for each $(x,x^{\prime},y)\in I_{1}\times
I_{2}\times I_{3}$, there is precisely one $y^{\prime}\in I_{4}$ so that
$F(x,x^{\prime},y,y^{\prime})=0$, and

$\begin{split}\mathcal{E}_{\delta}\big{(}(A_{1}\times A_{2}\times
A_{3}\times&A_{4})\cap Z(F)\big{)}\\\
&\leq\delta^{-\varepsilon}\mathcal{E}_{\delta}\\{(x,x^{\prime},y)\in
A_{1}\times A_{2}\times A_{3}\colon\ \exists\ y^{\prime}\in A_{4}\
\textrm{s.t.}\ F(x,x^{\prime},y,y^{\prime})=0\\}.\end{split}$

In particular, to establish (2.34) it suffices to prove the estimate

$\mathcal{E}_{\delta}\Big{(}\pi\big{(}(A_{1}\times A_{2}\times A_{3}\times
A_{4})\cap Z(F)\big{)}\Big{)}\leq\delta^{-3\alpha+2\varepsilon}.$ (2.35)

Since $F$ has full projection, there is a smooth function $G\colon I_{1}\times
I_{2}\times I_{3}\to I_{4}$ so that $F(x,x^{\prime},y,G(x,x^{\prime},y))=0$
for all $(x,x^{\prime},y)\in I_{1}\times I_{2}\times I_{3}$. By the implicit
function theorem, (2.32), and (2.31), we have the bound

$\delta^{2\varepsilon}\leq\min_{I}\left|\frac{\partial_{x}F}{\partial_{y^{\prime}}F}\right|\leq\max_{I}\left|\frac{\partial_{x}F}{\partial_{y^{\prime}}F}\right|\leq\delta^{-2\varepsilon},$

and similarly for $\left|\partial_{x^{\prime}}/\partial_{y^{\prime}}F\right|$
and $\left|\partial_{y}F/\partial_{y^{\prime}}F\right|$. Thus

$\delta^{2\varepsilon}\leq\min_{(x,x^{\prime},y)\in I_{1}\times I_{2}\times
I_{3}}|\nabla G_{(y)}(x,x^{\prime})|.$ (2.36)

By continuing to differentiate our expression for $G$ and using (2.31) for
$\partial_{y^{\prime}}F$ and (2.32), we obtain the bound

$\|G_{(y)}\|_{C^{2}(I_{1}\times I_{2}\times
I_{3})}\lesssim\delta^{-10\varepsilon}.$ (2.37)

Define

$\varphi_{(x,x^{\prime})}(y):=\frac{\partial_{x^{\prime}}G(x,x^{\prime},y)}{\partial_{x}G(x,x^{\prime},y)}=\tan(\theta_{(x,x^{\prime})}(y)).$

We have

$\varphi_{(x,x^{\prime})}(y)=\frac{\partial_{x^{\prime}}F}{\partial_{x}F},$
(2.38)

and

$\begin{split}\partial_{y}&\varphi_{(x,x^{\prime})}(y)\\\
&=\frac{(\partial_{x}F)(\partial_{y^{\prime}}F)(\partial_{x^{\prime}y}F)-(\partial_{x}F)(\partial_{y}F)(\partial_{x^{\prime}y^{\prime}}F)-(\partial_{x^{\prime}}F)(\partial_{y^{\prime}}F)(\partial_{xy}F)+(\partial_{x^{\prime}}F)(\partial_{y}F)(\partial_{xy^{\prime}}F)}{(\partial_{y^{\prime}}F)(\partial_{x}F)^{2}}\\\
&=\frac{H_{F}}{(\partial_{y^{\prime}}F)(\partial_{x}F)^{2}},\end{split}$
(2.39)

where in (2.38) and (2.39), $F$ (and its partial derivatives) and $H_{F}$ are
evaluated at the point $(x,y,G(x,x^{\prime},y))$. Thus by (2.31) and (2.32),
we have

$|\varphi_{(x,x^{\prime})}(y)|\leq\delta^{-2\varepsilon},$ (2.40)

while by (2.31) and (2.39), we have

$|\partial_{y}\varphi_{(x,x^{\prime})}(y)|\geq\delta^{6\varepsilon}|H_{F}|\geq\delta^{7\varepsilon}.$
(2.41)

Next, we compute

$\begin{split}|\partial_{y}\varphi_{(x,x^{\prime})}(y)|&=|\partial_{y}(\tan(\theta_{(x,x^{\prime})}))|\\\
&=|(1+\tan^{2}(\theta_{(x,x^{\prime})}(y))||\partial_{y}\theta_{(x,x^{\prime})}(y)|,\end{split}$

and thus

$|\partial_{y}\theta_{(x,x^{\prime})}(y)|=\frac{|\partial_{y}\varphi_{(x,x^{\prime})}(y)|}{1+(\varphi_{(x,x^{\prime})}(y))^{2}}\geq\delta^{11\varepsilon}.$
(2.42)

The estimates (2.36), (2.37), and (2.42) are precisely the requirements needed
to apply Lemma 2.2 to $G$ and the sets $A_{1},A_{2},A_{3},$ and $R=A_{4}$. If
$\varepsilon\leq\varepsilon_{1}(\alpha,\kappa)>0$ and
$0<\delta<\delta_{0}(\alpha,\kappa)>0$ are selected sufficiently sufficiently
small, then there exists $\sigma=\sigma(\alpha,\kappa)>0$ so that

$\mathcal{E}_{\delta}\Big{(}\pi\big{(}(A_{1}\times A_{2}\times A_{3}\times
A_{4})\cap Z(F)\big{)}\Big{)}\leq\delta^{-3\alpha+\sigma}.$ (2.43)

We conclude that (2.35) hold for some
$\varepsilon=\varepsilon(\alpha,\kappa)>0$. ∎

##  3 The auxiliary function $K_{P}$

In this section we will define and study the auxiliary function $K_{P}$.

######  Definition 3.1.

Let $P(x,y)$ be smooth. Define

$K_{P}(x,y)=\nabla
P\wedge\nabla\Big{(}\frac{\partial_{x}P\partial_{y}P}{\partial_{xy}P}\Big{)}.$

A computation shows that (formally),

$\begin{split}\big{(}\partial_{x}P\partial_{y}P\big{)}^{2}&\partial_{xy}\left(\log\left(\frac{\partial_{x}P}{\partial_{y}P}\right)\right)\\\
&=(\partial_{y}P)^{2}\big{(}\partial_{x}P\partial_{xxy}P-\partial_{xx}P\partial_{xy}P\big{)}-(\partial_{x}P)^{2}\big{(}\partial_{y}P\partial_{xyy}P-\partial_{xy}P\partial_{yy}P\big{)}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\big{(}\partial_{xy}P\big{)}^{2}\Big{(}\nabla
P\wedge\nabla\Big{(}\frac{\partial_{x}P\partial_{y}P}{\partial_{xy}P}\Big{)}\Big{)}.\end{split}$
(3.1)

Then the middle expression is well-defined on the domain of $P$. The first
equality holds on the domain of the LHS of (3.1) (here we extend $\log$ to
$\mathbb{R}\backslash\\{0\\}$ by defining $\log(-x)=\log(x)$ for $x$
positive), while the second equality holds on the domain of the RHS of (3.1).
The auxiliary function $K_{P}$ is useful because of the following result from
Elekes and Ronyai [14]. A detailed proof can be found in [35, Lemma 10].

######  Lemma 3.2 (Elekes and Rónyai).

Let $U\subset\mathbb{R}^{2}$ be a connected open set and let $P\colon
U\to\mathbb{R}$ be analytic. If $P_{x}$ or $P_{y}$ vanishes identically on
$U$, then $P$ is an analytic special form, in the sense of Definition 1.17. If
neither $P_{x}$ nor $P_{y}$ vanishes identically on $U$, then $P$ is an
analytic special form if and only if
$\partial_{xy}\left(\log\left(\frac{\partial_{x}P}{\partial_{y}P}\right)\right)$
vanishes everywhere on $U$ that it is defined (i.e. everywhere $\partial_{x}P$
and $\partial_{y}P$ are non-zero).

The identity (3.1) and Lemma 3.2 have the following consequence.

######  Lemma 3.3.

Let $U\subset\mathbb{R}^{2}$ be a connected open set and let $P\colon
U\to\mathbb{R}$ be analytic (resp. polynomial). Suppose that none of
$\partial_{x}P$, $\partial_{y}P$, and $\partial_{xy}P$ vanishes identically on
$U$. Then $K_{P}$ vanishes identically on $U$ if and only if $P$ is an
analytic (resp. polynomial) special form.

Next, we will show that if $K_{P}$ is far from 0, then $P$ satisfies an energy
dispersion estimate.

######  Proposition 3.4.

For every $0<\alpha<1$ and $0<\kappa\leq\alpha,$ there exists
$\varepsilon=\varepsilon(\alpha,\kappa)>0$ and
$\delta_{0}=\delta_{0}(\alpha,\kappa)>0$ such that the following holds for all
$0<\delta<\delta_{0}$. Let $I\subset[0,1]^{2}$ be a rectangle, and let
$P\colon I\to[0,1]$ be smooth. Suppose that

$|\partial_{x}P|,\ |\partial_{y}P|,\ |\partial_{xy}P|,\
|K_{P}|\geq\delta^{\varepsilon}\quad\textrm{on}\ I,$ (3.2)

and

$\|P\|_{C^{3}(I)}\leq\delta^{-\varepsilon}.$ (3.3)

Let $A,B$ be sets with $A\times B\subset I$. Suppose that for all intervals
$J$ of length at least $\delta$, $A$ and $B$ satisfy the non-concentration
condition.

$\begin{split}\mathcal{E}_{\delta}(A\cap
J)&\leq|J|^{\kappa}\delta^{-\alpha-\varepsilon},\\\ \mathcal{E}_{\delta}(B\cap
J)&\leq|J|^{\kappa}\delta^{-\alpha-\varepsilon}.\end{split}$ (3.4)

Then

$\mathcal{E}_{\delta}\big{(}\\{(x,x^{\prime},y,y^{\prime})\in A\times A\times
B\times B\colon
P(x,y)=P(x^{\prime},y^{\prime})\\}\big{)}\leq\delta^{-3\alpha+\varepsilon}.$
(3.5)

###### Proof.

Define $F(x,x^{\prime},y,y^{\prime})=P(x,y)-P(x^{\prime},y^{\prime})$. If
$I=I_{1}\times I_{2}$, then $F$ has domain $I^{\prime}=I_{1}\times I_{1}\times
I_{2}\times I_{2}$. By (3.2), $\partial_{y^{\prime}}F$ never changes sign on
$I^{\prime}$, and thus for each $(x,x^{\prime},y)\in I_{1}\times I_{1}\times
I_{2}$, there is at most one $y^{\prime}\in I_{2}$ with
$F(x,x^{\prime},y,y^{\prime})=0$. Our next task is to cut the domain of $F$
into smaller rectangles, so that on most rectangles, for each
$x,x^{\prime},y$, there is precisely one $y^{\prime}$ with
$F(x,x^{\prime},y,y^{\prime})=0$. The size of (3.5) will be bounded by the
corresponding sum taken over these sets (plus a small error term corresponding
to the rectangles where the above property fails).

Let $t=t(\kappa)>0$ be a small constant to be specified below. Cover
$I^{\prime}$ by interior-disjoint rectangles, each of whose dimensions is
$~{}\frac{t}{6}\delta^{2\varepsilon+5\varepsilon/\kappa}$ (a slight adjustment
might be needed if $6|I_{i}|/(t\delta^{2\varepsilon+5\varepsilon/\kappa})$ is
not an integer). For each such rectangle $J=J_{1}\times J_{2}\times
J_{3}\times J_{4}$, let $\tilde{J}_{4}$ be an interval of length
$t\delta^{5\varepsilon/\kappa}$, with the same midpoint as $J_{4}$. Define
$\tilde{J}=J_{1}\times J_{2}\times J_{3}\times\tilde{J}_{4}$.

We claim that if $J\cap Z(F)\neq\emptyset$ and if $\tilde{J}\subset
I^{\prime},$ then for each $(x,x^{\prime},y)\in J_{1}\times J_{2}\times
J_{3}$, there is $y^{\prime}\in\tilde{J}_{4}$ with
$F(x,x^{\prime},y,y^{\prime})=0$. To prove this claim, observe that the
assumption $J\cap Z(F)\neq\emptyset$ implies there exists
$(x_{0},x_{0}^{\prime},y_{0},y_{0}^{\prime})\in J$ with
$P(x_{0},y_{0})=P(x_{0}^{\prime},y_{0}^{\prime})$. By (3.3), we thus have

$\displaystyle|P(x,y)-P(x^{\prime},y_{0}^{\prime})|$
$\displaystyle\leq|P(x,y)-P(x_{0},y_{0})|+|P(x_{0},y_{0})-P(x_{0}^{\prime},y_{0}^{\prime})|+|P(x_{0}^{\prime},y_{0}^{\prime})-P(x^{\prime},y_{0}^{\prime})|$
$\displaystyle\leq\delta^{-\varepsilon}|(x,y)-(x_{0},y_{0})|+0+\delta^{-\varepsilon}|x_{0}-x^{\prime}|$
$\displaystyle\leq\delta^{-\varepsilon}\operatorname{diam}(J)\leq\frac{t}{2}\delta^{\varepsilon+5\varepsilon/\kappa}.$

By (3.2), there exists $y$ with
$|y_{0}-y|\leq\frac{(t/2)\delta^{\varepsilon+5\varepsilon/\kappa}}{\delta^{\varepsilon}}=\frac{t}{2}\delta^{5\varepsilon/\kappa}$
with $P(x,y)=P(x^{\prime},y^{\prime})$; by our definition of $\tilde{J}_{4}$
we have $y\in\tilde{J}_{4}$. This completes the proof of the claim.

