# New improvement to Falconer distance set problem in higher dimensions

Xiumin Du, Yumeng Ou, Kevin Ren and Ruixiang Zhang

###### Abstract.

We show that if a compact set $E\subset\mathbb{R}^{d}$ has Hausdorff dimension
larger than $\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 3$, then
there is a point $x\in E$ such that the pinned distance set $\Delta_{x}(E)$
has positive Lebesgue measure. This improves upon bounds of Du–Zhang and
Du–Iosevich–Ou–Wang–Zhang in all dimensions $d\geq 3$. We also prove lower
bounds for Hausdorff dimension of pinned distance sets when
$\dim_{H}(E)\in(\frac{d}{2}-\frac{1}{4}-\frac{3}{8d+4},\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4})$,
which improves upon bounds of Harris and Wang–Zheng in dimensions $d\geq 3$.

##  1\. Introduction

A classical question in geometric measure theory, introduced by Falconer in
the early 80s ([9]) is, how large does the Hausdorff dimension of a compact
subset of ${\mathbb{R}}^{d}$, $d\geq 2$ need to be to ensure that the Lebesgue
measure of the set of its pairwise Euclidean distances is positive.

Let $E\subset\mathbb{R}^{d}$ be a compact set, its _distance set_ $\Delta(E)$
is defined by

$\Delta(E):=\\{|x-y|:x,y\in E\\}\,.$

######  Conjecture.

[Falconer] Let $d\geq 2$ and $E\subset\mathbb{R}^{d}$ be a compact set. Then

${\dim_{H}}(E)>\frac{d}{2}\Rightarrow|\Delta(E)|>0.$

Here $|\cdot|$ denotes the Lebesgue measure and ${\dim_{H}}(\cdot)$ is the
Hausdorff dimension.

The main result in this paper improves the best-known dimensional threshold
towards the Falconer conjecture in dimensions $d\geq 3$.

######  Theorem 1.1.

Let $d\geq 3$ and $E\subset\mathbb{R}^{d}$ be a compact set. Then

${\dim_{H}}(E)>\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}\Rightarrow|\Delta(E)|>0.$

Falconer’s conjecture remains open in all dimensions as of today. It has
attracted a great amount of attention in the past decades. To name a few
landmarks: in 1985, Falconer [9] showed that $|\Delta(E)|>0$ if
${\dim_{H}}(E)>\frac{d}{2}+\frac{1}{2}$. Bourgain [1] was the first to lower
the threshold $\frac{d}{2}+\frac{1}{2}$ in dimensions $d=2,d=3$ and to use the
theory of Fourier restriction in the Falconer problem. The thresholds were
further improved by Wolff [29] to $\frac{4}{3}$ in the case $d=2$, and by
Erdoğan [8] to $\frac{d}{2}+\frac{1}{3}$ when $d\geq 3$. These records were
only very recently rewritten:

$\begin{cases}\frac{5}{4},&d=2,\quad\qquad\text{(Guth--Iosevich--Ou--Wang
\cite[cite]{[\@@bibref{}{guth2020falconer}{}{}]})}\\\
\frac{9}{5},&d=3,\quad\qquad\text{(Du--Guth--Ou--Wang--Wilson--Zhang
\cite[cite]{[\@@bibref{}{DGOWWZ}{}{}]})}\\\
\frac{d}{2}+\frac{1}{4}+\frac{1}{8d-4},&d\geq 3,\quad\qquad\text{(Du--Zhang
\cite[cite]{[\@@bibref{}{du2019sharp}{}{}]})}\\\
\frac{d}{2}+\frac{1}{4},&d\geq 4\text{ even},\quad\text{(Du--Iosevich--Ou--
Wang--Zhang \cite[cite]{[\@@bibref{}{du2021improved}{}{}]})}.\end{cases}$

Our main result in this paper further improves the thresholds in all
dimensions $d\geq 3$. Now, the gap between the best-known threshold and the
conjectured one is $\frac{1}{4}-\frac{1}{8d+4}$ when $d\geq 3$. This is the
first time that the gap in higher dimensions is smaller than that in dimension
$d=2$. Similar to [10, 5], we in fact prove a slightly stronger version of the
main theorem regarding the pinned distance set.

######  Theorem 1.2.

Let $d\geq 3$ and $E\subset\mathbb{R}^{d}$ be a compact set. Suppose that
$\dim_{H}(E)>\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$. Then there is a point
$x\in E$ such that the pinned distance set $\Delta_{x}(E)$ has positive
Lebesgue measure, where

$\Delta_{x}(E):=\\{|x-y|:\,y\in E\\}.$

We also get a Hausdorff dimension version if $\dim_{H}(E)$ is lower than the
threshold. Let $f(\alpha)=\alpha\cdot\frac{2d+1}{d+1}-(d-1)$. Then the
following theorem gives a non-trivial result for
$\dim_{H}(E)\in(\frac{d^{2}-1}{2d+1},\frac{d^{2}+d}{2d+1}]$.

######  Theorem 1.3.

Let $d\geq 2$, $0<\alpha\leq d-1$, and $E\subset\mathbb{R}^{d}$ be a compact
set. Suppose that $\dim_{H}(E)=\alpha$. Then for any $\varepsilon>0$,

$\dim_{H}(\\{x\in
E:\dim_{H}(\Delta_{x}(E))\leq\min(f(\alpha),1)-\varepsilon\\})<\alpha.$

In particular, $\sup_{x\in E}\dim_{H}(\Delta_{x}(E))\geq\min(f(\alpha),1)$,
and if $\mathcal{H}^{\alpha}(E)>0$, then
$\dim_{H}(\Delta_{x}(E))\geq\min(f(\alpha),1)$ for
$\mathcal{H}^{\alpha}$-almost all $x\in E$.

In dimension $d=2$, a lot of recent progress has been made. Liu [15] showed
that if $\dim_{H}(E)=\alpha>1$, then $\sup_{x\in
E}\dim_{H}(\Delta_{x}(E))\geq\min(\frac{4\alpha}{3}-\frac{2}{3},1)$. For small
values of $\alpha>1$, Shmerkin [24] improved this bound, and later, Stull [27]
made a further improvement, he showed that
$\dim_{H}(\Delta_{x}(E))\geq\frac{\alpha}{4}+\frac{1}{2}$ for many $x$.
Theorem 1.3 deals with the case $\dim_{H}(E)\leq 1$.

When $d\geq 3,\dim_{H}(E)=\alpha>\frac{d}{2}$, the previous best-known results
are as follows: Harris [12] proved that $\sup_{x\in
E}\dim_{H}(\Delta_{x}(E))\geq\min(\frac{2d-1}{d}\alpha-(d-1),1)$; for even
dimensions $d\geq 4$, Wang–Zheng [28] improved this result to $\sup_{x\in
E}\dim_{H}(\Delta_{x}(E))\geq\min(\frac{2d}{d+1}\alpha-\frac{2d^{2}-d-2}{2(d+1)},1)$.
Theorem 1.3 further improves upon these two results.

However, getting a lower bound for $\dim_{H}(\Delta_{x}(E))$ given
$\dim_{H}(E)=\frac{d}{2}$ is much more challenging. Falconer in [9] proved
that $\dim_{H}(\Delta(E))\geq\frac{1}{2}$, and a pinned version $\sup_{x\in
E}\dim_{H}(\Delta_{x}(E))\geq\frac{1}{2}$ was proved in [17]. The reason
$\frac{1}{2}$ is a natural barrier is that if there existed a
$\frac{1}{2}$-dimensional ring $R\subset[1,2]$, then the distance set of
$R^{d}$ is contained in $\sqrt{R}$, so it has Hausdorff dimension
$\frac{1}{2}$. Even though the works of Bourgain [2] and Katz–Tao [13] showed
a quantitative discretized sum-product theorem that rules out the existence of
$R$, it is still a challenging problem to obtain explicit bounds for the
discretized sum-product problem, which asks for the largest exponent
$\gamma>0$ such that $\min(|A+A|_{\delta},|A\cdot
A|_{\delta})\gtrsim|A|^{1+\gamma}$ for any Katz–Tao $(\delta,\frac{1}{2})$-set
$A\subset\mathbb{R}^{1}$ (see [13] for the definition of such sets). See [11],
[20], [16], and [23] for the best known bounds for the discretized sum-product
problem.