Finally, we will estimate the contribution to (3.5) coming from rectangles $J$
for which $\tilde{J}$ is not contained in $I^{\prime}$. Such rectangles must
be of the form $J_{1}\times J_{2}\times J_{3}\times J_{4}$, where $J_{4}$ is
near one of the endpoints of $I_{2}$; specifically, if $I_{4}=[a,b]$, then
$J\subset[a,a+t\delta^{5\varepsilon/\kappa}]\cup[b-t\delta^{5\varepsilon/\kappa},b]$.
By (3.4), if the constant $t=t(\kappa)>0$ is selected sufficiently small, then

$\mathcal{E}_{\delta}\big{(}B\cap[a,a+t\delta^{5\varepsilon/\kappa}]\big{)}\leq\delta^{-\varepsilon}(t\delta^{5\varepsilon/\kappa})^{\kappa}\delta^{-\alpha}\leq\frac{1}{4}\delta^{4\varepsilon-\alpha},$

and similarly for $B\cap[b-t\delta^{5\varepsilon/\kappa},b].$ Using the bound
$\mathcal{E}_{\delta}(A)\leq\delta^{-\varepsilon-\alpha}$ and
$\mathcal{E}_{\delta}(B)\leq\delta^{-\varepsilon-\alpha}$ from (3.4), we have

$\mathcal{E}_{\delta}(\\{(x^{\prime},y,y^{\prime})\in A\times B\times B\colon
y^{\prime}\in[a,a+t\delta^{5\varepsilon/\kappa}]\cup[b-t\delta^{5\varepsilon/\kappa},b]\\})\leq\frac{1}{2}\delta^{2\varepsilon-3\alpha}.$

By (3.3) (specifically the bound on
$|\partial_{x}P|\leq\delta^{-\varepsilon})$, this implies

$\mathcal{E}_{\delta}(\\{(x,x^{\prime},y,y^{\prime})\in A\times A\times
B\times B\colon P(x,y)=P(x^{\prime},y^{\prime}),\
y^{\prime}\in[a,a+t\delta^{5\varepsilon/\kappa}]\cup[b-t\delta^{5\varepsilon/\kappa},b]\\})\leq\frac{1}{2}\delta^{\varepsilon-3\alpha}.$

Thus the contribution from rectangles $J$ with $\tilde{J}\not\subset
I^{\prime}$ is at most half our desired final bound (3.5).

The number of rectangles $J$ with $\tilde{J}\subset I^{\prime}$ is at most
$6^{4}(t\delta^{2\varepsilon+5\varepsilon/\kappa})^{-4}\leq
6^{4}t^{-4}\delta^{-28\varepsilon/\kappa}$ (a better estimate is possible
since not all rectangles $J$ intersect $Z(F)$, but this will not matter for
our arguments). Thus to establish the bound (3.5), it suffices to establish
the bound

$\mathcal{E}_{\delta}\big{(}\\{(x,x^{\prime},y,y^{\prime})\in J\cap(A\times
A\times B\times B)\colon
P(x,y)=P(x^{\prime},y^{\prime})\\}\big{)}\leq\delta^{-3\alpha+30\varepsilon/\kappa}$
(3.6)

for each rectangle $J$ with $J\cap Z(F)\neq\emptyset$ and $\tilde{J}\subset
I^{\prime}$.

For the remainder of the argument we will fix such a rectangle $J=J_{1}\times
J_{2}\times J_{3}\times J_{4}$. Unless stated otherwise, when we refer to
variables $x,x^{\prime},y,y^{\prime}$, we will assume that
$(x,x^{\prime},y,y^{\prime})\in J$. By construction, there is a smooth
function $G\colon J_{1}\times J_{2}\times J_{3}\to\tilde{J}_{4}$ so that
$F(x,x^{\prime},y^{\prime},G(x,x^{\prime},y))=0$.

Recall Definition 2.3, which defines the auxiliary function $H_{F}$. By (2.30)
we have

$\begin{split}H_{F}(x,x^{\prime},y,y^{\prime})&=\partial_{xy}P(x,y)\partial_{x^{\prime}y^{\prime}}P(x^{\prime},y^{\prime})\Big{(}\frac{\partial_{x}P(x,y)\partial_{y}P(x,y)}{\partial_{xy}P(x,y)}-\frac{\partial_{x^{\prime}}P(x^{\prime},y^{\prime})\partial_{y^{\prime}}P(x^{\prime},y^{\prime})}{\partial_{x^{\prime}y^{\prime}}P(x^{\prime},y^{\prime})}\Big{)}.\end{split}$

Our next task is to control the size of the region where
$H_{F}(x,x^{\prime},y,G(x,x^{\prime},y))$ is small. Fix $s_{0}>0$.

Fix $(x_{0},y_{0})\in J_{1}\times J_{2}$ and suppose there is a point
$x_{0}^{\prime}\in J_{3}$ with
$|H(x_{0},x_{0}^{\prime},y_{0},G(x_{0},x_{0}^{\prime},y_{0}))|\leq s_{0}$.
Consider the curve

$\gamma=\\{(x^{\prime},y^{\prime})\in J_{3}\times\tilde{J}_{4}\colon
P(x_{0},y_{0})=P(x^{\prime},y^{\prime})\\}=\\{(x^{\prime},G(x_{0},x^{\prime},y_{0}))\colon
x^{\prime}\in J_{3}\\}.$

Then $\gamma$ is a smooth curve, and, by (3.2) and (3.3), at each point the
tangent to this curve has slope $\leq\delta^{-2\varepsilon}$. Let
$\gamma(t)\colon S\to\mathbb{R}^{2}$ be a unit speed parameterization of
$\gamma$, where $S\subset\mathbb{R}$ is an interval,
$y_{0}^{\prime}=G(x_{0},x_{0}^{\prime},y_{0})$, and
$\gamma(0)=(x_{0}^{\prime},y_{0}^{\prime})$.

For each $t\in S$, the unit tangent vector to $\gamma$ at $t$ is given by
$\frac{\nabla P^{\perp}(\gamma(t))}{|\nabla P(\gamma(t))|}$. Thus for all
$t\in S$ we have

$\begin{split}\Big{|}\partial_{t}\Big{(}\frac{\partial_{x}P(\gamma(t))\partial_{y}P(\gamma(t))}{\partial_{xy}P(\gamma(t))}\Big{)}\Big{|}&=\Big{|}\frac{\nabla
P^{\perp}}{|\nabla
P|}\cdot\nabla\big{(}\frac{(\partial_{x}P)(\partial_{y}P)}{\partial_{xy}P}\big{)}(\gamma(t))\Big{|}\\\
&=\frac{1}{|\nabla P|}\Big{|}\Big{(}\nabla
P\wedge\nabla\big{(}\frac{(\partial_{x}P)(\partial_{y}P)}{\partial_{xy}P}\big{)}\Big{)}(\gamma(t))\Big{|}\\\
&=\frac{|K_{P}(\gamma(t))|}{|\nabla P(\gamma(t))|}\\\
&\geq\delta^{2\varepsilon}.\end{split}$

Therefore if $\gamma(t)=(x^{\prime}(t),y^{\prime}(t))$, then

$\Big{|}\frac{\partial_{x^{\prime}}P(x^{\prime}(0),y^{\prime}(0))\partial_{y^{\prime}}P(x^{\prime}(0),y^{\prime}(0))}{\partial_{x^{\prime}y^{\prime}}P(x^{\prime}(0),y^{\prime}(0)}-\frac{\partial_{x^{\prime}}P(x^{\prime}(t),y^{\prime}(t))\partial_{y^{\prime}}P(x^{\prime}(t),y^{\prime}(t))}{\partial_{x^{\prime}y^{\prime}}P(x^{\prime}(t),y^{\prime}(t)}\Big{|}\geq\delta^{2\varepsilon}t,$

and hence

$\begin{split}&\Big{|}\frac{\partial_{x}P(x_{0},y_{0})\partial_{y}P(x_{0},y_{0})}{\partial_{xy}P(x_{0},y_{0})}-\frac{\partial_{x^{\prime}}P(x^{\prime}(t),y^{\prime}(t))\partial_{y^{\prime}}P(x^{\prime}(t),y^{\prime}(t))}{\partial_{x^{\prime}y^{\prime}}P(x^{\prime}(t),y^{\prime}(t))}\Big{|}\\\
&\geq\Big{|}\frac{\partial_{x^{\prime}}P(x^{\prime}(t),y^{\prime}(t))\partial_{y^{\prime}}P(x^{\prime}(t),y^{\prime}(t))}{\partial_{x^{\prime}y^{\prime}}P(x^{\prime}(t),y^{\prime}(t))}-\frac{\partial_{x^{\prime}}P(x^{\prime}(0),y^{\prime}(0))\partial_{y^{\prime}}P(x^{\prime}(0),y^{\prime}(0))}{\partial_{x^{\prime}y^{\prime}}P(x^{\prime}(0),y^{\prime}(0))}\Big{|}\\\
&\qquad-\Big{|}\frac{\partial_{x}P(x_{0},y_{0})\partial_{y}P(x_{0},y_{0})}{\partial_{xy}P(x_{0},y_{0})}-\frac{\partial_{x^{\prime}}P(x^{\prime}(0),y^{\prime}(0))\partial_{y^{\prime}}P(x^{\prime}(t),y^{\prime}(t))}{\partial_{x^{\prime}y^{\prime}}P(x^{\prime}(0),y^{\prime}(0))}\Big{|}\\\
&\geq\delta^{2\varepsilon}t-s_{0}.\end{split}$ (3.7)

We conclude that

$|H_{F}(x_{0},x^{\prime}(t),y_{0},y^{\prime}(t))|\geq\big{|}\partial_{xy}P(x,y)\partial_{x^{\prime}y^{\prime}}P(x^{\prime}(t),y^{\prime}(t))\big{|}\big{(}\delta^{2\varepsilon}t-s\big{)}\geq\delta^{2\varepsilon}(\delta^{2\varepsilon}t-s_{0}).$
(3.8)

Using (3.3), one can easily verify that $|\nabla H_{F}|\leq
12\delta^{-3\varepsilon}$. Combined with (3.8), we have that if $(x,y)\in
J_{1}\times J_{2}$ with $|x-x_{0}|\leq\delta$ and $|y-y_{0}|\leq\delta$, then

$|H_{F}(x,x^{\prime}(t),y,y^{\prime}(t))|\geq\delta^{2\varepsilon}(\delta^{2\varepsilon}t-s_{0})-12\delta^{1-3\varepsilon}.$
(3.9)

Recall that $\gamma(t)$ is a unit speed parameterization, and so
$|(x_{0}^{\prime},y_{0}^{\prime})-\gamma(t)|\leq t$. We conclude that for each
$(x_{0},y_{0})\in J_{1}\times J_{2}$, if there exists a point
$x_{0}^{\prime}\in J_{3}$ with
$|H(x_{0},x_{0}^{\prime},y_{0},G(x_{0},x_{0}^{\prime},y_{0}))|\leq s_{0}$,
then for each $s\geq s_{0}$ we have that the set

$\\{x^{\prime}\in J_{3}\colon|H_{F}(x,x^{\prime},y,G(x,x^{\prime},y))|\leq s\
\quad\textrm{for some}\ x\in J_{1}\cap[x_{0}-\delta,x_{0}+\delta],\ y\in
J_{2}\cap[y_{0}-\delta,y_{0}+\delta]\\}$

is contained in an interval of length at most $2\delta^{-4\varepsilon}s$ (this
interval necessarily contains $x_{0}^{\prime}$).

Since there are at most $\delta^{-2\alpha-2\varepsilon}$ $\delta$-separated
points $(x_{0},y_{0})\in J_{1}\times J_{2}$ for which there exists
$x_{0}^{\prime}\in J_{3}$ with
$|H(x_{0},x_{0}^{\prime},y_{0},G(x_{0},x_{0}^{\prime},y_{0}))|\leq s_{0}$, we
can use (3.4) to bound the covering number of each corresponding interval;
setting $s=2s_{0}$, we conclude that

$\begin{split}\mathcal{E}_{\delta}\Big{\\{}(x,x^{\prime},y)\colon|H(x,x^{\prime},y,G(x,x^{\prime},y))|\leq
s_{0}\Big{\\}}&\lesssim\delta^{-2\alpha-2\varepsilon}\big{(}(\delta^{-4\varepsilon}(2s_{0}))^{\kappa}\delta^{-\alpha-\varepsilon}\big{)}\\\
&\lesssim\delta^{-3\alpha-7\varepsilon}s_{0}^{\kappa},\end{split}$ (3.10)

and hence

$\mathcal{E}_{\delta}\big{(}\\{(x,x^{\prime},y,y^{\prime})\in J_{1}\times
J_{2}\times J_{3}\times J_{4}\colon P(x,y)=P(x^{\prime},y^{\prime}),\
|H(x,x^{\prime},y,y^{\prime})|\leq
s_{0}\\}\big{)}\lesssim\delta^{-3\alpha-7\varepsilon}s_{0}^{\kappa}.$ (3.11)

Next, let

$W=\Big{\\{}(x,x^{\prime},y)\in J_{1}\times J_{2}\times
J_{3}\colon|H(x,x^{\prime},y,G(x,x^{\prime},y))|\leq
c_{0}\delta^{5\varepsilon}s_{0}\Big{\\}},$ (3.12)

where $c_{0}>0$ is a small constant to be specified below. Cover
$\mathbb{R}^{3}\backslash W$ by interior-disjoint cubes, with the property
that each cube $Q$ in the cover has side-length comparable to its distance
from $W$. The existence of such a collection of cubes is guaranteed by the
Whitney cube decomposition (see Theorem 4.4 below). Let $\mathcal{Q}_{1}$
denote the set of cubes $Q$ satisfying $\operatorname{dist}(Q,W)\leq
c_{0}\delta^{5\varepsilon}s_{0}$, and let $\mathcal{Q}_{2}$ denote the set of
cubes $Q$ that intersect $J_{1}\times J_{2}\times J_{3}$, but which are not in
$\mathcal{Q}_{1}$.

We claim that

$\bigcup_{Q\in\mathcal{Q}_{1}}Q\subset\Big{\\{}(x,x^{\prime},y)\colon|H(x,x^{\prime},y,G(x,x^{\prime},y))|\leq
s_{0}\Big{\\}}.$ (3.13)

Indeed, if $Q\in\mathcal{Q}_{1}$, then $\operatorname{dist}(Q,W)\leq
c_{0}\delta^{5\varepsilon}s_{0}$. Since the diameter of $Q$ is
$O(c_{0}\delta^{5\varepsilon}s_{0})$, we have that each point in $Q$ has
distance $O(c_{0}\delta^{5\varepsilon}s_{0})$ from $W$. Using (3.2), and (3.3)
one can verify that the gradient of the trivariate function

$(x,x^{\prime},y)\mapsto H(x,x^{\prime},y,G(x,x^{\prime},y))$

has norm $O(\delta^{-5\varepsilon})$. Thus for every $(x,x^{\prime},y)\in Q$,
we have

$|H(x,x^{\prime},y,G(x,x^{\prime},y))|\lesssim\delta^{-5\varepsilon}(c_{0}\delta^{5\varepsilon}s_{0}).$
(3.14)

If we select $c_{0}$ from (3.12) sufficiently small, then the LHS of (3.14) is
at most $s_{0}$. Fixing this choice of $c_{0}$, we have

$J_{1}\times J_{2}\times J_{3}\subset\bigcup_{Q\in\mathcal{Q}_{2}}Q\ \cup\
\Big{\\{}(x,x^{\prime},y,)\colon|H(x,x^{\prime},y,G(x,x^{\prime},y))|\leq
s_{0}\Big{\\}}.$ (3.15)

Recall that our goal is to establish (3.6). We will do this by bounding the
number of quadruples $(x,x^{\prime},y,y^{\prime})$ with
$(x,x^{\prime},y)\in\bigcup_{Q\in\mathcal{Q}_{2}}Q$, and the number of
quadruples with $(x,x^{\prime},y)\in(J_{1}\times J_{2}\times J_{3})\
\backslash\ \bigcup_{Q\in\mathcal{Q}_{2}}Q$. By (3.15), we have
$|H(x,x^{\prime},y,G(x,x^{\prime},y))|\leq s_{0}$ for each $(x,x^{\prime},y)$
of the latter kind, and thus we can use (3.11) to bound the contribution to
(3.6) of quadruples arising from such $(x,x^{\prime},y)$. We will select

$s_{0}=\delta^{100\varepsilon/\kappa^{2}}.$ (3.16)

With this choice of $s_{0}$, the RHS of (3.11) is
$O(\delta^{-3\alpha-7\varepsilon+100\varepsilon/\kappa})$, and thus in turn is
at most $\frac{1}{2}\delta^{-3\alpha+30\varepsilon/\kappa}$. Comparing with
(3.6), this contribution is acceptable.

Our next task is to obtain an analagous bound for

$\sum_{Q\in\mathcal{Q}_{2}}\mathcal{E}_{\delta}\big{(}\\{(x,x^{\prime},y,y^{\prime})\in
J\cap(A\times A\times B\times B)\colon P(x,y)=P(x^{\prime},y^{\prime}),\
(x,x^{\prime},y)\in Q_{2}\\}\big{)}.$ (3.17)

Each cube in $\mathcal{Q}_{2}$ intersects $J_{1}\times J_{2}\times
J_{3}\subset[0,1]^{3}$, and each cube has side-length
$\gtrsim\delta^{5\varepsilon}s_{0}$. For technical reasons, it will be
convenient to further partition the cubes in $\mathcal{Q}_{2}$ into a set of
$O(\delta^{-48\varepsilon}s_{0}^{-3})$ smaller rectangular prisms, each of
which has diameter at most $\delta^{16\varepsilon}s_{0}$. Denote this new set
of prisms by $\mathcal{Q}_{3}.$

Fix $Q=Q_{1}\times Q_{2}\times Q_{3}\in\mathcal{Q}_{3}$, so
$Q\times{\tilde{J}}_{4}=Q_{1}\times Q_{2}\times Q_{3}\times{\tilde{J}}_{4}$.
Note that $F$ has full projection on $Q\times{\tilde{J}}_{4}$. We claim that
there exists a subinterval $K_{Q}\subset\tilde{J}_{4}$ such that $F$ has a
full projection on $Q\times K_{Q}$ and $K_{Q}$ has length at most
$\delta^{14\varepsilon}s_{0}$. Indeed, first note that, by (3.2) and (3.3),
$|\nabla G|$ is bounded by $\delta^{-2\varepsilon}$ on $Q\times\tilde{J}_{4}$.
Thus, for
$(x_{1},x_{1}^{\prime},y_{1},y_{1}^{\prime}),(x_{2},x_{2}^{\prime},y_{2},y_{2}^{\prime})\in(Q\times\tilde{J}_{4})\cap
Z(F)$, we have

$|y_{1}^{\prime}-y_{2}^{\prime}|\lesssim\delta^{-2\varepsilon}(\delta^{16\varepsilon}s_{0}),$
(3.18)

where here we used the property that the diameter of $Q$ is at most
$\delta^{16\varepsilon}s_{0}$.

Finally, fix $Q=Q_{1}\times Q_{2}\times Q_{3}\in\mathcal{Q}_{3}$ and let
$K_{Q}$ be as above. By the definition of the collection $\mathcal{Q}_{2}$, we
have
$|H_{F}(x,x^{\prime},y,G(x,x^{\prime},y))|\gtrsim\delta^{5\varepsilon}s_{0}$
on each point in $Q$. Since $(Q\times K_{Q})\cap Z(F)\neq\emptyset$, there
exists $(x_{1},x_{1}^{\prime},y_{1},y_{1}^{\prime})\in Q\times K_{Q}$, such
that $y_{1}^{\prime}=G(x_{1},x_{1}^{\prime},y_{1})$. Recalling that $|\nabla
H_{F}|\leq 12\delta^{-3\varepsilon}$, we conclude that

$|H_{F}(x,x^{\prime},y,y^{\prime})|\geq\delta^{10\varepsilon}s_{0}-12\delta^{-3\varepsilon}(\delta^{14\varepsilon}s_{0})\geq\tfrac{1}{2}\delta^{10\varepsilon}s_{0},$

for every $(x,x^{\prime},y,y^{\prime})\in Q\times K_{Q}$.

Let $\varepsilon^{\prime}$ and $\delta_{0}^{\prime}$ be the value of
$\varepsilon$ and $\delta_{0}$ coming from Lemma 2.5 (with the value of
$\alpha$ and $\kappa$ specified as input to the present lemma). If
$\varepsilon$ is selected sufficiently small so that
$10\varepsilon+100\varepsilon/\kappa^{2}\leq\varepsilon^{\prime}$ (i.e.
$\tfrac{1}{2}\delta^{10\varepsilon}s_{0}\geq\delta^{\varepsilon^{\prime}}$),
then $F$ satisfies hypotheses (2.31) and (2.32), and the sets $A\cap
Q_{1},B\cap Q_{2},A\cap Q_{3}$ and $B\cap K_{Q}$ satisfy the hypothesis
(2.33). By construction, the function $F\colon Q_{1}\times Q_{2}\times
Q_{3}\times K_{Q}\to\mathbb{R}$ has full projection. We conclude that for all
$0<\delta<\delta_{0}$,

$\mathcal{E}_{\delta}\Big{(}(A\times A\times B\times B)\cap(Q_{1}\times
Q_{2}\times Q_{3}\times{\tilde{J}}_{4})\cap
Z(F)\Big{)}\leq\delta^{-3\alpha+\varepsilon^{\prime}}.$ (3.19)

Summing over all $O(\delta^{-48\varepsilon}s_{0}^{-3})$ such cubes, we
conclude that

$\eqref{boundSumOverQ2}\lesssim\delta^{-3\alpha-48\varepsilon+\varepsilon^{\prime}}s_{0}^{-3}.$
(3.20)

Finally, using our choice of $s_{0}$ from (3.16), we conclude that if
$\varepsilon>0$ is selected sufficently small (depending on
$\varepsilon^{\prime}$, which in turn depends on $\alpha$ and $\kappa$), then
$\eqref{boundSumOverQ2}\leq\frac{1}{2}\delta^{-3\alpha+30\varepsilon/\kappa}$.
We conclude that (3.6) holds, and we are done. ∎

##  4 Blaschke curvature and discretized projections

In this section we will prove Theorem 1.11. First, we will show that if a non-
concentrated set has small image under three different maps whose gradients
are pairwise linearly independent at each point, then the image of this set
under these maps must also be non-concentrated.

######  Lemma 4.1.

Let $Q\subset[0,1]^{2}$ be a square, let $\alpha,\eta>0$, $C\geq 1$. Then the
following is true for all $\delta>0$ sufficiently small. Let
$f_{1},f_{2},f_{3}\in C^{2}(Q)$. Suppose that for each $i=1,2,3$ we have

$\|f_{i}\|_{C^{2}(Q)}\leq C,$ (4.1)

and for each $i\neq j$ and each $z\in Q$ we have

$|\nabla f_{i}(z)\wedge\nabla f_{j}(z)|\geq C^{-1}.$ (4.2)

Let $X\subset Q$ with with $\mathcal{E}_{\delta}(X)\geq\delta^{-\alpha}$.
Suppose that for all balls $B$ of radius $r\geq\delta$ we have

$\mathcal{E}_{\delta}(X\cap B)\leq r^{\alpha}\delta^{-\alpha-\eta}.$ (4.3)

Suppose that for each $i=1,2,3$ we have

$\mathcal{E}_{\delta}(f_{i}(X))\leq\delta^{-\alpha/2-\eta}.$ (4.4)

Then there exists $X^{\prime}\subset X$, with

$\mathcal{E}_{\delta}(X^{\prime})\gtrsim(\delta^{\eta}C^{-1})^{O(1)}\mathcal{E}_{\delta}(X),$
(4.5)

such that for all intervals $J$ of length at least $\delta$, and each index
$i=1,2,3$, we have

$\begin{split}&\mathcal{E}_{\delta}(f_{i}(X^{\prime})\cap
J)\leq(C|\log\delta|\delta^{-\eta})^{O(1)}|J|^{\alpha/2}\delta^{-\alpha/2}.\end{split}$
(4.6)

###### Proof.

First, we claim that the set $X$ cannot be too concentrated near the boundary
of $Q$, i.e.

$\mathcal{E}_{\delta}\big{(}X\cap
N_{C\delta}(\operatorname{bdry}(Q))\big{)}\leq\frac{1}{2}\mathcal{E}_{\delta}(X).$
(4.7)

Indeed, if (4.7) failed, then there is a line segment $\ell\subset[0,1]^{2}$
(one of the four edges of the square $Q$) with $\mathcal{E}_{\delta}(X\cap
N_{C\delta}(\ell))\geq\frac{1}{8}\mathcal{E}_{\delta}(X)$. But then by (4.1)
and (4.2), we can find a segment $\ell^{\prime}\subset\ell$ with
$\mathcal{E}_{\delta}(X\cap N_{C\delta}(\ell^{\prime}))\geq
C^{-O(1)}\mathcal{E}_{\delta}(X)$ and an index $i$ so that $|\nabla f\cdot
v|\geq C^{-O(1)})$ on $\ell$, where $v$ is the unit vector parallel to $\ell$.
But this implies

$\mathcal{E}_{\delta}(f_{i}(X\cap N_{C\delta}(\ell^{\prime})))\geq
C^{-O(1)}\mathcal{E}_{\delta}(X),$

which contradicts (4.4), provided $\delta>0$ is sufficiently small (depending
on $\alpha,\eta$, and $C$).

Next, by (4.1) and (4.2), we can select a square $Q^{\prime}\subset
Q\backslash N_{C\delta}(\operatorname{bdry}(Q))$ with

$\mathcal{E}_{\delta}(X\cap Q^{\prime})\geq C^{-O(1)}\mathcal{E}_{\delta}(X)$

so that for each $i\neq j$, $f_{i}\otimes f_{j}\colon
Q^{\prime}\to\mathbb{R}^{2}$ is injective on $N_{C\delta}(Q^{\prime})\subset
Q$, where $N_{C\delta}(Q^{\prime})$ is the $C\delta$-neighborhood of
$Q^{\prime}$. In particular, since (4.1) and (4.2) continue to hold on
$Q^{\prime}$, if $z,z^{\prime}\in Q^{\prime}$, then

$|(f_{i}(z),f_{j}(z))-(f_{i}(z^{\prime}),f_{j}(z^{\prime}))|\geq
C^{-O(1)}|z-z^{\prime}|.$ (4.8)

Next, let $X_{0}\subset X\cap Q^{\prime}$ be a maximal $C^{O(1)}\delta$
separated subset of $X\cap Q^{\prime}$, where the implicit constant is chosen
so that if $z,z^{\prime}\in X_{0}$, then for each $i\neq j$ we have

$|(f_{i}(z),f_{j}(z))-(f_{i}(z^{\prime}),f_{j}(z^{\prime}))|\geq 2\delta.$
(4.9)

Note that $\\#X_{0}\geq C^{-O(1)}\mathcal{E}_{\delta}(X),$ so

$\mathcal{E}_{\delta}(X_{0})\gtrsim(\delta^{\eta}C^{-1})^{O(1)}\delta^{-\alpha}.$
(4.10)

In the argument that follows, we will show that if $X_{0}$ is a set satisfying
(4.3), (4.4), (4.8), (4.9), and (4.10), then there is a set $X_{1}\subset
X_{0}$ that continues to satisfy these inequalities (with slightly worse
implicit constants for (4.10)), and that also satisfies (4.6) for $i=1$. We
can then repeat this argument to find a set $X_{2}\subset X_{1}$ that
continues to satisfy the above inequalities, and also satisfies (4.6) for
$i=2$, and then a set $X_{3}\subset X_{2}$ that satisfies (4.6) for $i=3$. To
complete the proof of the lemma, we define $X^{\prime}=X_{3}$.