For the Falconer distance set problem, the only explicit improvements known
over $\frac{1}{2}$ for $\dim_{H}(\Delta(E))$ when $\dim_{H}(E)=\frac{d}{2}$
were derived in [25] and [26] for $d=2,d=3$, and the latter paper also proved
box dimension results for $d\geq 4$. In personal communication, Shmerkin–Wang
extended Stull’s bound in $d=2$ to the case $\dim_{H}(E)=1$. Furthermore, by
plugging in the sharp radial projection estimates of [19] into the proofs of
Theorems 1.2 and 1.3 of [26], we get improved bounds over those stated in
these theorems for $d\geq 3$. In summary, the previously known results are:

  * •

if $d=2$, $\dim_{H}(E)=1$, then $\sup_{x\in
E}\dim_{H}(\Delta_{x}(E))\geq\frac{3}{4}$;

  * •

if $d=3$, $\dim_{H}(E)=\frac{3}{2}$, then $\sup_{x\in
E}\dim_{H}(\Delta_{x}(E))\geq\frac{5}{8}$;

  * •

if $d\geq 4$, $\dim_{H}(E)=\frac{d}{2}$, then $\sup_{x\in
E}\dim_{B}(\Delta_{x}(E))\geq\frac{d+2}{2(d+1)}$.

A key obstruction to Hausdorff dimension estimates in [26] when $d\geq 4$ is
that $\frac{d}{2}$-dimensional measures don’t necessarily have decay around
small neighborhoods of $(d-1)$-planes when $d\geq 4$, so one cannot apply
Bourgain’s discretized projection theorem to such measures.

Note that in Theorem 1.3, $f(\frac{d}{2})=\frac{d+2}{2(d+1)}>\frac{1}{2}$.
This matches the best-known bound $\frac{5}{8}$ when $d=3$ using an entirely
different approach. And when $d\geq 4$, this is the first time one obtains an
explicit improved bound over $\frac{1}{2}$ for the Hausdorff dimension of
pinned distance sets of a $\frac{d}{2}$-dimensional set, (and this matches the
box dimension bound as in the above).

Finally, let us remark what is known when $\alpha<\frac{d}{2}$. Falconer [9]
proved that $\dim_{H}(\Delta(E))\geq\min(\alpha-\frac{d-1}{2},0)$; this was
improved in dimensions $d=2$ and $3$ to $\sup_{x\in
E}\dim_{H}(\Delta_{x}(E))\geq\frac{\alpha+1}{d+1}$ for
$\alpha\in(\frac{d-1}{2},\frac{d}{2})$ by [19], [26]. In dimension $d\geq 4$
however, their approach only recovers box dimension estimates. Our bound
$\sup_{x\in E}\dim_{H}(\Delta_{x}(E))\geq f(\alpha)$ from Theorem 1.3 is
weaker than $\frac{\alpha+1}{d+1}$ for all $\alpha<\frac{d}{2}$, but it works
for Hausdorff dimension for all $d\geq 3$.

### Old and new ideas

We will adapt the good tube/bad tube and decoupling method pioneered by [10]
for dimension $d=2$ and continued in [5] for even dimensions $d$. In both
papers, Orponen’s radial projection theorem [18] plays a key role. However,
the argument does not perform well for odd dimensions $d$, and the result of
[7] provides a better bound for distance sets. The reason is that Orponen’s
radial projection theorem only works for sets with dimension $>d-1$, where $d$
is the dimension of the ambient space. To overcome this issue, [5] projected
the set onto a generic $(\frac{d}{2}+1)$-dimensional subspace of
$\mathbb{R}^{d}$ (assuming $d$ is even). While this orthogonal projection
trick works well in even dimensions, for odd dimensions we are forced to
project to a $(\frac{d+1}{2})$-dimensional subspace instead, which creates
some loss. To avoid this loss, a natural approach is to avoid the initial
orthogonal projection; but then, we need a radial projection theorem that
works for sets of dimension $\leq d-1$.

The starting point for this paper is a new radial projection result, Theorem
3.1, by the third author [22]. For concreteness, let us work in
$\mathbb{R}^{3}$ (so $d=3$) at a single scale $r$. Then Theorem 3.1 tells us
given $\alpha$-dimensional Frostman measures $\mu_{1},\mu_{2}$, that most
$r$-tubes either have $\mu_{2}$-mass $\lesssim r^{\alpha-\varepsilon}$ or lie
in some heavy $(r^{\kappa},2)$-plate, which is the $r^{\kappa}$-neighborhood
of a 2-dimensional hyperplane with $(\mu_{1}+\mu_{2})$-mass $\gtrsim r^{\eta}$
(here, $\eta\ll\kappa\ll\varepsilon$). We point out that the bound
$r^{\alpha-\varepsilon}$ is a significant improvement over the bound
$r^{(d-1)/2-\varepsilon}$ one would get through combining Orponen’s radial
projection theorem and orthogonal projections in odd dimensions; the cost is
that we need to deal with heavy $(r^{\kappa},2)$-plates. The technical novelty
of this paper is how to deal with these plates in the context of the
decoupling framework.

Following the setup in [5], we let $\mu_{1},\mu_{2}$ be $\alpha$-dimensional
Frostman measures supported on subsets $E_{1},E_{2}\subset E$ with
$\text{dist}(E_{1},E_{2})\gtrsim 1$. Let $R_{0}$ be a large number and define
the scales $R_{j}=2^{j}R_{0}$. Fix a scale $j$; we shall work with
$R_{j}^{-1/2+\beta}$-tubes. Let $G(x)$ to be the set of $y\in E_{1}$ such that
$x,y$ don’t both lie in some heavy $(R_{j}^{-\kappa},2)$-plate. Define a good
$R_{j}^{-1/2+\beta}$-tube to be one with mass $\lesssim
R_{j}^{-\alpha/2+\varepsilon}$ and doesn’t lie in any heavy plate, and
$\mu_{1,g}$ to be a part of $\mu_{1}$ from all good tubes. With these
definitions, we can try to follow the framework of [5] and show that
$\mu_{1}|_{G(x)}$ and $\mu_{1,g}$ are close in $L^{1}$-norm. There are two
subtleties with this approach. First, the error in
$\|\mu_{1}|_{G(x)}-\mu_{1,g}\|_{L^{1}}$ may contain contributions from tubes
$T$ at the border of $G(x)$, i.e. $T\subset G(x)$ but $2T\not\subset G(x)$. To
overcome this issue, we introduce a probabilistic wiggle. If we work with
heavy $(aR_{j}^{-\kappa},2)$-plates for a uniformly random $a\in[1,2]$, then
the borderline error will be small on average, so there exists a choice for
$a$ to make the borderline error small.