Our next goal is to construct the set $X_{1}$ described above. Let
$A_{0}\subset f_{1}(X_{0})$ and $B_{0}\subset f_{2}(X_{0})$ be maximal
$\delta$-separated sets. Define $E_{0}\subset A_{0}\times B_{0}$, with
$(x,y)\in E_{0}$ if there exists a point $z\in X_{0}$ with
$|x-f_{1}(z)|\leq\delta,\ |y-f_{2}(z)|\leq\delta$. Let $\pi_{x},\pi_{y}$ be
the projections from $E_{0}$ to $A_{0}$ and $B_{0}$, respectively. By (4.4) we
have $\\#A_{0}\leq\delta^{-\alpha/2-\eta}$ and
$\\#B_{0}\leq\delta^{-\alpha/2-\eta}$, and by construction we have
$\\#E_{0}=\\#X_{0}\geq(\delta^{\eta}C^{-1})^{O(1)}\delta^{-\alpha}$.

Using a standard popularity argument (see e.g. [10, Lemma 2.8]), we can select
sets $A\subset A_{0}$, $B\subset B_{0}$ and an edge set $E\subset E_{0}$ so
that $\\#E\geq(\\#E_{0})/2$; each element of $A$ is adjacent to at least
$(\\#E_{0})/(4\\#A_{0})\gtrsim(\delta^{\eta}C^{-1})^{O(1)}\delta^{-\alpha/2}$
and at most $\\#B=\delta^{-\alpha/2-\eta}$ elements of $B$. Similarly, each
element of $B$ is adjacent to at least
$\gtrsim(\delta^{\eta}C^{-1})^{O(1)}\delta^{-\alpha/2}$ and at most
$\delta^{-\alpha/2-\eta}$ elements of $A$.

Define $X_{1}=X_{0}\cap N_{\delta}((f_{1}\otimes f_{2})^{-1}(E)).$ We claim
that $X_{1}$ satisfies (4.6) for $i=1$. To verify this, let
$J\subset\mathbb{R}$ be an interval of length $t\geq\delta$ and suppose

$\mathcal{E}_{\delta}(f_{1}(X_{1})\cap J)=Kt^{\alpha/2}\delta^{-\alpha/2},$

for some $K\geq 1$. Our goal is to show that
$K\leq(C|\log\delta|\delta^{-\eta})^{O(1)}$.

By construction we have $\\#(A\cap J)=\mathcal{E}_{\delta}(f_{1}(X_{1})\cap
J)=Kt^{\alpha/2}\delta^{-\alpha/2}$. For each $z=(x,y)\in E$, define

$m(z)=\\#\pi_{y}^{-1}(\pi_{y}(z))=\\#\\{x^{\prime}\in A\colon(x^{\prime},y)\in
E\\}.$

We have $1\leq m(z)\leq\\#(A\cap J)\leq\delta^{-1}$.

By dyadic pigeonholing, there is a number $1\leq m\leq\\#(A\cap J)$ and a set
$E^{\prime}\subset E\cap f_{1}^{-1}(J))$ with

$\\#E^{\prime}\geq|\log\delta|^{-1}\\#(E\cap
f_{1}^{-1}(J))\gtrsim\big{(}\\#(A\cap
J)\big{)}\big{(}(\delta^{\eta}C^{-1})^{O(1)}\delta^{-\alpha/2}\big{)}=(\delta^{\eta}C^{-1}|\log\delta|^{-1})^{O(1)}Kt^{\alpha/2}\delta^{-\alpha},$

so that $m(z)\sim m$ for all $z\in E^{\prime}$. Note that if $z=(x,y)\in
E^{\prime}$ and $x^{\prime}\in A$, then $(x^{\prime},y)\in E^{\prime}$.

We have $\pi_{y}(E^{\prime})\subset B$, and thus
$\\#(\pi_{y}(E^{\prime}))\leq\\#B\leq\delta^{-\alpha/2-\eta}$. On the other
hand $(\\#E^{\prime})/m\leq\\#(\pi_{y}(E^{\prime}))$, so by (4.4) we have

$m\gtrsim\frac{(\\#E^{\prime})}{\\#(\pi_{y}(E^{\prime})}\gtrsim\frac{(\delta^{\eta}C^{-1}|\log\delta|^{-1})^{O(1)}Kt^{\alpha/2}\delta^{-\alpha}}{\delta^{-\alpha/2-\eta}}=(\delta^{\eta}C^{-1}|\log\delta|^{-1})^{O(1)}Kt^{\alpha/2}\delta^{-\alpha/2}.$

By (4.3) we have that for each $T\geq 1$,

$\mathcal{E}_{C^{T}t}(E^{\prime})\geq\frac{\\#E^{\prime}}{(C^{T}t)^{\alpha}\delta^{-\alpha-\eta}}\gtrsim
C^{-T\alpha}(\delta^{\eta}C^{-1}|\log\delta|^{-1})^{O(1)}Kt^{-\alpha/2}.$

Thus we can select a $C^{T}t$-separated set $B^{\prime}\subset B$ with

$\\#B^{\prime}\gtrsim
C^{-T\alpha}(\delta^{\eta}C^{-1}|\log\delta|^{-1})^{O(1)}Kt^{-\alpha/2},$

so that for each $b\in B^{\prime}$, there is a set $E^{\prime}_{b}\subset
E^{\prime}$ with $\\#E^{\prime}_{b}\sim m$, and $\pi_{y}(z)=b$ for each $z\in
E^{\prime}_{b}$. Let $X_{b}=(f_{1}\otimes f_{2})^{-1}(E^{\prime}_{b})$. Since
$f_{1}(X_{b})$ is contained in an interval of length $t$ and since
$f_{2}(X_{b})$ is contained in an interval of length $\delta$ for each $z\in
E^{\prime}_{b}$, by (4.8) (with $i=1,j=2$) we have that $X_{b}$ is contained
in a ball of radius $C^{O(1)}t$. Thus by (4.1), $f_{3}(X_{b})$ is contained in
an interval of length $C^{O(1)}t$. By (4.8) (with $i=2,j=3)$, we have that
$\mathcal{E}_{\delta}(f_{3}(X_{b}))\gtrsim C^{-O(1)}m$. Next, note that for
all $b\in B^{\prime}$, $f_{1}(X_{b})\subset J$, which is an interval of length
$t$. Thus if the constant $T=O(1)$ is chosen sufficiently large, then by (4.8)
(with $i=1,j=3)$, we have that if $b,b^{\prime}\in B^{\prime}$ are distinct,
then the sets $f_{3}(X_{b})$ and $f_{3}(X_{b})$ are contained in disjoint
intervals (of length $C^{O(1)}t$ ).

Thus

$\mathcal{E}_{\delta}(f_{3}((f_{1}\circ f_{2})^{-1}(E^{\prime}))\geq\sum_{b\in
B^{\prime}}\mathcal{E}_{\delta}(f_{3}(E^{\prime}_{b}))\geq
C^{-O(1)}(\\#B^{\prime})m.$

We conclude that

$\begin{split}\mathcal{E}_{\delta}(f_{3}(X_{1}))&\geq
C^{-O(1)}(\\#B^{\prime})m\\\
&\gtrsim(\delta^{\eta}C^{-1}|\log\delta|^{-1})^{O(1)}\big{(}Kt^{-\alpha/2}\big{)}\big{(}Kt^{\alpha/2}\delta^{-\alpha/2}\big{)}\\\
&=K^{2}(\delta^{\eta}C^{-1}|\log\delta|^{-1})^{O(1)}\delta^{-\alpha/2}.\end{split}$
(4.11)

Comparing this with (4.4) for $i=3$, we conclude that

$K\leq(C|\log\delta|\delta^{-\eta})^{O(1)}.$

Thus the set $X_{1}$ satisfies (4.6) for $i=1$, which completes the proof. ∎

Next, we will show that if a non-concentrated set is contained in the small
neighborhood of a curve, and if two functions have linearly independent
gradients on this curve, and these gradients are not normal to the curve, then
at least one of the functions must have large image on the set.

######  Lemma 4.2.

Let $0<\delta<s$, and let $\eta>0$. Let $\gamma\subset[0,1]^{2}$ be a simple
(no self-intersections) smooth curve. Let $f_{1},f_{2}\colon
N_{s}(\gamma)\to\mathbb{R}$ be functions, with

$\|f_{i}\|_{C^{1}(N_{s}(\gamma)}\leq\delta^{-\eta},$ (4.12)

$|\nabla f_{1}(x)\wedge\nabla f_{2}(x)|\geq\delta^{\eta},\quad x\in\gamma,$
(4.13)

$|\nabla f_{i}(x)\cdot v(x)|\geq\delta^{\eta},\quad x\in\gamma,$ (4.14)

where $v(x)$ is the unit vector tangent to $\gamma$ at $x$.

Let $A\subset N_{s}(\gamma)$ be a set, with
$\mathcal{E}_{\delta}(A)\geq\delta^{-\alpha+\eta}$, and suppose that for all
balls $B$ of radius $r\geq\delta$, $A$ satisfies the non-concentration
condition

$\mathcal{E}_{\delta}(A\cap B)\leq r^{\alpha}\delta^{-\alpha-\eta}.$ (4.15)

Then for at least one index $i$, we have

$\mathcal{E}_{\delta}(f_{i}(A))\gtrsim
s^{-\alpha/2}\delta^{-\alpha/2+5\eta}|\log\delta|^{-1}.$

###### Proof.

Cover $N_{s}(\gamma)$ by finitely overlapping balls of radius $O(s)$. After
dyadic pigeonholing, we can select a set of $N$ balls, each of which satisfy
$\mathcal{E}_{\delta}(A\cap
B)\gtrsim|\log\delta|^{-1}\delta^{-\alpha+\eta}N^{-1}$. Denote this set of
balls by $\mathcal{B}$. By (4.15), we have

$N\gtrsim|\log\delta|^{-1}\delta^{2\eta}s^{-\alpha}.$ (4.16)

By (4.12), for each ball $B\in\mathcal{B}$ and each index $i$, we have that
$f_{i}(B)$ is contained in an interval of length $O(s\delta^{-\eta})$. Let
$\gamma(t)$ be a unit speed parameterization of $\gamma$. By (4.14), if
$|t-t^{\prime}|\geq Cs\delta^{-2\eta}$ for a sufficiently large constant
$C=O(1)$, then if $B,B^{\prime}\in\mathcal{B}$ are balls containing
$\gamma(t)$ and $\gamma(t^{\prime})$ respectively, then the intervals
$f_{i}(B)$ and $f_{i}(B^{\prime})$ are disjoint. Fix such a choice of $C$.
After refining our collection of balls by a multiplicative factor of
$O(\delta^{2\eta})$, we can suppose that for each pair of distinct balls $B$
and $B^{\prime}$ and each index $i$, the intervals $f_{i}(B)$ and
$f_{i}(B^{\prime})$ are disjoint. Let $\mathcal{B}^{\prime}\subset\mathcal{B}$
be such a refinement.

By (4.13), for each ball $B\in\mathcal{B}$ there exists an index $i$ so that

$\mathcal{E}_{\delta}(f_{i}(A\cap
B))\gtrsim\delta^{\eta}\mathcal{E}_{\delta}(A\cap
B)^{1/2}\gtrsim|\log\delta|^{-1/2}\delta^{-\alpha/2+(3/2)\eta}N^{-1/2}$

Choose an index $i$ so that the above inequality holds for at least half of
the balls $B\in\mathcal{B}^{\prime}$; denote this set of balls
$\mathcal{B}^{\prime\prime}$. We have

$\begin{split}\mathcal{E}_{\delta}(f_{i}(A))&\geq\sum_{B\in\mathcal{B}^{\prime\prime}}\mathcal{E}_{\delta}(f_{i}(A\cap
B))\\\
&\gtrsim(\delta^{2\eta}N)|\log\delta|^{-1/2}\delta^{-\alpha/2+(3/2)\eta}N^{-1/2}\\\
&\gtrsim N^{1/2}\delta^{-\alpha/2+(7/2)\eta}|\log\delta|^{-1/2}\\\ &\gtrsim
s^{-\alpha/2}\delta^{-\alpha/2+5\eta}|\log\delta|^{-1}.\qed\end{split}$

With these lemmas, we are now ready to begin the proof of Theorem 1.11. We
will actually state and prove the following slightly more general (and
technical) version of the theorem.

######  Theorem 1.11∗.

For each $0<\alpha<2$, there exists $\eta=\eta(\alpha)>0$ and
$\delta_{0}=\delta_{0}(\alpha)>0$ so that the following is true for all
$0<\delta<\delta_{0}$. Let $Q\subset[0,1]^{2}$ be a square, and let
$\phi_{1},\phi_{2},\phi_{3}\colon Q\to\mathbb{R}$ be smooth, with

$\|\phi_{i}\|_{C^{2}(Q)}\leq\delta^{-\eta}.$ (4.17)

Suppose that for each $i\neq j$, we have

$|\nabla\phi_{i}\wedge\nabla\phi_{j}|\geq\delta^{\eta}\quad\textrm{on}\ Q,$
(4.18)

and

$\Big{|}\frac{\partial}{\partial\phi_{1}}\frac{\partial}{\partial\phi_{2}}\log\frac{\partial\phi_{3}/\partial\phi_{1}}{\partial\phi_{3}/\partial\phi_{2}}\Big{|}\geq\delta^{\eta}\quad\textrm{on}\
Q.$ (4.19)

Let $X\subset Q$ be a set with $\mathcal{E}_{\delta}(X)\geq\delta^{-\alpha}$,
and suppose that for all balls $B$ of diameter $r\geq\delta$, $X$ satisfies
the non-concentration condition

$\begin{split}\mathcal{E}_{\delta}(X\cap B)&\leq
r^{\alpha}\delta^{-\alpha-\eta}.\end{split}$ (4.20)

Then for at least one index $i\in\\{1,2,3\\}$ we have

$\mathcal{E}_{\delta}(\phi_{i}(X))\geq\delta^{-\alpha/2-\eta}.$ (4.21)

######  Remark 4.3.

To see that Theorem 1.11∗ implies Theorem 1.11, cover $K\subset U$ by open
squares, whose closures are contained in $U$. Since $K$ is compact, there is a
square $S\subset U$ with $\mathcal{E}_{\delta}(X\cap
S)\gtrsim\delta^{-\alpha}$. Since $\phi_{1},\phi_{2},\phi_{3}$ have gradients
that are pairwise linearly independent, and since the Blaschke curvature form
is non-vanishing on $U$, we have that (4.18) and (4.19) holds for all
$\delta>0$ sufficiently small.

###### Proof.

Suppose that (4.21) is false for $i=1,2,3$. If $\eta=\eta(\alpha)>0$ and
$\delta_{0}=\delta_{0}(\alpha)>0$ are selected sufficiently small, we will
obtain a contradiction.

First, we will select a sub-square $Q^{\prime}\subset Q$ with
$\mathcal{E}_{\delta}(X\cap Q^{\prime})\gtrsim\delta^{-\alpha+O(\eta)}$ so
that for each $i\neq j$, $\phi_{i}\otimes\phi_{j}$ is injective on
$Q^{\prime}$. Next, we apply Lemma 4.1 to $X\cap Q^{\prime}$ and the functions
$\phi_{1},\phi_{2},\phi_{3}$. We obtain a set $X^{\prime}\subset Q^{\prime}$
with $\mathcal{E}_{\delta}(X^{\prime})\gtrsim\delta^{-\alpha+O(\eta)}$ so that
for all intervals $J$ of length at least $\delta$, and for each index $i$, we
have

$\mathcal{E}_{\delta}(\phi_{i}(X^{\prime}\cap
J))\lesssim|J|^{\alpha/2}\delta^{-\alpha/2+O(\eta)}.$ (4.22)

The boundary of $Q^{\prime}$ is a union of four line segments. We will first
prove the following claim: Claim 1: If the constant $C=C(\alpha)$ is chosen
sufficiently large, then for each of these line segments $L$, we have

$\mathcal{E}_{\delta}(N_{\delta^{C\eta}}(L)\cap
X^{\prime})\leq\frac{1}{8}\mathcal{E}_{\delta}(X^{\prime}).$

Suppose not. Cover $L$ by finitely overlapping balls of radius
$\delta^{100\eta}$. By pigeonholing, there must exist a ball $B$ with

$\mathcal{E}_{\delta}(N_{\delta^{C\eta}}(L)\cap B\cap
X^{\prime})\geq\frac{1}{8}\delta^{100\eta}\mathcal{E}_{\delta}(X^{\prime})\gtrsim\delta^{-\alpha+O(\eta)}.$

Let $v$ be the unit vector parallel to $L$. By (4.18), there exist indices
$i\neq j$ so that $|v\cdot\nabla\phi_{i}(x_{0})|\gtrsim\delta^{\eta}$ and
$|v\cdot\nabla\phi_{j}(x_{0})|\gtrsim\delta^{\eta}$, where $x_{0}$ is the
center of the ball $B$. But by (4.17) and the fact that $B$ has radius
$\delta^{100\eta}$, we have that
$|v\cdot\nabla\phi_{i}(x_{0})|\gtrsim\delta^{\eta}$ and
$|v\cdot\nabla\phi_{j}(x_{0})|\gtrsim\delta^{\eta}$ for all $x\in B$. We now
apply Lemma 4.2 to conclude that (after interchanging the indices $i$ and $j$
if necessary)

$\mathcal{E}_{\delta}(\phi_{i}(N_{\delta^{C\eta}}(L)\cap B\cap
X^{\prime}))\gtrsim\delta^{-C\eta\alpha/2}\delta^{-\alpha/2+O(\eta)}|\log\delta|^{-1}.$

If $C=C(\alpha)$ is selected sufficiently large, then (4.21) holds for
$\phi_{i}$ and we are done. This concludes the proof of Claim 1.

Next, let $\psi=\phi_{1}\otimes\phi_{2}$ and let $V=\psi(Q^{\prime})$. By
(4.17) and (4.18), we have that $|D\psi|=O(\delta^{-\eta})$, and $|\det
D\psi|\gtrsim\delta^{O(\eta)}$. This implies that the
$\delta^{C\eta}$-neighborhood of the boundary of $Q^{\prime}$ is mapped to the
$\delta^{C+O(\eta)}$ neighborhood of the boundary of $V$. In particular, by
Claim 1 we can select $C^{\prime}=C^{\prime}(\alpha)$ so that if
$Y\subset\psi(X^{\prime})$ denotes the subset of $\psi(X^{\prime})$ that has
distance at least $\delta^{C^{\prime}\eta}$ from the boundary of $V$, then
$\mathcal{E}_{\delta}(Y)\gtrsim\delta^{-\alpha+O(\eta)}$. To simplify notion,
in what follows we will use $O_{\alpha}(\eta)$ to denote a quantity that is
bounded by $C_{0}\eta$, where $C_{0}=C_{0}(\alpha)$ may depend on $\alpha$.

Next we will introduce the Whitney cube decomposition. The version stated here
can be found in [11, Appendix J].

######  Theorem 4.4 (Whitney).

Let $\Omega$ be an open non-empty proper subset of $\mathbb{R}^{d}$. Then
there exists a countable family $\mathcal{Q}$ of closed cubes such that

  * •

$\bigcup_{Q\in\mathcal{Q}}Q=\Omega$, and the cubes in $\mathcal{Q}$ have
disjoint interiors.

  * •

For each cube $Q\in\mathcal{Q}$, we have $2Q\not\subset\Omega$, where $2Q$
denotes the 2-fold dilate of $Q$.

Apply Theorem 4.4 to $V$. While the set of cubes $\mathcal{Q}$ might be
infinite, the second item from Theorem 4.4 implies that
$\delta^{O_{\alpha}(\eta)}$ cubes from $\mathcal{Q}$ intersect $Y$. Hence we
can find a cube $Q_{0}\subset V$ of with $\mathcal{E}_{\delta}(Q_{0}\cap
Y)\gtrsim\delta^{-\alpha+O_{\alpha}(\eta)}$. This cube has side-length at most
$\delta^{-\eta}$. Cutting this cube into sub-cubes if necessary, we can find a
cube $Q_{1}\subset Q_{0}$ with side-length at most one, with
$\mathcal{E}_{\delta}(Q_{1}\cap Y)\gtrsim\delta^{-\alpha+O_{\alpha}(\eta)}.$
Let $Y^{\prime}=(Y\cap Q_{1})-(x_{1},y_{1}),$ where $(x_{1},y_{1})$ is the
bottom left corner of $Q_{1}$. Let $A=\pi_{x}(Y^{\prime})$ and
$B=\pi_{y}(Y^{\prime})$. Then $A,B\subset[0,1]$. Since $X^{\prime}$ satisfies
the conclusion of Lemma 4.1, for all intervals $J$ of length at least
$\delta$, we have the non-concentration estimates

$\begin{split}\mathcal{E}_{\delta}(A\cap
J)&\leq|J|^{\alpha/2}\delta^{-\alpha/2+O(\eta)},\\\ \mathcal{E}_{\delta}(B\cap
J)&\leq|J|^{\alpha/2}\delta^{-\alpha/2+O(\eta)}.\end{split}$

Define $P=\phi_{3}(\psi^{-1})$. We have that

$\|P\|_{C^{2}(Q_{1})}\lesssim\delta^{-O(\eta)},$ (4.23)

and

$|\partial_{x}P|\gtrsim\delta^{O(\eta)},\quad|\partial_{y}P|\gtrsim\delta^{O(\eta)}\quad\textrm{on}\
Q_{1}.$ (4.24)

Since the Blaschke curvature form is invariant under change of coordinates, by
(4.18) and (4.19), we have

$\Big{|}\partial_{xy}\Big{(}\log\left(\frac{\partial_{x}P}{\partial_{y}P}\right)\Big{)}\gtrsim\delta^{O(\eta)}\quad\textrm{on}\
Q_{1}.$

By (3.1), this implies

$|\partial_{xy}P|\gtrsim\delta^{O(\eta)}\quad\textrm{on}\ Q_{1}.$

Define $A^{\prime}\times B^{\prime}=A\times B\cap Q_{1}$. The function $P$ and
the sets $A^{\prime},B^{\prime},Q_{1}\cap Y^{\prime}$ satisfy the hypotheses
of Proposition 3.4 (with $\kappa=\alpha$). We conclude that if
$\eta=\eta(\alpha)>0$ is chosen sufficiently small, then there exists
$\varepsilon=\varepsilon(\alpha)>0$ so that

$\mathcal{E}_{\delta}\big{(}\\{(x,x^{\prime},y,y^{\prime})\in A^{\prime}\times
A^{\prime}\times B^{\prime}\times B^{\prime}\colon
P(x,y)=P(x^{\prime},y^{\prime})\\}\big{)}\leq\delta^{-(3/2)\alpha+\varepsilon}.$

By Lemma 8.7, this implies

$\mathcal{E}_{\delta}(P(Q_{2}\cap
Y^{\prime}))\gtrsim\frac{\delta^{-2\alpha+O(\eta)}}{\delta^{-(3/2)\alpha+\varepsilon}}=\delta^{-\alpha/2-\varepsilon+O(\eta)},$

and thus

$\mathcal{E}_{\delta}(\phi_{3}(X))\gtrsim\delta^{-\alpha/2-\varepsilon+O(\eta)}.$

If $\eta=\eta(\alpha)>0$ is sufficiently small, then we have
$\mathcal{E}_{\delta}(\phi_{3}(X))\geq\delta^{-\alpha/2-\eta},$ as desired. ∎

##  5 Pinned distinct distances: a single-scale estimate

In this section we will prove Theorem 1.8. To begin, we will perform some
computations to calculate the Blaschke curvature for the (squared) distance
functions from three points.

###  5.1 Blaschke curvature estimates for pinned distances

In this section we will compute the Blaschke curvature for the pinned distance
functions from three points. Define

$p_{1}=(t,0),\quad p_{2}=(-t,0),\quad p_{3}=(a,b).$

Define

$\begin{split}\phi_{1}(x,y)&=(x-t)^{2}+y^{2},\\\
\phi_{2}(x,y)&=(x+t)^{2}+y^{2},\\\
\phi_{3}(x,y)&=(x-a)^{2}+(y-b)^{2}.\end{split}$ (5.1)

We have

$\begin{split}x&=\frac{\phi_{2}-\phi_{1}}{4t},\\\
y&=\Big{(}\frac{\phi_{1}}{2}+\frac{\phi_{2}}{2}-t^{2}-\frac{(\phi_{2}-\phi_{1})^{2}}{16t^{2}}\Big{)}^{1/2},\end{split}$
(5.2)

and thus

$\phi_{3}=\frac{1}{16}\Big{(}\Big{(}\frac{4at+\phi_{1}-\phi_{2}}{t}\Big{)}^{2}+\Big{(}-4b+\sqrt{8\phi_{1}+8\phi_{2}-(\phi_{1}-\phi_{2})^{2}/t^{2}-16t^{2}}\
\ \Big{)}^{2}\Big{)}.$

We therefore have

$\begin{split}\frac{\partial\phi_{3}}{\partial\phi_{1}}&=\frac{-bx+(t+a)y-bt}{2ty},\\\
\frac{\partial\phi_{3}}{\partial\phi_{2}}&=\frac{\phantom{-}bx+(t-a)y-bt}{2ty},\end{split}$
(5.3)

i.e. $\frac{\partial\phi_{3}}{\partial\phi_{1}}$ vanishes on the line passing
through $p_{1}=(-t,0)$ and $p_{3}=(a,b)$, and
$\frac{\partial\phi_{3}}{\partial\phi_{2}}$ vanishes on the line passing
through $p_{2}=(t,0)$ and $p_{3}=(a,b)$. Both functions are finite off the
line passing through $p_{1}=(-t,0)$ and $p_{2}(t,0)$.

Next, we compute

$d\phi_{1}\wedge d\phi_{2}=(-8ty)dxdy,$

and finally, we compute

$\Big{(}\frac{\partial}{\partial\phi_{1}}\frac{\partial}{\partial\phi_{2}}\log\frac{\partial\phi_{3}/\partial\phi_{1}}{\partial\phi_{3}/\partial\phi_{2}}\Big{)}\
d\phi_{1}\wedge d\phi_{2}=b\frac{f(x,y)}{g^{2}(x,y)}dxdy,$ (5.4)

where

$\begin{split}f(x,y)=&\big{(}ab^{2}t^{4}-b^{2}t^{4}x-2ab^{2}t^{2}x^{2}+2b^{2}t^{2}x^{3}+ab^{2}x^{4}-b^{2}x^{5}\big{)}\\\
&+\big{(}-4abt^{4}+4a^{2}bt^{2}x+4bt^{4}x-4a^{2}bx^{3}-4bt^{2}x^{3}+4abx^{4}\big{)}y\\\
&+\big{(}-3a^{3}t^{2}-ab^{2}t^{2}+3at^{4}+3a^{2}t^{2}x+3b^{2}t^{2}x-3t^{4}x+3a^{3}x^{2}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad-3ab^{2}x^{2}-3at^{2}x^{2}-3a^{2}x^{3}+b^{2}x^{3}+3t^{2}x^{3}\big{)}y^{2}\\\
&+\big{(}2abt^{2}+2a^{2}bx-2b^{3}x-6bt^{2}x+2abx^{2}\big{)}y^{3}\\\
&+\big{(}a^{3}+2ab^{2}-at^{2}-3a^{2}x+2b^{2}x+3t^{2}x\big{)}y^{4}\\\
&+\big{(}-2ab\big{)}y^{5},\end{split}$ (5.5)

and

$g(x,y)=y\big{(}-bx+(t+a)y-bt\big{)}\big{(}bx+(t-a)y-bt\big{)}.$

We see that $|g|=O(1)$ on $[0,1]^{2}$, and $g$ vanishes on the three lines
spanned by the points $(-t,0),\ (t,0)$, and $(a,b)$. We see that $f$ is a
polynomial of degree at most 5 in the variables $x$ and $y$, and the
coefficient of $x^{5}$ is $-b^{2}$.