The second issue is that we do not have the luxury of working at a single
scale, so we need to introduce some ideas from [25, Appendix B]. Specifically,
we will construct a decreasing sequence $\mathbb{R}^{3}\supset G_{0}(x)\supset
G_{1}(x)\supset\cdots$ such that $\mathbb{R}^{3}\setminus G_{0}(x)$ is
contained in a $(R_{0}^{-\kappa/2},2)$-plate, $\mu(G_{0}(x)\setminus
G_{\infty}(x))\leq R_{0}^{-\kappa/2}$, and $G_{j}(x)$ is disjoint from all
heavy $(aR_{j}^{-\kappa},2)$-plates containing $x$ (where $a\in[1,2]$ is the
probabilistic wiggle). Since all the $G_{j}$’s are close in measure, we can
combine multiscale information to show that $\mu_{1}|_{G_{0}(x)}$ and
$\mu_{1,g}$ are close in $L^{1}$-norm.

The final step is to show $G_{0}(x)$ is large. This is equivalent to showing
that any $(R_{0}^{-\kappa/2},2)$-plate has small ($\leq\frac{1}{2}$) measure
for $R_{0}$ large enough. If this is not true, then by compactness we can find
a hyperplane with nonzero measure, and then we reduce to Falconer in 2D. In
this case, we have strong bounds for distance sets (e.g. see Wolff [29])
because $\dim_{H}(E)>1.5$ is large.

######  Remark 1.4.

In dimensions $d\geq 4$, a slightly simpler and more intuitive choice of good
measures exists. We refer the interested reader to [6] for more details, where
such a direction is pursued and same results in Theorems 1.1 and 1.2 (except
for the $d=3$ case) are obtained as a consequence of new weighted refined
decoupling estimates. In dimension $d=3$ though, the strategy still works but
doesn’t seem to yield as good a result as the current paper.

######  Remark 1.5.

What is the limitation of the methods of this paper and the companion paper
[6]? The radial projection theorem of [22] is sharp in the following sense:
let $E_{1},E_{2}$ each be unions of $r^{-\alpha}$ many $r$-balls satisfying an
$\alpha$-dimensional spacing condition such that
$\mathrm{dist}(E_{1},E_{2})\geq\frac{1}{2}$, and let $\mu_{1},\mu_{2}$ be
probability measures supported on $E_{1},E_{2}$ respectively such that each
$r$-ball in $E_{i}$ has $\mu_{i}$-measure $r^{\alpha}$, $i=1,2$. Then, we know
that many $r$-tubes can intersect at least one $r$-ball in $E_{1}$ and one
$r$-ball in $E_{2}$. Thus, many $r$-tubes can have both $\mu_{1}$ and
$\mu_{2}$-mass at least $r^{\alpha}$, so $r^{\alpha}$ is the best possible
threshold for the good tubes in this paper. To make further progress, we
suggest to look at improving the decoupling framework.

### Outline of the paper

In Section 2, we outline two main estimates and prove Theorem 1.2 using them.
In Section 3, we list several results that we will use to bound the bad part,
including the new radial projection estimate by the third author [22] and two
geometric lemmas governing the physical location of small plates with large
mass. In Section 4, we construct the good measure and prove the first main
estimate - Proposition 2.1. In Section 5, we prove the second main estimate -
Proposition 2.2 using refined decoupling. In Section 6, we prove Theorem 1.3
using the two main estimates and a framework of Liu [15]. In Section 7, we
give some remarks about the extension of Theorem 1.2 to more general norms and
its connection with the Erdős distance problem.

### Notations

Throughout the article, we write $A\lesssim B$ if $A\leq CB$ for some absolute
constant $C$; $A\sim B$ if $A\lesssim B$ and $B\lesssim A$;
$A\lesssim_{\varepsilon}B$ if $A\leq C_{\varepsilon}B$; $A\lessapprox B$ if
$A\leq C_{\varepsilon}R^{\varepsilon}B$ for any $\varepsilon>0$, $R>1$.

For a large parameter $R$, ${\rm RapDec}(R)$ denotes those quantities that are
bounded by a huge (absolute) negative power of $R$, i.e. ${\rm RapDec}(R)\leq
C_{N}R^{-N}$ for arbitrarily large $N>0$. Such quantities are negligible in
our argument.

For $x\in\mathbb{R}^{d}$ and $t>0$, $B(x,t)$ is the ball in $\mathbb{R}^{d}$
of radius $t$ centered at $x$. A $\delta$-tube is the intersection of an
infinite cylinder of radius $\delta$ and $B(0,10)$. This is not standard
(usually, a tube is defined to be a finite $\delta$-cylinder with length $1$),
but this definition won’t cause problems and is slightly more convenient for
us.

For a set $E\subset\mathbb{R}^{d}$, let $E^{c}=\mathbb{R}^{d}\setminus E$, and
$E^{(\delta)}$ the $\delta$-neighborhood of $E$. For subsets
$E_{1},E_{2}\subset\mathbb{R}^{d}$, $\mathrm{dist}(E_{1},E_{2})$ is their
Euclidean distance.

For $A\subset X\times Y$ and $x\in X$, define the slice $A|_{x}=\\{y\in
Y:(x,y)\in A\\}$. Similar definition for $A|_{y}$, when $y\in Y$.

For a measure $\mu$ and a measurable set $G$, define the restricted measure
$\mu|_{G}$ by $\mu|_{G}(A)=\mu(G\cap A)$ for all measurable
$A\subset\mathbb{R}^{d}$.

We say a measure $\mu$ supported in $\mathbb{R}^{d}$ is an
$\alpha$-dimensional measure with constant $C_{\mu}$ if it is a probability
measure satisfying that

$\mu(B(x,t))\leq C_{\mu}t^{\alpha},\qquad\forall x\in\mathbb{R}^{d},\,\forall
t>0.$

An $(r,k)$-plate $H$ in $\mathbb{R}^{d}$ is the $r$-neighborhood of a
$k$-dimensional affine plane in the ball $B^{d}(0,10)$. More precisely,

$H:=\\{z\in B(0,10):\mathrm{dist}(z,P_{H})<r\\},$

where $P_{H}$ is a $k$-dimensional affine plane, which is called the central
plane of $H$. We can also write $H=P_{H}^{(r)}\cap B(0,10)$. A $C$-scaling of
$H$ is

$CH=\\{z\in B(0,10):\mathrm{dist}(z,P_{H})<Cr\\}.$

Denote the surface of $H$ by

$\mathrm{Surf}(H):=\\{z\in B(0,10):\mathrm{dist}(z,P_{H})=r\\}\,.$

Since a $\delta$-tube is a $(\delta,1)$-plate, the same conventions apply to
tubes.

To prevent the reader from feeling too attached to $B(0,10)$, we will see in
the proofs that $B(0,10)$ can be replaced by any smooth bounded convex body
that contains $B(0,10)$.

We say that an $(r,k)$-plate $H$ is $\gamma$-concentrated on $\mu$ if
$\mu(H)\geq\gamma$.

Given a collection $\mathcal{H}$ of plates in $\mathbb{R}^{d}$ and a point
$x\in\mathbb{R}^{d}$, define $\mathcal{H}(x):=\\{H\in\mathcal{H}:x\in aH\\}$,
where $a$ is a “master parameter” that will be defined later.

We will work with a collection $\mathcal{E}_{r,k}$ of essentially distinct
$(r,k)$-plates with the following properties:

  * •

Each $(\frac{r}{2},k)$-plate intersecting $B(0,1)$ lies in at least one plate
of $\mathcal{E}_{r,k}$;

  * •

For $s\geq r$, every $(s,k)$-plate contains
$\lesssim_{k,d}\left(\frac{s}{r}\right)^{(k+1)(d-k)}$ many $(r,k)$-plates of
$\mathcal{E}_{r,k}$.

For example, when $k=1$ and $d=2$, we can simply pick $\sim r^{-1}$ many
$r$-tubes in each of an $r$-net of directions. This generalizes to higher $k$
and $d$ via a standard $r$-net argument, which can be found in [22, Section
2.2].

######  Acknowledgements.

XD is supported by NSF DMS-2107729 (transferred from DMS-1856475), NSF
DMS-2237349 and Sloan Research Fellowship. YO is supported by NSF DMS-2142221
and NSF DMS-2055008. KR is supported by a NSF GRFP fellowship. RZ is supported
by NSF DMS-2207281 (transferred from DMS-1856541), NSF DMS-2143989 and the
Sloan Research Fellowship.

##  2\. Main estimates

In this section, we outline two main estimates, from which Theorems 1.2 and
1.3 follow.

Let $E\subset\mathbb{R}^{d}$ be a compact set with positive
$\alpha$-dimensional Hausdorff measure. Without loss of generality, assume
that $E$ is contained in the unit ball, and there are subsets $E_{1}$ and
$E_{2}$ of $E$, each with positive $\alpha$-dimensional Hausdorff measure, and
$\mathrm{dist}(E_{1},E_{2})\gtrsim 1$. Then there exist $\alpha$-dimensional
probability measures $\mu_{1}$ and $\mu_{2}$ supported on $E_{1}$ and $E_{2}$,
respectively.

To relate the measures to the distance set, we consider their pushforward
measures under the distance map. For a fixed point $x\in E_{2}$, let
$d^{x}:E_{1}\to\mathbb{R}$ be the pinned distance map given by
$d^{x}(y):=|x-y|$. Then, the pushforward measure $d^{x}_{\ast}(\mu_{1})$,
defined as

$\int_{\mathbb{R}}\psi(t)\,d^{x}_{\ast}(\mu_{1})(t)=\int_{E_{1}}\psi(|x-y|)\,d\mu_{1}(y),$

is a natural measure that is supported on $\Delta_{x}(E_{1})$.

In the following, we will construct another complex-valued measure
$\mu_{1,g}^{x}$ that is the _good_ part of $\mu_{1}$ with respect to $\mu_{2}$
depending on $x$, and study its pushforward under the map $d^{x}$. The main
estimates are the following.

######  Proposition 2.1.

Let $d\geq 2$, $k\in\\{1,2,\cdots,d-1\\}$, $k-1<\alpha\leq k$, and
$\varepsilon>0$. Then there exists a small $\beta(\varepsilon)>0$ such that
the following holds for sufficiently large $R_{0}(\beta,\varepsilon)$. Assume
$\mu_{1,g}^{x}$ has been constructed following the procedure in Section 4.2
below. Then there is a subset $E_{2}^{\prime}\subset E_{2}$ with
$\mu_{2}(E_{2}^{\prime})\geq 1-R_{0}^{-\beta}$ and for each $x\in
E_{2}^{\prime}$, there exists a set $G(x)\subset B^{d}(0,10)$ where
$B^{d}(0,10)\setminus G(x)$ is contained within some
$(R_{0}^{-\beta},k)$-plate, such that the following estimate holds:

(2.1) $\|d_{*}^{x}(\mu_{1}|_{G(x)})-d_{*}^{x}(\mu_{1,g}^{x})\|_{L^{1}}\leq
R_{0}^{-\beta}.$

######  Proposition 2.2.

Let $d\geq 2$, $0<\alpha<d$, and $\varepsilon>0$. Then for sufficiently small
$\beta(\varepsilon)\in(0,\varepsilon)$ in the construction of $\mu_{1,g}^{x}$
in Section 4.2 below,

$\int_{E_{2}}\|d^{x}_{*}(\mu_{1,g}^{x})\|_{L^{2}}^{2}d\mu_{2}(x)\lesssim_{d,\alpha,\varepsilon}\int_{\mathbb{R}^{d}}|\xi|^{-\frac{\alpha
d}{d+1}+\varepsilon}|\widehat{\mu}_{1}(\xi)|^{2}\,d\xi+R_{0}^{d}.$

######  Remark 2.3.

Broadly speaking, $G(x)$ removes the contributions from small plates that
contain large mass. This makes it possible to efficiently apply the new radial
projection theorem, Theorem 3.1.

###### Proof of Theorem 1.2 using Propositions 2.1 and 2.2.

For $d\geq 3$ and $\frac{d}{2}<\alpha<\frac{d+1}{2}$, let
$k=\lfloor\frac{d}{2}\rfloor+1$ so that $k-1<\alpha\leq k$. If $\mu_{1}$ gives
nonzero mass to some $k$-dimensional affine plane, then we are done by
applying [9] to that $k$-plane since $\alpha>\frac{d}{2}\geq\frac{k+1}{2}$.
Thus, assume $\mu_{1}$ gives zero mass to every $k$-dimensional affine plane.
By a compactness argument, there exists $r_{0}>0$ such that
$\mu_{1}(H)<\frac{1}{1000}$ for any $(r_{0},k)$-plate $H$. Now the two
propositions tell us that there is a point $x\in E_{2}$ such that

$\displaystyle\|d_{*}^{x}(\mu_{1}|_{G(x)})-d_{*}^{x}(\mu_{1,g}^{x})\|_{L^{1}}\leq\frac{1}{1000},$
$\displaystyle\|d_{*}^{x}(\mu_{1,g}^{x})\|^{2}_{L^{2}}\lesssim
I_{\lambda}(\mu_{1})+R_{0}^{d}<\infty,$

by choosing $R_{0}$ sufficiently large. Here $I_{\lambda}(\mu_{1})$ is the
$\lambda$-dimensional energy of $\mu_{1}$ and $\lambda=d-\frac{\alpha
d}{d+1}+\varepsilon$, by a Fourier representation for $I_{\lambda}$:

$I_{\lambda}(\mu)=\int\int|x-y|^{-\lambda}d\mu(x)d\mu(y)=C_{d,\lambda}\int_{\mathbb{R}^{d}}|\xi|^{\lambda-d}|\widehat{\mu}(\xi)|^{2}\,d\xi.$

One has $I_{\lambda}(\mu_{1})<\infty$ if $\lambda<\alpha$, which is equivalent
to $\alpha>\frac{d(d+1)}{2d+1}=\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$. Now
$B^{d}(0,10)\setminus G(x)$ is contained in some $(R_{0}^{-\beta},k)$-plate
$H$. If $R_{0}$ is chosen sufficiently large such that
$R_{0}^{\beta}>r_{0}^{-1}$, then

$\mu_{1}(G(x))\geq 1-\mu_{1}(H)>1-\frac{1}{1000}.$

Since $d_{*}^{x}(\mu_{1}|_{G(x)})$ is a positive measure, its $L^{1}$ norm is
$\mu_{1}(G(x))>1-\frac{1}{1000}$. Thus,

$\int_{\Delta_{x}(E)}|d_{*}^{x}(\mu_{1,g}^{x})|=\int|d_{*}^{x}(\mu_{1,g}^{x})|-\int_{\Delta_{x}(E)^{c}}|d_{*}^{x}(\mu_{1,g}^{x})|\\\
\geq
1-\frac{2}{1000}-\int|d_{*}^{x}(\mu_{1}|_{G(x)})-d_{*}^{x}(\mu_{1,g}^{x})|\geq
1-\frac{3}{1000}.$

On the other hand,

$\int_{\Delta_{x}(E)}|d_{*}^{x}(\mu_{1,g}^{x})|\leq|\Delta_{x}(E)|^{1/2}\left(\int|d_{*}^{x}(\mu_{1,g}^{x})|^{2}\right)^{1/2}.$

Therefore, $|\Delta_{x}(E)|>0$. ∎

The proof of Theorem 1.3 is deferred until Section 6.