###  5.2 Pinned distances from three points

In this section we will prove Theorem 1.8. Suppose the proposition fails for
some $c>0$. If $c$ is sufficiently small (depending on $\alpha$), we will
obtain a contradiction. Applying a rotation, translation, and reflection, we
can suppose that $p_{1}=(t,0)$, $p_{2}=(-t,0)$ and $p_{3}=(a,b)$, with
$t\gtrsim\delta^{c}$ and $b\gtrsim\delta^{c}$.

Define $\phi_{i}(z)=|p_{i}-z|^{2}$. $|\nabla\phi_{i}|$ is bounded away from 0
on $[0,1]^{2}$ except in a small neighborhood of $p_{i}$. $\phi_{i}$ and
$\phi_{j}$ have linearly independent gradients except on the line $\ell_{ij}$
that contains $p_{i}$ and $p_{j}$. If $v_{ij}$ is a unit vector parallel to
$\ell_{ij}$ and if $k$ is the remaining index, then $\nabla\phi_{k}(z)\cdot
v_{ij}$ vanishes at one point on $\ell_{ij}$, which we will denote by $q_{k}$.

Our next task is to find a connected open region $U$ so that
$\mathcal{E}_{\delta}(X\cap U)\geq\frac{1}{2}\delta^{-\alpha+c}$, and $U$ is
far from the points $p_{i}$ and $q_{i}$, and the lines $\ell_{ij}$. The set
$U$ will be $[0,1]^{2}$, with certain neighborhoods of $p_{i},q_{i}$ and
$\ell_{ij}$ removed. To begin, define

$U_{0}=[0,1]^{2}\backslash\Big{(}\bigcup_{i}N_{C^{-1}\delta^{2c/\alpha}}(p_{i})\
\cup\ \bigcup_{i}N_{C^{-1}\delta^{2c/\alpha}}(q_{i})\Big{)}.$

If $C=O(1)$ is selected sufficiently large, then by (1.8), we have
$\mathcal{E}_{\delta}(X\cap U_{0})\geq\frac{3}{4}\delta^{-\alpha+c}$. We have

$\delta^{2c/\alpha}\lesssim|\nabla\phi_{i}|\lesssim 1\quad\textrm{on}\ U_{0}.$

Furthermore, since $B(q_{i},\ C^{-1}\delta^{2c/\alpha})$ is disjoint from
$U_{0}$ for each index $i$, we have

$\begin{split}&|\nabla\phi_{1}(x)\wedge\nabla\phi_{2}(x)|,\
|\nabla\phi_{2}(x)\wedge\nabla\phi_{3}(x)|\gtrsim\delta^{O_{\alpha}(c)}\quad\textrm{for
all}\ x\in\ell_{p_{1},p_{3}}\cap U_{0},\\\
&|\nabla\phi_{1}(x)\wedge\nabla\phi_{3}(x)|,\
|\nabla\phi_{2}(x)\wedge\nabla\phi_{3}(x)|\gtrsim\delta^{O_{\alpha}(c)}\quad\textrm{for
all}\ x\in\ell_{p_{1},p_{2}}\cap U_{0},\\\
&|\nabla\phi_{1}(x)\wedge\nabla\phi_{2}(x)|,\
|\nabla\phi_{1}(x)\wedge\nabla\phi_{3}(x)|\gtrsim\delta^{O_{\alpha}(c)}\quad\textrm{for
all}\ x\in\ell_{p_{2},p_{3}}\cap U_{0}.\end{split}$ (5.6)

Applying Lemma 4.2 to $\phi_{1}$ and $\phi_{2}$ with $s=\delta^{cC}$ and
$\eta\sim\delta^{O_{\alpha}(c)}$, then either

$\mathcal{E}_{\delta}(\phi_{i}\big{(}X\cap U_{0}\cap
N_{\delta^{Cc}}(\ell_{p_{1},p_{2}}))\big{)}\gtrsim\delta^{O_{\alpha}(c)-\alpha
cC/2-\alpha/2}|\log\delta|^{-1}$ (5.7)

for at least one index $i\in\\{1,2\\}$, or

$\mathcal{E}_{\delta}(X\cap U_{0}\cap
N_{\delta^{Cc}}(\ell_{p_{1},p_{2}}))\leq\frac{1}{12}\delta^{-\alpha+c}.$ (5.8)

If $C=C(\alpha)$ is chosen sufficiently large and $\delta>0$ is sufficiently
small, then (5.7) implies (1.9) and we are done. Thus we may suppose that
(5.8) holds. We argue similarly for $\ell_{p_{2},p_{3}}$ and
$\ell_{p_{1},p_{3}}$. Define

$U_{1}=U_{0}\backslash\\{N_{\delta^{Cc}}(\ell_{p_{1},p_{2}})\ \cup\
N_{\delta^{Cc}}(\ell_{p_{2},p_{3}})\ \cup\
N_{\delta^{Cc}}(\ell_{p_{1},p_{3}})\\}.$

Then for each pair of indices $i\neq j$, we have
$|\nabla\phi_{i}(x)\wedge\nabla\phi_{j}(x)|\gtrsim\delta^{O_{\alpha}(c)}$ for
all $x\in U_{1}.$ In fact, the above inequality holds on a slightly larger
set; if $C_{1}=C_{1}(\alpha)$ is selected sufficiently large, then

$|\nabla\phi_{i}(x)\wedge\nabla\phi_{j}(x)|\gtrsim\delta^{O_{\alpha}(c)}\quad\textrm{for
all}\ x\in N_{\delta^{C_{1}c}}(U_{1}).$ (5.9)

Our next task is to bound the size of the subset of $X\cap U_{1}$ where the
function $f$ from (5.5) is small. Let $s>0$ be a small parameter. Since $f$ is
a polynomial of degree at most 5, which contains the monomial $-b^{2}x^{5}$,
and since $b\gtrsim\delta^{c}$, there exists an absolute constant $C_{0}$ so
that

$\sup_{(x,y)\in[0,1]^{2}}|f(x,y)|\geq C_{0}^{-1}\delta^{C_{0}c}.$

Next, we will introduce the following Remez-type inequality. See [8, Theorem
2]

######  Theorem 5.1 (Remez).

Let $P\colon\mathbb{R}^{d}\to\mathbb{R}$ be a polynomial of degree at most
$D$. Let $\Omega\subset\mathbb{R}^{d}$ be an open convex set. Let
$m=\sup_{z\in\Omega}|P(z)|$. Then for all $0<\lambda<1$,

$|\\{z\in\Omega\colon|P(z)|\leq\lambda m\\}|\leq 4d|\Omega|\lambda^{1/D}.$

Applying Theorem 5.1, we have

$|\\{(x,y)\in[0,1]^{2}\colon|f(x,y)|<s\\}|\leq
8C_{0}^{1/2}\delta^{-C_{0}c/2}s^{1/2}.$ (5.10)

Now, the set $S=\\{(x,y)\in[0,1]^{2}\colon|f(x,y)|<s\\}$ is relatively open in
$[0,1]^{2}$. The boundary of this set is contained in

$Z=Z(x)\cup Z(y)\cup Z(x-1)\cup Z(y-1)\cup Z(f-s)\cup Z(f+s).$

In particular, if
$w=\big{(}8C_{0}^{1/2}\delta^{-C_{0}c/2}s^{1/2}\big{)}^{1/2}$, then

$S\subset N_{w}(Z).$ (5.11)

To verify (5.11), note that if $z\in S\backslash N_{w}(Z)$, then
$\operatorname{dist}(z,Z)\geq w$, and hence $B(z,w)\subset S$, and thus
$|S|\geq|B(z,w)|>w^{2},$ which contradicts (5.10).

Define $Q=xy(x-1)(y-1)(f-s)(f+s)$. Without loss of generality we will suppose
$Q$ is square-free (if not, we will remove any repeated components). Then $Q$
is a polynomial of degree at most 14, and $Z=Z(Q)$. At each point $p\in Z$ for
which $\nabla Q$ is non-zero, the unit tangent vector to $Z$ is given by
$v_{p}=\frac{(\partial_{y}Q,\ -\partial_{x}Q)}{|\nabla Q|}$. By (5.9), for
each $p\in Z\cap U_{1}$, there are at least two indices $i$ for which
$|v_{p}\cdot\nabla\phi_{i}(p)|\gtrsim\delta^{O_{\alpha}(c)}$, i.e. there
exists a constant $C_{3}=C(\alpha)$ (that is independent of $p$ and $i$) so
that

$\big{(}(\partial_{y}Q)(\partial_{x}\phi_{i})-(\partial_{x}Q)(\partial_{y}\phi_{i})\big{)}^{2}\geq
C_{3}\delta^{C_{3}c}\big{(}(\partial_{x}Q)^{2}+(\partial_{y}Q)^{2}\big{)}.$

Cut the curve $Z$ at each point where it intersects $B(0,2)$; at each point
where $|\nabla Q|$ vanishes; and at each point satisfying

$\big{(}(\partial_{y}Q)(\partial_{x}\phi_{i})-(\partial_{x}Q)(\partial_{y}\phi_{i})\big{)}^{2}=C_{3}\delta^{C_{\alpha}c}\big{(}(\partial_{x}Q)^{2}+(\partial_{y}Q)^{2}\big{)}\quad\textrm{for
some index}\ i.$

By slightly perturbing the constant $C_{3}$ if necessary, we can ensure that
the latter set of points is finite. All together, we have cut $Z$ at $O(1)$
points, and after removing these points, $Z\cap B(0,1)$ is a union of $M=O(1)$
simple smooth curves. For each such curve $\gamma$, there are at least two
indices $i,j$ so that $|v_{p}\cdot\nabla\phi_{i}(p)|\geq C_{3}\delta^{C_{3}c}$
for all $p\in\gamma$, and similarly for $\phi_{j}$.

We now apply Lemma 4.2 to each of these simple smooth curves. We conclude that
if $C_{4}=C_{4}(\alpha)$ is chosen sufficiently large, then for each of these
curves $\gamma$, we have

$\mathcal{E}_{\delta}(X\cap
N_{\delta^{C_{4}c}}\gamma)\leq(10M)^{-1}\delta^{-\alpha+c}.$

Define $U=U_{1}\backslash\bigcup_{\gamma}\cap N_{\delta^{C_{4}c}}\gamma$. If
$C_{5}=C_{5}(\alpha)$ is chosen sufficiently large, then

$|f(z)|\gtrsim\delta^{O_{\alpha}(c)}\quad\textrm{for all}\ z\in
N_{\delta^{C_{5}c}}(U).$ (5.12)

Finally, let $C_{6}=\max(C_{1},C_{5})$, where $C_{1}$ is the constant from
(5.9) and $C_{5}$ is the constant from (5.12). By Theorem 4.4, we can cover
$U$ by interior-disjoint squares, each of which has side-length
$\lesssim\delta^{C_{6}c}$, and each of which are contained in
$N_{\delta^{C_{6}c}}(U)$.

By pigeonholing, we can select a square $Q$ with $\mathcal{E}_{\delta}(Q\cap
X)\gtrsim\delta^{-\alpha+O_{\alpha}(c)}$. We have

$|\nabla\phi_{i}\wedge\nabla\phi_{j}|\gtrsim\delta^{O_{\alpha}(c)},\quad|f|\gtrsim\delta^{O_{\alpha}(c)}\quad\textrm{on}\
Q.$ (5.13)

Thus if $c$ and $\delta_{0}$ are selected sufficiently small (depending on
$\alpha)$, then by (5.4) and (5.13), the functions
$\phi_{1},\phi_{2},\phi_{3}$ satisfy the hypotheses of Theorem 1.11∗ on $Q$.
We conclude that there exists an index $i$ so that

$\mathcal{E}_{\delta}(\phi_{i}(X))\geq\mathcal{E}_{\delta}(\phi_{i}(X\cap
Q))\geq\delta^{-\alpha/2-2c}.$ (5.14)

Finally, since $\phi_{i}(x)=(\Delta_{p_{i}}(x))^{2}$ and $|\phi_{i}(x)|\leq 2$
for $x\in[0,1]^{2}$, (5.14) implies that

$\mathcal{E}_{\delta}(\Delta_{p_{i}}(X))\gtrsim\delta^{-\alpha/2-2c}.$ (5.15)

Decreasing $\delta_{0}$ slightly if necessary to account for the implicit
constant in the quasi-inequality (5.15), we obtain (1.9).

##  6 Convexity and discretized projections

In this section we will prove Theorem 1.12. We will actually state and prove
the following slightly more general (and technical) version of the theorem.

######  Theorem 1.12∗.

For each $0<\alpha<2,$ there exists $c=c(\alpha)>0$ and
$\delta_{0}=\delta_{0}(\alpha)>0$ so that the following holds for all
$0<\delta<\delta_{0}$. Let $I,J\subset[0,1]$ be intervals and let $u\colon
I\to\mathbb{R},\ v\colon J\to\mathbb{R}$ be smooth. Suppose that

$\begin{split}\delta^{c}\leq\Big{|}\frac{d}{dx}u(x)\Big{|}\leq\delta^{-c},\quad\delta^{c}\leq\Big{|}\frac{d^{2}}{(dx)^{2}}u(x)\Big{|}\leq\delta^{-c},&\quad\textrm{for}\
x\in I,\\\
\delta^{c}\leq\Big{|}\frac{d}{dx}v(x)\Big{|}\leq\delta^{-c},\quad\quad\quad\;\Big{|}\frac{d^{2}}{(dx)^{2}}v(x)\Big{|}\leq\delta^{-c}&\quad\textrm{for}\
x\in J.\end{split}$ (6.1)

Let $\phi_{1}(x,y)=x,$ $\phi_{2}(x,y)=y,$ $\phi_{3}(x,y)=\frac{1}{2}(x+y)$,
and $\phi_{4}(x,y)=u(x)+v(y)$. Let $X\subset I\times J$, with
$\mathcal{E}_{\delta}(X)\geq\delta^{-\alpha+c}$, and suppose that for all
squares $Q$ of sidelength $r\geq\delta$, $X$ satisfies the non-concentration
condition

$\mathcal{E}_{\delta}(X\cap Q)\leq r^{\alpha}\delta^{-\alpha-c}.$ (6.2)

Then for at least one index $i$ we have

$\mathcal{E}_{\delta}(\phi_{i}(X))\geq\delta^{-\alpha/2-c}.$ (1.13)

###### Proof.

Let $X$ be a union of $\delta$-squares such that
$\mathcal{E}_{\delta}(X)=\delta^{-\alpha+c}$ and $X$ satisfies (6.2). Put
$A=\phi_{1}(X)$, $B=\phi_{2}(X)$, $C=\frac{1}{2}\phi_{3}(X)$, and
$D=\phi_{4}(X)$. Suppose that

$\mathcal{E}_{\delta}(A),\;\mathcal{E}_{\delta}(B),\;\mathcal{E}_{\delta}(C),\;\mathcal{E}_{\delta}(D)\leq\delta^{-\alpha/2-c}.$
(6.3)

If $c,\delta>0$ are sufficiently small, we will reach a contradiction.
Replacing $X$ by a large subset of $X$, we may assume without loss of
generality that

$\begin{split}&\mathcal{E}_{\delta}(\phi_{i}(X)\cap
K)\leq\delta^{-c}|K|^{\alpha/2}\delta^{-\alpha/2},\end{split}$ (6.4)

for all intervals $K$ of length at least $\delta$ and for each index
$i=1,2,3$. Indeed, this follows from Lemma 4.1 applied to $\phi_{1}$,
$\phi_{2}$, $\phi_{3}$ and the set $X\subset I\times J$.

Note that our assumption (6.3) and the fact that $X\subset A\times B$ imply

$\mathcal{E}_{\delta}(A),\mathcal{E}_{\delta}(B)\geq\delta^{-\alpha/2+2c}.$
(6.5)

Similarly, we have

$\mathcal{E}_{\delta}(C)\gtrsim\delta^{-\alpha/2+2c}.$ (6.6)

Indeed, let $\Gamma$ be the graph of $(x,y)\mapsto\frac{1}{2}(x+y)$ restricted
to $(x,y)\in X$. Then
$\mathcal{E}_{\delta}(\Gamma)\approx\mathcal{E}_{\delta}(X)$. Up to reorder of
the coordinates, $\Gamma$ is also the graph of $(x,z)\mapsto 2z-x$ restricted
to the set $Z=\\{(x,\tfrac{1}{2}(x+y))\mid(x,y)\in X\\}\subset A\times C$.
Thus

$\delta^{-\alpha+c}\approx\mathcal{E}_{\delta}(\Gamma)\approx\mathcal{E}_{\delta}(Z)\lesssim\mathcal{E}_{\delta}(A\times
C)\leq\delta^{-\alpha/2-c}\mathcal{E}_{\delta}(C)$

which gives (6.6).

Let

$Y:=\\{(t,x,y)\in A\times B\times B\mid(t,x),\;(t,y)\in X\\}.$

We have

$\mathcal{E}_{\delta}(Y)\gtrsim\delta^{-3\alpha/2+3c}.$ (6.7)

Indeed, by Cauchy–Schwarz inequality,

$|Y|=\int_{t\in A}|\phi_{1}^{-1}(t)\cap X|^{2}dt\geq\left(\int_{t\in
A}|\phi_{1}^{-1}(t)\cap
X|dt\right)^{2}/|A|=|X|^{2}/|A|\gtrsim\delta^{3-3\alpha/2+3c}.$

Consider the variety $V\subset[0,1]^{4}\times\mathbb{R}$ given by

$\displaystyle z$ $\displaystyle=\tfrac{1}{2}(t+x)$ $\displaystyle w$
$\displaystyle=u(t)+v(y).$

Let $\Gamma^{\prime}\subset V$ denote the graph of
$(t,x,y)\mapsto(\tfrac{1}{2}(t+x),u(t)+v(y))$ restricted to $Y$. Then

$\mathcal{E}_{\delta}(\Gamma^{\prime})\geq\mathcal{E}_{\delta}(Y)\gtrsim\delta^{-3\alpha/2+3c}.$

Let $S\subset B\times B\times C$ denote the projection of $\Gamma^{\prime}$ to
the $xyz$-space. We can view $\Gamma^{\prime}$ as the graph of
$(x,y,z)\mapsto(t,w)$ restricted to $S$. Indeed, for $(x,y,z)\in S$ we have
$(t,w)=(2z-x,u(2z-x)+v(y))$. Then

$\mathcal{E}_{\delta}(S)\gtrsim\delta^{c}\mathcal{E}_{\delta}(\Gamma^{\prime})\gtrsim\delta^{-3\alpha/2+4c},$
(6.8)

where for this we used the upper bounds from (6.1).

Let

$G(x,y,z)=u(2z-x)+v(y),$

which is defined on

${\rm dom}(G)=[0,1]^{3}\cap\\{2z-x\in I,\ y\in J\\}.$

Then $S\subset{\rm dom}(G)\cap(B\times B\times C)$, $G(S)\subset D$, and
$\mathcal{E}_{\delta}(S)\gtrsim\delta^{-3\alpha/2+4c}$.

We are almost ready to apply Lemma 2.2 to $G$. As a final preparation step we
would like to restrict $G$ to a cube that contains a portion of $S$. For this
we fix a parameter $0<r<1$, and consider the family ${\mathcal{Q}}$ of cubes
in $[0,r]^{3}+r\mathbb{Z}^{3}$ that intersect $S$. Using
$\mathcal{E}_{\delta}(S)\leq\mathcal{E}_{\delta}(B\times B\times C)$, we have

$\\#{\mathcal{Q}}\leq r^{-3\alpha/2-3c}.$ (6.9)

Recall that ${\rm dom}(G)=[0,1]^{3}\cap\\{2z-x\in I,\ y\in J\\}$. Write
$I=[a_{0},a_{1}]$ and let $\partial{\mathcal{Q}}\subset{\mathcal{Q}}$ denote
the subset of cubes that intersect one of the planes $2z-x=a_{0}$ and
$2z-x=a_{1}$. We claim that

$\mathcal{E}_{\delta}\left(S\cap\bigcup_{Q\in\partial{\mathcal{Q}}}Q\right)\lesssim
r^{\alpha/2}\delta^{-3\alpha/2-3c}.$ (6.10)

Indeed, let $\mathcal{T}$ denote the projection of the cubes in
$\partial\mathcal{Q}$ to the $xy$-plane. Then $\mathcal{T}$ is a family of
interior-disjoint squares of side length $r$ that intersect $B\times B$. Each
square in $\mathcal{T}$ is the projection of at most four cubes from
$\partial\mathcal{Q}$, as is easy to verify. Thus, we have

$\displaystyle|S\cap\bigcup_{Q\in\partial{\mathcal{Q}}}Q|$
$\displaystyle\leq\sum_{Q\in\partial{\mathcal{Q}}}|(B\times B\times C)\cap Q|$
$\displaystyle=\sum_{Q=T\times K\in\partial{\mathcal{Q}}}|(B\times B)\cap
T||C\cap K|$
$\displaystyle\leq\delta^{-c}r^{\alpha/2}\delta^{1-\alpha/2}\sum_{Q=T\times
K\in\partial{\mathcal{Q}}}|(B\times B)\cap T|$ $\displaystyle\lesssim
4\delta^{-c}r^{\alpha/2}\delta^{1-\alpha/2}|B\times B|$
$\displaystyle\lesssim\delta^{-3c}r^{\alpha/2}\delta^{3-3\alpha/2},$

where for the third line we use the non-concentration inequality (6.4). This
proves (6.10).

Letting $r\approx\delta^{14c/\alpha}$ and in view of (6.8) and (6.10), we get

$\mathcal{E}_{\delta}\left(S\cap\bigcup_{Q\in{\mathcal{Q}}\setminus\partial{\mathcal{Q}}}Q\right)\geq\frac{1}{2}\mathcal{E}_{\delta}(S).$
(6.11)

By the pigeonhole principle and (6.9), there exists a cube
$Q_{0}\in{\mathcal{Q}}\setminus\partial{\mathcal{Q}}$ such that

$|S\cap Q_{0}|\geq\delta^{3-3\alpha/2+\eta},$ (6.12)

where $\eta=c(37+42c/\alpha)$.

Finally, note that $Q_{0}\cap{\rm dom}(G)$ is a rectangular box, and write
$Q_{0}\cap{\rm dom}(G)=I_{1}\times I_{2}\times I_{3}$. We apply Lemma 2.2 to
$G:I_{1}\times I_{2}\times I_{3}\to\mathbb{R}$; let $\sigma=\sigma(\alpha)$ be
the quantity given by the lemma. Write $A_{1}=B\cap I_{1}$, $A_{2}=B\cap
I_{2}$ and $A_{3}=C\cap I_{3}$. Combining (6.3) and (6.12), we see that

$\mathcal{E}_{\delta}(A_{i})\geq\delta^{-\alpha/2+\eta+2c},~{}~{}\text{for
$i=1,2,3$}.$

By (6.4), we have the non-concentration inequality

$\mathcal{E}_{\delta}(A_{i}\cap
K)\leq\delta^{-c}r^{\alpha/2}\delta^{-\alpha/2},$

where $K$ is an interval of length $r\geq\delta$, for $i=1,2,3$.

We have

$\displaystyle G_{x}$
$\displaystyle=-u^{\prime}(2z-x),~{}~{}G_{y}=v^{\prime}(y),~{}~{}G_{z}=2u^{\prime}(2z-x)$

and

$\displaystyle G_{xx}$
$\displaystyle=u^{\prime\prime}(2z-x),~{}~{}G_{xy}=G_{yz}=0,~{}~{}G_{xz}=-2u^{\prime\prime}(2z-x)$
$\displaystyle G_{yy}$
$\displaystyle=v^{\prime\prime}(y),~{}~{}G_{zz}=4u^{\prime\prime}(2z-x).$

In view of (6.1), we have

$\|G\|_{C^{2}(I_{1}\times I_{2}\times I_{3})}\lesssim\delta^{-c}$

and

$\inf_{I_{1}\times I_{2}\times I_{3}}|\nabla G_{(z)}(x,y)|\geq\delta^{c}.$

So $G$ satisfies (2.6) and (2.7).

We next verify (2.8). We have

$\displaystyle\partial_{z}\theta_{(x,y)}(z)$
$\displaystyle=\partial_{z}\arctan\frac{G_{y}}{G_{x}}$
$\displaystyle=\frac{G_{yz}G_{x}-G_{y}G_{xz}}{G_{x}^{2}+G_{y}^{2}}$
$\displaystyle=\frac{2u^{\prime\prime}(2z-x)v^{\prime}(y)}{(u^{\prime}(2z-x))^{2}+(v^{\prime}(y))^{2}}.$

Thus, in view of (6.1), we have

$\displaystyle\inf_{I_{1}\times I_{2}\times
I_{3}}|\partial_{z}\theta_{(x,y)}(z)|$ $\displaystyle\gtrsim\delta^{4c}.$

Thus Lemma 2.2 tells us that

$\mathcal{E}_{\delta}(S\cap Q_{0})\leq\mathcal{E}_{\delta}\left(\\{(x,y,z)\in
A_{1}\times A_{2}\times A_{3}\mid G(x,y,z)\in
D\\}\right)\leq\delta^{-3\alpha/2+\sigma}.$

However, choosing $c=c(\alpha)>0$ sufficiently small, this contradicts (6.12).
Thus our assumption (6.3) is false, which proves the theorem. ∎

##  7 Exceptional vantage points for pinned distances

In this section we will prove Theorem 1.5 using its single-scale variant,
Theorems 1.8. Suppose that the set

$K=\\{p\in\mathbb{R}^{2}\colon\dim\Delta_{p}(A)\leq(\dim A)/2+c\\}$

is curved. If $c$ is sufficiently small (depending on $\alpha$) then we will
obtain a contradiction. Our arguments will be similar to those in [4, Section
7]. Our first task will be to reduce to the case where $A$ and $K$ are
contained in $[0,1]^{2}$.

Let $\mu$ be a Borel probability measure on $K$ that satisfies (1.5). Select
$R$ sufficiently large so that $\dim(A\cap B(0,R))\geq\dim A-c/2$ and
$\mu(B(0,R))\geq 1/2$. Define $h(x,y)=(2R)^{-1}(x,y)+(1/2,1/2)$, so
$h(B(0,R))\subset[0,1]^{2}$. Abusing notation slightly, we will replace $K$ by
$h(K\cap B(0,R))$, and we will replace $A$ by $h(A\cap B(0,R))$. Then
$K,A\subset[0,1]^{2}$, $\dim(A)\geq\alpha-c/2$, and
$\dim\Delta_{p}(A)\leq\alpha/2+c<(\dim A)/2+2c$ for each $p\in K$. Continuing
our abuse of notation, we will let $\mu$ be a Borel probability measure on $K$
that satisfies (1.5).