##  3\. Radial projections and heavy plates

In this section, we list several results that we will use in Sections 4.4 and
4.6 to bound the bad part of $\mu_{1}$.

We’ll use the following new radial projection estimate, which follows from
[22, Theorem 1.13].

######  Theorem 3.1.

Let $d\geq 2$, $k\in\\{1,2,\cdots,d-1\\}$, $k-1<\alpha\leq k$, and fix
$\eta,\varepsilon>0$, and two $\alpha$-dimensional measures $\mu_{1},\mu_{2}$
with constants $C_{\mu_{1}},C_{\mu_{2}}$ supported on $E_{1},E_{2}\subset
B(0,1)$ respectively. There exists $\gamma>0$ depending on
$\eta,\varepsilon,\alpha,k$ such that the following holds. Fix $\delta<r<1$.
Let $A$ be the set of pairs $(x,y)\in E_{1}\times E_{2}$ satisfying that $x$
and $y$ lie in some $\delta^{\eta}$-concentrated $(r,k)$-plate on
$\mu_{1}+\mu_{2}$. Then there exists a set $B\subset E_{1}\times E_{2}$ with
$\mu_{1}\times\mu_{2}(B)\leq\delta^{\gamma}$ such that for every $x\in E_{1}$
and $\delta$-tube $T$ through $x$, we have

$\mu_{2}(T\setminus(A|_{x}\cup
B|_{x}))\lesssim\frac{\delta^{\alpha}}{r^{\alpha-(k-1)}}\delta^{-\varepsilon}.$

The implicit constant may depend on
$\eta,\varepsilon,\alpha,k,C_{\mu_{1}},C_{\mu_{2}}$.

######  Remark 3.2.

(a) The roles of $\mu_{1}$ and $\mu_{2}$ in Theorem 3.1 are interchangeable,
so the conclusion also holds for $\mu_{1}$ instead of $\mu_{2}$.

(b) If $\alpha>d-1$, then the numerology of Theorem 3.1 doesn’t apply.
Instead, Orponen’s radial projection theorem [18] in dimension $d$ applies.
The result (stated in [10, Lemma 3.6] for $d=2$, but can be generalized to all
dimensions $d$) is that for $\gamma=\varepsilon/C$, there exists a set
$B\subset E_{1}\times E_{2}$ with $\mu_{1}\times\mu_{2}(B)\leq\delta^{\gamma}$
such that for every $x\in E_{1}$ and $\delta$-tube $T$ through $x$, we have

$\mu_{2}(T\setminus B|_{x})\lesssim\delta^{d-1-\varepsilon}.$

Note that the set $A$ of “concentrated pairs” is not needed here.

(c) If $r\sim\delta$, we can obtain a slightly better result by projecting to
a $k$-dimensional subspace and following the argument in [5, Section 3.2]. The
result is that for $\gamma=\varepsilon/C$, there exists a set $B\subset
E_{1}\times E_{2}$ with $\mu_{1}\times\mu_{2}(B)\leq\delta^{\gamma}$ such that
for every $x\in E_{1}$ and $\delta$-tube $T$ through $x$, we have

$\mu_{2}(T\setminus B|_{x})\lesssim\delta^{k-1-\varepsilon}.$

The set $A$ is again not needed in this case. The main novelty of Theorem 3.1
comes when $r>\delta$.

We will also need the following two lemmas from [22] (Lemmas 7.5 and 7.8)
governing the physical location of small plates with large mass.

######  Lemma 3.3.

Let $k-1<s\leq k$ and $0<r\leq 1$. There is $N=N(s,k,d)$ such that the
following holds: let $\nu$ be an $s$-dimensional measure with constant
$C_{\nu}\geq 1$, and let $\mathcal{E}_{r,k}$ be the collection of essentially
distinct $(r,k)$-plates from the Notations part of Section 1. Let
$\mathcal{H}=\\{H\in\mathcal{E}_{r,k}:\nu(H)\geq a\\}$. Then
$|\mathcal{H}|\lesssim(\frac{C_{\nu}}{a})^{N}$. (The implicit constant only
depends on $k,d$ and is independent of $a,r$.)

######  Lemma 3.4.

Let $0<r<r_{0}\lesssim 1$ and $s>k-1$. Let $\mathcal{H}$ be a collection of
$(r,k)$-plates, and let $\mu$ be a compactly supported $s$-dimensional measure
with constant $C_{\mu}$. Then for all $x\in\mathrm{spt}\,(\mu)$ except a set
of $\mu$-measure $\lesssim
C_{\mu}\left(\frac{r}{r_{0}}\right)^{s-(k-1)}|\mathcal{H}|^{2}$, there exists
an $(r_{0},k)$-plate that contains every $(r,k)$-plate in $\mathcal{H}$ that
passes through $x$.

##  4\. Construction of good measure and Proposition 2.1

In this section, we will construct the good measure $\mu_{1,g}^{x}$ and prove
Proposition 2.1. We will henceforth treat $\alpha,k,d,\varepsilon$ in the
hypothesis of Proposition 2.1 as fixed constants. To assist in the proof, we
will eventually be choosing the following parameters:
$\varepsilon_{0}(\varepsilon)$, $\kappa(\varepsilon_{0})$,
$\eta(\kappa,\varepsilon_{0})$, $\beta(\kappa,\eta,\varepsilon_{0})$. In terms
of size, they satisfy

$0<\beta\ll\eta\ll\kappa\ll\varepsilon_{0}\ll\varepsilon.$

Here, $A\ll B$ means “$A$ is much smaller than $B$.” Unwrapping the
dependences, we see that $\beta$ ultimately only depends on $\varepsilon$,
which is what we want.

###  4.1. Smooth Partitions of Unity

We follow the first part of [5, Section 3.1]. Let $R_{0}$ be a large power of
$2$ that will be determined later, and let $R_{j}=2^{j}R_{0}$. Construct a
partition of unity

$1=\sum_{j\geq 0}\psi_{j},$

where $\psi_{0}$ is supported in the ball $|\omega|\leq 2R_{0}$ and each
$\psi_{j}$ for $j\geq 1$ is supported on the annulus $R_{j-1}\leq|\omega|\leq
R_{j+1}$. Importantly, we may choose $\psi_{j}$ such that
$\|\check{\psi}_{j}\|_{L^{1}}\leq C$ for some absolute constant $C$ and all
$j\geq 1$. For example, choose $\psi_{j}$ to be Littlewood-Paley functions
$\chi(x/R_{j})-\chi(x/R_{j-1})$, where $\chi$ is a smooth bump function that
is $1$ on $B(0,1)$ and $0$ outside $B(0,2)$.

In $\mathbb{R}^{d}$, cover the annulus $R_{j-1}\leq|\omega|\leq R_{j+1}$ by
rectangular blocks $\tau$ of dimensions approximately
$R_{j}^{1/2}\times\cdots\times R_{j}^{1/2}\times R_{j}$, with the long
direction of each block $\tau$ being the radial direction. Choose a smooth
“partition of unity” with respect to this cover such that

$\psi_{j}=\sum_{\tau}\psi_{j,\tau}(\omega).$

The functions $\psi_{j,\tau}$ satisfy the following properties:

  * •

$\psi_{j,\tau}$ is supported on $\tau$ and $\|\psi_{j,\tau}\|_{L^{\infty}}\leq
1$;

  * •

$\check{\psi}_{j,\tau}$ is essentially supported on a
$R_{j}^{-1/2}\times\cdots\times R_{j}^{-1/2}\times R_{j}^{-1}$ box $K$
centered at $0$, in the sense that
$|\check{\psi}_{j,\tau}(x)|\leq\mathrm{RapDec}(R_{j})$ if
$\mathrm{dist}(x,K)\gtrsim R_{j}^{-1/2+\beta}$ (and the implicit constant in
the decay $C_{N}R_{j}^{-N}$ is universal only depending on $N$);

  * •

$\|\check{\psi}_{j,\tau}\|_{L^{1}}\lesssim 1$ (the implicit constant is
universal).