Let $\nu$ be a Borel probability measure on $A$ that satisfies the Frostman
condition

$\nu(A\cap B(x,r))<Cr^{\alpha-c}.$ (7.1)

Let $\delta_{0}=\delta_{0}(\dim A)$ be the quantity from Theorem 1.8, and let
$k_{0}\in\mathbb{Z}$ be sufficiently large so that $2^{-k_{0}}<\delta_{0}$.
Let $p\in K$. Since $\dim(\Delta_{p}(A))<\dim(A)/2+2c$, we can cover
$\Delta_{p}(A)$ by a union of intervals
$\bigcup_{I\in\mathcal{G}_{p}}I=\bigcup_{k\geq
k_{0}}\bigcup_{I\in\mathcal{G}_{p,k}}I$, where $\mathcal{G}_{p,k}$ is a set of
at most $\delta^{-\alpha/2-2c}$ intervals of length $2^{-k}$. Since $\nu$ is a
probability measure, we have

$\sum_{k\geq k_{0}}\sum_{I\in\mathcal{G}_{p,k}}\nu(d_{p}^{-1}(I))\geq 1.$

Integrating with respect to $\mu$, we conclude that

$\sum_{k\geq
k_{0}}\int\sum_{I\in\mathcal{G}_{p,k}}\nu(d_{p}^{-1}(I))d\mu(p)=\int\sum_{k\geq
k_{0}}\sum_{I\in\mathcal{G}_{p,k}}\nu(d_{p}^{-1}(I))d\mu(p)\geq 1,$

and thus there exists $k\geq k_{0}$ so that

$\int\sum_{I\in\mathcal{G}_{p,k}}\nu(d_{p}^{-1}(I))d\mu(p)\geq\frac{6}{\pi^{2}}k^{-2}.$

Fix this choice of $k$. Define $\delta=2^{-k}$ (so
$k^{-2}\sim|\log\delta|^{-2}$). Cover $[0,1]^{2}$ by squares of the form
$[n\delta,\ (n+1)\delta]\times[m\delta,\ (m+1)\delta]$. After dyadic
pigeonholing, we can choose a set $\mathcal{S}$ of such squares, and a number
$w>0$ so that if we define $A^{\prime}=A\cap\bigcup_{S\in\mathcal{S}}S,$ then

$\int\sum_{I\in\mathcal{G}_{p,k}}\nu(A^{\prime}\cap
d_{p}^{-1}(I))d\mu(p)\gtrsim|\log\delta|^{-3},$ (7.2)

and $w\leq\nu(A^{\prime}\cap S)<2w$ for each $S\in\mathcal{S}$. By (7.2), we
have $|\log\delta|^{-3}w^{-1}\lesssim\\#\mathcal{S}\leq w^{-1}$. By (7.1) we
have $w\leq C\delta^{\alpha-c}$. Define
$A_{\delta}=\bigcup_{S\in\mathcal{S}}S$. By (7.2) we have
$\nu(A_{\delta})\gtrsim|\log\delta|^{-3}$, and thus
$\\#\mathcal{S}\gtrsim\delta^{-3}/(C\delta^{\alpha-c})\gtrsim\delta^{-\alpha+c}|\log\delta|^{-2}.$

Note that if $p\in\mathbb{R}^{2}$, $I$ is an interval of length $\delta$, and
$d_{p}(z)\in I$, then by the triangle inequality, $\Delta_{p}(S)\subset 3I$,
where $S\in\mathcal{S}$ is a square containing $z$, and $3I$ is the interval
of length $3\delta$ with the same midpoint as $I$. In particular,

$\nu(d_{p}^{-1}(I))\lesssim w\\#\\{S\in\mathcal{S}\colon d_{p}(S)\subset
3I\\}\lesssim(\\#\mathcal{S})^{-1}\\#\\{S\in\mathcal{S}\colon d_{p}(S)\subset
3I\\}.$

For each $p\in K,$ define $\mathcal{I}_{p}=\\{3I\colon
I\in\mathcal{G}_{p,k}\\}$ (note that
$\\#\mathcal{I}_{p}\leq\delta^{-\alpha/2-2c})$ and define

$\mathcal{S}_{p}=\bigcup_{I\in\mathcal{I}_{p}}\\{S\in\mathcal{S}\colon\Delta_{p}(S)\subset
I\\}.$

The sets $\\{S\in\mathcal{S}\colon\Delta_{p}(S)\subset I\\}$ are boundedly
overlapping as $I$ ranges over the elements in $\mathcal{I}_{p}$, and thus
(7.2) becomes

$\int(\\#\mathcal{S}_{p})d\mu(p)\gtrsim|\log\delta|^{-3}(\\#\mathcal{S}).$

For each $S\in\mathcal{S}$, define $m(S)=\mu(\\{p\in K\colon
S\in\mathcal{S}_{p}\\})$. We have

$\displaystyle|\log\delta|^{-9}(\\#\mathcal{S})^{3}$
$\displaystyle\lesssim\Big{(}\sum_{S\in\mathcal{S}}m(S)\Big{)}^{3}\leq(\\#\mathcal{S})^{2}\Big{(}\sum_{S\in\mathcal{S}}m(S)^{3}\Big{)},$
(7.3)

and thus

$\displaystyle\int\\#(\mathcal{S}_{p_{1}}\cap\mathcal{S}_{p_{2}}\cap\mathcal{S}_{p_{3}})d\mu(p_{1})d\mu(p_{2})d\mu(p_{3})\gtrsim|\log\delta|^{-9}(\\#\mathcal{S}),$
(7.4)

Define

$\displaystyle\mathcal{T}$ $\displaystyle=\big{\\{}(p_{1},p_{2},p_{3})\in
K^{3}\colon\\#(\mathcal{S}_{p_{1}}\cap\mathcal{S}_{p_{2}}\cap\mathcal{S}_{p_{3}})\geq
C_{0}^{-1}|\log\delta|^{-9}(\\#\mathcal{S})\big{\\}},$ (7.5)

If $C_{0}$ is a sufficiently large absolute constant, then
$\mu^{3}(\mathcal{T})\geq C_{0}^{-1}|\log\delta|^{-9}$.

Next, by (1.5), for each $r>0$ we have

$\displaystyle\mu^{3}\\{(p_{1},p_{2},p_{3})\colon|T(p_{1},p_{2},p_{3})|\leq
r\\}\leq Cr^{\beta},$ (7.6)

Comparing (7.5) and (7.6), if the constant $C_{1}$ is chosen sufficiently
large, then there exists $(p_{1},p_{2},p_{3})\in K^{3}$ so that

$\displaystyle\\#(\mathcal{S}_{p_{1}}\cap\mathcal{S}_{p_{2}}\cap\mathcal{S}_{p_{3}})\geq
C_{0}^{-1}|\log\delta|^{-9}(\\#\mathcal{S}),$ (7.7)

and

$\displaystyle|T(p_{1},p_{2},p_{3})|>C_{1}^{-1}|\log\delta|^{-9/\beta},$ (7.8)

Define
$\mathcal{S}^{\prime}=\mathcal{S}_{p_{1}}\cap\mathcal{S}_{p_{2}}\cap\mathcal{S}_{p_{3}}$.
Then
$\mathcal{E}_{\delta}(\Delta_{p_{i}}(\mathcal{S}^{\prime})\lesssim\delta^{-\alpha/2-2c}$
for each index $i$. If $c=c(\alpha)>0$ and $\delta_{0}=\delta_{0}(\alpha)>0$
are selected sufficiently small, then this violates Theorem 1.8.

##  8 Discretized expanding polynomials

In this section we will prove Theorem 1.18. Recall that Proposition 3.4
controls the number of solutions to $P(a,b)=P(a^{\prime},b^{\prime})$ on a
region where $P_{x},P_{y},P_{xy}$ and $K_{P}$ are large. If any of these
quantities vanish identically, then $P$ is a special form. But it is possible
that none of these quantities vanish identically, but nonetheless they are
small on a substantial portion of $X\subset A\times B$. In the next sub-
section, we will show that after restricting to a suitable sub-square of
$[0,1]^{2}$, we can suppose that $P_{x},P_{y},P_{xy}$ and $K_{P}$ are large.

###  8.1 Analytic varieties do not concentrate on fractal sets

In this section we will introduce several technical results showing how
products of non-concentrated sets can intersect thin neighborhoods of
varieties $Z(f)$, where $f$ is an analytic function. Our setup will be as
follows. Let $A_{1},\ldots,A_{d}\subset[0,1]$ be sets and let
$A=A_{1}\times\cdots\times A_{d}$. Let $\delta>0$ and suppose that for all
intervals $J$ of length at least $\delta$, each set $A_{i}$ satisfies the non-
concentration condition

$\mathcal{E}_{\delta}(A_{i}\cap
J)\leq\delta^{-\eta}|J|^{\kappa}\mathcal{E}_{\delta}(A_{i}).$

We would like to show that if $f\colon\mathbb{R}^{d}\to\mathbb{R}$ is
analytic, then $|N_{t}(Z(f))\cap A|$ must be much smaller than $|A|$. Results
of this type, combined with Whitney’s cube decomposition, will allow us to
find a (reasonably) small collection of interior disjoint cubes whose union
covers most of $A$, so that $f$ is not too small on each cube. Results of this
type will help us control the size of various “bad” regions where discretized
expansion might fail.

######  Theorem 8.1 (Stratification of the vanishing locus of an analytic
function [28]).

Let $U\subset\mathbb{R}^{d}$ be an open set that contains $[0,1]^{d}$ and let
$\phi\colon U\to\mathbb{R}$ be real analytic and not identically zero. Then
there is a finite set $\mathcal{M}$ of smooth (proper) submanifolds of
$\mathbb{R}^{d}$ so that
$Z(\phi)\cap[0,1]^{d}\subset\bigcup_{M\in\mathcal{M}}M$.

######  Lemma 8.2.

Let $M$ be a smooth (proper) submanifold of $\mathbb{R}^{d}$, and suppose
$M\cap[0,1]^{d}$ is compact. Then there is a constant $C>0$ so that the
following holds.

Let $A_{1},\ldots,A_{d}\subset[0,1]$ be sets and let
$A=A_{1}\times\cdots\times A_{d}$. Let $\delta>0$ and suppose that for all
intervals $J$ of length at least $\delta$, each set $A_{i}$ satisfies the non-
concentration condition

$\mathcal{E}_{\delta}(A_{i}\cap
J)\leq\delta^{-\eta}|J|^{\kappa}\mathcal{E}_{\delta}(A_{i}).$

Then for each $s>\delta$ we have

$\mathcal{E}_{\delta}\big{(}A\cap N_{s}(M)\big{)}\leq
C\delta^{-\eta}s^{\kappa}\mathcal{E}_{\delta}(A).$ (8.1)

###### Proof.

For each $p\in M\cap[0,1]^{d}$, let $i(p)\in\\{1,\ldots,d\\}$ be an index so
that the coordinate unit vector $e_{i(p)}$ is not contained in the tangent
space $T_{p}(M)$. If $t>0$ is selected sufficiently small, then the set
$M_{p}=M\cap B(p,t)$ is a smooth manifold, and there exists a constant $C_{p}$
so that for all $q,q^{\prime}\in M_{p}$, we have

$|e_{i(p)}\cdot q-e_{i(p)}\cdot q^{\prime}|\leq
C_{p}|\pi_{p}(q)-\pi_{p}(q^{\prime})|,$ (8.2)

where $\pi_{p}\colon\mathbb{R}^{d}\to\mathbb{R}^{d-1}$ is the orthogonal
projection to the $(d-1)$-dimensional subspace
$e_{i(p)}^{\perp}\subset\mathbb{R}^{d}$. Indeed, the above inequality follows
from the fact that $\inf_{v\in T_{p}(M)}\angle(e_{i(p)},v)>0$, and the map
$q\mapsto T_{q}(M)$ is continuous.

Since $M\cap[0,1]^{d}$ is compact, we can cover $M\cap[0,1]^{d}$ by a finite
set $\mathcal{M}$ of manifolds of the form $M_{p}$. It suffices to prove (8.1)
for each of these manifolds. Let $C_{0}=\max_{M_{p}\in\mathcal{M}}C_{p}$.

Let $M_{p}\in\mathcal{M}$. Let $s\geq\delta$ and let $\mathcal{Q}$ be the set
of cubes of side-length $s$ aligned with the grid $(s\mathbb{Z})^{d}$ that
intersect $M_{p}$. We claim that at most $C_{1}$ cubes from $\mathcal{Q}$ can
have the same projection under $\pi_{p}$, for some $C_{1}$ that depends only
on $d$ and $C_{0}$. Indeed, if $C_{1}>2$ cubes have the same projection under
$e_{i(p)}$, then select points $q,q^{\prime}\in M_{p}$ from two cubes that
have greatest distance. Since there are at least $C_{1}-2$ cubes in-between
the cubes containing $q$ and $q^{\prime}$, we have $|e_{i(p)}\cdot
q-e_{i(p)}\cdot q^{\prime}|\geq(C_{1}-2)s$. On the other hand, since all of
the cubes (and in particular, the cubes containing $q$ and $q^{\prime}$) have
the same projection under $\pi_{p}$, we have that
$|\pi_{p}(q)-\pi_{p}(q^{\prime})|\leq\sqrt{d-1}s$. Thus by (8.2), we must have
$(C_{1}-2)s\leq C_{0}\sqrt{d-1}s,$ so $C_{1}\leq C_{0}\sqrt{d-1}+2$, which
proves our claim.

For each such cube $Q\in\mathcal{Q}$ we have the estimate

$\mathcal{E}_{\delta}(Q\cap
A)\leq\delta^{-\eta}s^{\kappa}\mathcal{E}_{\delta}(A_{i(p)})\mathcal{E}_{\delta}(\pi_{p}(A\cap
Q)).$ (8.3)

Summing (8.3) over all cubes in $\mathcal{Q}$ and using the fact that at most
$C_{1}$ cubes have the same projection under $\pi_{p}$, we obtain

$\begin{split}\mathcal{E}_{\delta}(N_{s}(M)\cap A)&\leq
C_{1}\delta^{-\eta}s^{\kappa}\mathcal{E}_{\delta}(A_{i(p)})\mathcal{E}_{\delta}(\pi_{p}(A))\\\
&\leq C_{1}\delta^{-\eta}s^{\kappa}\mathcal{E}_{\delta}(A).\qed\end{split}$

######  Remark 8.3.

Note that in the special case where where exists an index $i$ so that $e_{i}$
is not contained in $T_{p}(M)$, for every $p\in M$, it is sufficient in Lemma
8.2 to require the non-concentration condition only for $A_{i}$.

Combining Lemma 8.2 and Theorem 8.1, we obtain the following.

######  Corollary 8.4.

Let $U\subset\mathbb{R}^{d}$ be an open set that contains $[0,1]^{d}$ and let