For each $(j,\tau)$, cover the unit ball in $\mathbb{R}^{d}$ with tubes $T$ of
dimensions approximately $R_{j}^{-1/2+\beta}\times\cdots\times
R_{j}^{-1/2+\beta}\times 20$ with the long axis parallel to the long axis of
$\tau$. The covering has uniformly bounded overlap, each $T$ intersects at
most $C(d)$ other tubes. We denote the collection of all these tubes as
$\mathbb{T}_{j,\tau}$. Let $\eta_{T}$ be a smooth partition of unity
subordinate to this covering, so that for each choice of $j$ and $\tau$,
$\sum_{T\in\mathbb{T}_{j,\tau}}\eta_{T}$ is equal to 1 on the ball of radius
10 and each $\eta_{T}$ is smooth.

For each $T\in\mathbb{T}_{j,\tau}$, define an operator

$M_{T}f:=\eta_{T}(\psi_{j,\tau}\hat{f})^{\vee},$

which, morally speaking, maps $f$ to the part of it that has Fourier support
in $\tau$ and physical support in $T$. Define also
$M_{0}f:=(\psi_{0}\hat{f})^{\vee}$. We denote
$\mathbb{T}_{j}=\cup_{\tau}\mathbb{T}_{j,\tau}$ and $\mathbb{T}=\cup_{j\geq
1}\mathbb{T}_{j}$. Hence, for any $L^{1}$ function $f$ supported on the unit
ball, one has the decomposition

$f=M_{0}f+\sum_{T\in\mathbb{T}}M_{T}f+\text{RapDec}(R_{0})\|f\|_{L^{1}}.$

See [10, Lemma 3.4] for a justification of the above decomposition. (Even
though [10, Lemma 3.4] is stated in two dimensions, the argument obviously
extends to higher dimensions.)

###  4.2. Heavy Plates and Good Tubes

In this subsection, we define good tubes. Actually, we will use three
categories: good, acceptable, and non-acceptable. The idea is that Theorem 3.1
tells us that $R_{j}^{-1/2+\beta}$-tubes fall into one of three categories:

  * •

For $R_{j}^{-\eta}$-concentrated $(R_{j}^{-\kappa},k)$-plates, tubes in them
can have large $\mu_{2}$-mass. Call a tube inside one of these plates non-
acceptable.

  * •

Many of the acceptable tubes $T$ are good, i.e. $\mu_{2}(4T)\lesssim
R_{j}^{-\alpha/2+\varepsilon_{0}}$.

  * •

By Theorem 3.1, there are not many tubes that are neither non-acceptable nor
good.

The idea is that to form our good measure $\mu_{1,g}^{x}$, we keep
contributions only from good tubes. By the third bullet, we are allowed to
remove tubes that are neither non-acceptable nor good. To remove the non-
acceptable tubes, we will instead remove the heavy plates. Next, we formalize
this idea.

Let $k$ be the integer such that $k-1<\alpha\leq k$. Let $\mathcal{E}_{r,k}$
be the cover of $B(0,1)$ with $(r,k)$-plates constructed in Section
LABEL:subsec:r-net; every $(r/2,k)$-plate is contained within some element of
$\mathcal{E}_{r,k}$. Let $\mathcal{H}_{j}$ be the set of
$(R_{j}^{-\kappa},k)$-plates in $\mathcal{E}_{R_{j}^{-\kappa},k}$ that are
$R_{j}^{-\eta}$-concentrated on $\mu_{1}+\mu_{2}$; then Lemma 3.3 tells us
$|\mathcal{H}_{j}|\lesssim R_{j}^{N\eta}$. Let $\mathcal{H}_{\leq
j}=\cup_{i=1}^{j}\mathcal{H}_{i}$ and
$\mathcal{H}=\cup_{i=1}^{\infty}\mathcal{H}_{i}$. Note that
$|\mathcal{H}_{\leq j}|\lesssim R_{j}^{N\eta}$.

Let $C_{\text{sep}}\geq 1$ be a constant such that
$\mathrm{dist}(E_{1},E_{2})\geq C_{\text{sep}}^{-1}\gtrsim 1$. We will
eventually choose a “master parameter”
$a\in[99C_{\text{sep}},100C_{\text{sep}}]$. For $H\in\mathcal{H}$, we will use
$aH$ as proxies for $H$ when defining acceptable tubes and the good measure.

We briefly attempt to motivate the construction. The role of $C_{\text{sep}}$
is to make sure that tubes intersecting $H\cap E_{1}$ and $H\cap E_{2}$
actually lie inside $aH$. The role of $a$ is to introduce a probabilistic
wiggle to make a key technical condition hold (see the control of
$\text{Bad}_{j}^{2}$ in Lemma 4.8).

Now, we fix a choice of $a$ and define the following. We say an
$R_{j}^{-1/2+\beta}$-tube $T\in\mathbb{T}_{j}$ is _non-acceptable_ if there
exists some $H\in\mathcal{H}_{\leq\max(j,j_{*})}$ such that $2T$ is contained
in $aH$, where $j_{*}=\log_{2}R_{0}$ (the motivation for introducing $j_{*}$
will be explained in the proof of Lemma 4.2). Otherwise, we say it is
_acceptable_. Define a _good_ $R_{j}^{-1/2+\beta}$-tube $T$ to be an
acceptable tube with $\mu_{2}(4T)\leq R_{j}^{-\alpha/2+\varepsilon_{0}}$. And
define the good part of $\mu_{1}$ with respect to $\mu_{2}$ and $x\in E_{2}$
by

$\mu_{1,g}^{x}:=M_{0}(\mu_{1}|_{G_{0}(x)})+\sum_{T\in\mathbb{T},\,T\text{
good}}M_{T}\mu_{1}.$

The only dependence on $x$ comes in the $M_{0}(\mu_{1}|_{G_{0}(x)})$, which is
crucial when we try to prove Proposition 2.2 later. We will define $G_{0}(x)$
in the next subsection, roughly speaking, $G_{0}(x)$ is obtained by removing
heavy plates through $x$ at several scales.

As constructed, we may not get a good bound of the form
$\|d^{x}_{*}(\mu_{1,g}^{x})-d^{x}_{*}(\mu_{1})\|\leq R_{0}^{-\varepsilon}$.
This is because $\mu_{1,g}^{x}$ doesn’t include contributions from non-
acceptable tubes while $\mu_{1}$ does. Instead, we need to work with a measure
$\mu_{1}^{x}$ depending on $x$ that removes the contributions of the non-
acceptable tubes through $x$. In fact, since non-acceptable tubes are
contained in heavy plates, we should define $\mu_{1}^{x}$ by removing these
heavy plates “at different scales $R_{j}$” and make sure that summing over
different scales still leads to good behavior (see Lemma 4.3). We make things
rigorous in the next subsection.

###  4.3. Construction of $G(x)$ and $\mu_{1}^{x}$

Recall that $\mathrm{dist}(E_{1},E_{2})\geq C_{\text{sep}}^{-1}\gtrsim 1$.
Thus, for $x\in E_{2}$, we have $E_{1}\subset B(x,C_{\text{sep}}^{-1})^{c}$, a
fact that underlies the rest of the paper.

Recall that $a\in[99C_{\text{sep}},100C_{\text{sep}}]$ is the “master
parameter” to be chosen later. For $x\in E_{2}$ and $H\in\mathcal{H}_{j}(x)$,
let $F(x,aH)$ be given by

$F(x,aH):=\\{y\in B(x,C_{\text{sep}}^{-1})^{c}\cap
B(0,10):l(x,y)\cap\mathrm{Surf}(aH)=\emptyset\\}\,,$

where $l(x,y)$ is the line through $x$ and $y$. It is true that
$F(x,aH)\subset aH$; this will be proved in Lemma 4.1. Define
$j_{*}:=\log_{2}R_{0}$ such that $R_{j_{*}}=R_{0}^{2}$. For $j\geq 0$, let

(4.1) $G_{j}(x)=\left[B(x,C_{\text{sep}}^{-1})^{c}\cap
B(0,10)\right]\setminus\cup_{H\in\mathcal{H}_{\leq\max(j,j_{*})}(x)}F(x,aH)\,.$

Finally, we define $G(x)=G_{0}(x)\cup B(x,C_{\text{sep}}^{-1})$ and

(4.2) $\mu_{1}^{x}:=\sum_{j\geq 0}\mu_{1}|_{G_{j}(x)}*\check{\psi}_{j}.$

It will be proved that $G(x)$ satisfies the condition of Proposition 2.1 in
the next subsection, and this is the only reason why we include
$B(x,C_{\text{sep}}^{-1})$ in $G(x)$.

We now list some good properties of these definitions that will be critical
later. First, the construction of $G_{j}(x)$ ensures that $G_{j}(x)\supset
G_{j+1}(x)$ and

$G_{j}(x)\setminus G_{j+1}(x)\subset\cup_{H\in\mathcal{H}_{j+1}(x)}F(x,aH)\,,$

with $G_{j}(x)=G_{j+1}(x)$ for $j<j_{*}$. Next, $G_{j}(x)$ keeps the
contributions from the acceptable tubes through $x$ while discarding the non-
acceptable tubes through $x$.

######  Lemma 4.1.

Let $T\in\mathbb{T}_{j}$ and $x\in 2T\cap E_{2}$.

(a) If $2T\cap B(x,C_{\text{sep}}^{-1})^{c}\subset G_{j}(x)$, then $T$ is
acceptable;

(b) If $2T\cap B(x,C_{\text{sep}}^{-1})^{c}\subset F(x,aH)$ for some
$H\in\mathcal{H}_{\leq\max(j,j_{*})}(x)$, then $T$ is non-acceptable.

###### Proof.

Let $T\in\mathbb{T}_{j}$ and $x\in 2T$.

(a) If $2T\cap B(x,C_{\text{sep}}^{-1})^{c}\subset G_{j}(x)$, we show $T$ is
acceptable, i.e. $2T\not\subset aH$ for any
$H\in\mathcal{H}_{\leq\max(j,j_{*})}$. Fix
$H\in\mathcal{H}_{\leq\max(j,j_{*})}$. If $x\notin aH$, then $2T\not\subset
aH$. If $x\in aH$, then $H\in\mathcal{H}_{\leq\max(j,j_{*})}(x)$. Choose $y\in
2T\cap B(x,C_{\text{sep}}^{-1})^{c}$ such that the line $l(x,y)$ is parallel
to the central line of $T$. By assumption, we know that $y\in G_{j}(x)$, so
$l(x,y)$ intersects $\mathrm{Surf}(aH)$. In particular, $2T$ intersects
$\mathrm{Surf}(aH)$, and thus $2T\not\subset aH$.

(b) If $2T\cap B(x,C_{\text{sep}}^{-1})^{c}\subset F(x,aH)$ for some
$H\in\mathcal{H}_{\ell}(x)$ with $\ell\leq\max(j,j_{*})$, we show $T$ is non-
acceptable. More precisely, we can show that $2T$ is contained in $aH$.

As promised before, we will prove $F(x,aH)\subset aH$. Take $y\in F(x,aH)$,
and suppose $y\notin aH$. Then $d(y,P_{H})\geq aR_{\ell}^{-\kappa}$ and
$d(x,P_{H})<aR_{\ell}^{-\kappa}$. By continuity, we can find $z$ on the line
segment between $x$ and $y$ such that $d(z,P_{H})=aR_{\ell}^{-\kappa}$. By
convexity of $B(0,10)$, we have $z\in B(0,10)$. Thus, we have
$z\in\mathrm{Surf}(aH)$, which contradicts $y\in F(x,aH)$, and so in fact
$y\in aH$. This proves $F(x,aH)\subset aH$.

Now observe the following geometric fact: $2T$ is a tube through $x\in B(0,1)$
whose ends lie on $B(0,10)$, and so $2T\cap B(x,C_{\text{sep}}^{-1})^{c}$
contains both ends of $2T$. Also, by our initial assumption, we get

$2T\cap B(x,C_{\text{sep}}^{-1})^{c}\subset F(x,aH)\subset aH.$

Therefore, since $aH$ is convex, we get $2T\subset aH$. ∎

To estimate $\|d_{*}^{x}(\mu_{1}|_{G(x)})-d_{*}^{x}(\mu_{1,g}^{x})\|_{L^{1}}$,
it suffices to estimate $\|\mu_{1}|_{G(x)}-\mu_{1}^{x}\|_{L^{1}}$ and
$\|d_{*}^{x}(\mu_{1}^{x})-d_{*}^{x}(\mu_{1,g}^{x})\|_{L^{1}}$. This will be
the content of the next two subsections.

###  4.4. Pruning $E_{2}$ and estimating
$\|\mu_{1}|_{G(x)}-\mu_{1}^{x}\|_{L^{1}}$

If all the $G_{j}(x)$’s were the same, then
$\mu_{1}|_{G(x)}=\mu_{1}|_{G_{0}(x)}=\mu_{1}^{x}$. This isn’t quite true, but
in general, we will still get a good bound on
$\|\mu_{1}|_{G(x)}-\mu_{1}^{x}\|_{L^{1}}$ if $\mu_{1}(G_{j}(x)\setminus
G_{j+1}(x))$ is small for all $j$. This weaker assumption is in fact true for
“most” $x\in E_{2}$. To see this, we use a variant of Proposition 7.3 in [22]
(see also Proposition B.1 of [25]).

######  Lemma 4.2.

For every $\kappa>0$, there exists $\eta_{0}(\alpha,k,\kappa)>0$ such that for
all $\eta\in(0,\eta_{0}]$ and $R_{0}$ sufficiently large in terms of
$\alpha,k,\kappa,C_{\text{sep}},\eta$, the following holds. Then there exists
$E_{2}^{\prime\prime}\subset E_{2}$ with $\mu_{2}(E_{2}\setminus
E_{2}^{\prime\prime})\leq R_{0}^{-\eta}$ such that the following assertions
hold for $x\in E_{2}^{\prime\prime}$.

  1. (a)

The $100C_{\text{sep}}$-scalings of elements in $\mathcal{H}_{j}(x)$ are
contained within some $(R_{j}^{-\kappa/2}/3,k)$-plate through $x$;

  2. (b)

The $100C_{\text{sep}}$-scalings of elements in $\mathcal{H}_{\leq j_{*}}(x)$
are contained within some $(R_{0}^{-\kappa/2},k)$-plate through $x$;

  3. (c)

$\mu_{1}(G_{j}(x)\setminus G_{j+1}(x))\lesssim R_{j}^{-\eta/2}$, for any
$j\geq 0$.

###### Proof.

First pick $j\geq 0$. By Lemma 3.4 and $300C_{\text{sep}}<R_{0}^{\kappa/2}$
for sufficiently large $R_{0}$, we can find sets $F_{j}$ with
$\mu_{2}(F_{j})\lesssim R_{j}^{-\kappa(\alpha-k+1)/2}|\mathcal{H}_{j}|^{2}$
such that for $x\in E_{2}\setminus F_{j}$, the $100C_{\text{sep}}$-scalings of
elements in $\mathcal{H}_{j}(x)$ are contained within some
$(R_{j}^{-\kappa/2}/3,k)$-plate through $x$. Thus for $x\in E_{2}\setminus
F_{j}$, assertion (a) is true.

By the same Lemma, we can find a set $F_{0}$ with

$\mu_{2}(F_{0})\lesssim R_{0}^{-\kappa(\alpha-k+1)/2}|\mathcal{H}_{\leq
j_{*}}|^{2}$

such that for $x\in E_{2}\setminus F_{0}$, the $100C_{\text{sep}}$-scalings of
elements in $\mathcal{H}_{\leq j_{*}}(x)$ are contained within some
$(R_{0}^{-\kappa/2},k)$-plate through $x$. (Some of the plates in
$\mathcal{H}_{\leq j_{*}}$ are too small, but we simply thicken them to
$(R_{0}^{-\kappa},k)$-plates.) Thus for $x\in E_{2}\setminus F_{0}$, assertion
(b) is true.

Let
$E_{2}^{\prime\prime}=E_{2}\setminus(F_{0}\cup\bigcup_{j=j_{*}}^{\infty}F_{j})$;
then (using $R_{j_{*}}=R_{0}^{2}$ and Lemma 3.3):

$\mu_{2}(E_{2}\setminus E_{2}^{\prime\prime})\lesssim
R_{0}^{-\kappa(\alpha-k+1)/2}|\mathcal{H}_{\leq j_{*}}|^{2}+\sum_{j\geq
j_{*}}R_{j}^{-\kappa(\alpha-k+1)/2}|\mathcal{H}_{j}|^{2}\\\ \lesssim
R_{0}^{-\kappa(\alpha-k+1)/2+4N\eta}+\sum_{j\geq
j_{*}}R_{j}^{-\kappa(\alpha-k+1)/2+2N\eta}\lesssim\sum_{j\geq
0}R_{j}^{-2\eta}\lesssim_{\eta}R_{0}^{-2\eta},$

if $\eta\leq\eta_{0}$ is chosen sufficiently small in terms of $\kappa$.
Hence, if $R_{0}$ is chosen sufficiently large in terms of $\eta$, then
$R_{0}^{\eta}$ dominates the implicit constant and we get
$\mu_{2}(E_{2}\setminus E_{2}^{\prime\prime})\leq R_{0}^{-\eta}$. Thus,
$E_{2}^{\prime\prime}\subset E_{2}$ satisfies the desired bound and assertions
(a) and (b) are proved.

Let $x\in E_{2}^{\prime\prime}$. For assertion (c), we observe that if
$R_{j}<R_{0}^{2}$, then $G_{j}(x)=G_{j+1}(x)$. (This is the reason why the
parameter $j_{*}$ was introduced.) Thus, assume $R_{j}\geq R_{0}^{2}$. Then

$G_{j}(x)\setminus
G_{j+1}(x)\subset\bigcup_{H\in\mathcal{H}_{j+1}(x)}F(x,aH)\,,$

which is contained within some $(R_{j+1}^{-\kappa/2}/3,k)$-plate $V$ through
$x$ using assertion (a). Let $m$ be such that
$R_{j+1}\in[R_{m}^{2}/2,2R_{m}^{2}]$. Since $R_{j+1}^{-\kappa/2}/3\leq
R_{m}^{-\kappa}/2$, we know that $V$ is contained in some
$(R_{m}^{-\kappa},k)$-plate $H\in\mathcal{E}_{R_{m}^{-\kappa},k}$. Note that,
since $a\geq 99C_{\text{sep}}$, we have $H\cap
B(x,C_{\text{sep}}^{-1})^{c}\subset F(x,aH)$. Therefore, if
$H\in\mathcal{H}_{m}$, then $V\cap G_{m}(x)=\emptyset$. Otherwise, we have
$\mu_{1}(V)\leq R_{m}^{-\eta}$ by definition of $\mathcal{H}_{m}$. In either
case, we have

$\mu_{1}(G_{j}(x)\setminus G_{j+1}(x))\leq\mu_{1}(V\cap G_{m}(x))\lesssim
R_{m}^{-\eta}\lesssim R_{j}^{-\eta/2}\,,$

as desired. ∎

Lemma 4.2 has two ramifications. First, Lemma 4.2(b) tells us that
$B(0,10)\setminus G(x)$ is contained within some
$(R_{0}^{-\kappa/2},k)$-plate. Second, Lemma 4.2(c) tells us that
$\mu_{1}(G_{j}(x)\setminus G_{j+1}(x))$ is small for all $x\in
E_{2}^{\prime\prime}$, so we can estimate
$\|\mu_{1}^{x}-\mu_{1}|_{G(x)}\|_{L^{1}}$. We record these two observations
and provide detailed proofs in the following lemma.

######  Lemma 4.3.

Let $E_{2}^{\prime\prime}$ be the subset given in Lemma 4.2 and $x\in
E_{2}^{\prime\prime}$. Then $B(0,10)\setminus G(x)$ is contained within some
$(R_{0}^{-\kappa/2},k)$-plate and

$\|\mu_{1}|_{G(x)}-\mu_{1}^{x}\|_{L^{1}}\lesssim R_{0}^{-\eta/2}.$

###### Proof.

For the first assertion, we use the definition of $G(x),G_{0}(x)$ in (4.1) and
the fact $F(x,aH)\subset aH$ proved in Lemma 4.1 to write

$B(0,10)\setminus G(x)=\left[B(x,C_{\text{sep}}^{-1})^{c}\cap
B(0,10)\right]\setminus G_{0}(x)\subset\bigcup_{H\in\mathcal{H}_{\leq
j_{*}}(x)}aH.$

By Lemma 4.2(b), the rightmost expression is contained within some
$(R_{0}^{-\kappa/2},k)$-plate.

For the second assertion, let $G_{\infty}(x)=\bigcap_{j=0}^{\infty}G_{j}(x)$.
First, by Lemma 4.2(c), we establish that for all $j\geq 0$,

(4.3) $\mu_{1}(G_{j}(x)\setminus
G_{\infty}(x))=\sum_{i=j}^{\infty}\mu_{1}(G_{i}(x)\setminus
G_{i+1}(x))\lesssim\sum_{i=j}^{\infty}R_{i}^{-\eta/2}\lesssim
R_{j}^{-\eta/2}.$

Also, $\mu_{1}$ is supported on $B(x,C_{\text{sep}}^{-1})^{c}$, so
$\mu_{1}|_{G(x)}$ and $\mu_{1}|_{G_{0}(x)}$ are the same measure. Thus, by
(4.3), we get $\|\mu_{1}|_{G(x)}-\mu_{1}|_{G_{\infty}(x)}\|_{L^{1}}\lesssim
R_{0}^{-\eta/2}$, so it suffices to show that
$\|\mu_{1}|_{G_{\infty}(x)}-\mu_{1}^{x}\|_{L^{1}}\lesssim R_{0}^{-\eta/2}$.
Indeed, using $\sum_{j\geq 0}\psi_{j}=1$ and the definition (4.2) of
$\mu_{1}^{x}$, we have

$\mu_{1}^{x}-\mu_{1}|_{G_{\infty}(x)}=\sum_{j\geq
0}(\mu_{1}|_{G(x)}-\mu_{1}|_{G_{\infty}(x)})*\check{\psi}_{j}=\sum_{j\geq
0}\mu_{1}|_{G_{j}(x)\setminus G_{\infty}(x)}*\check{\psi}_{j},$

and so by $\|\check{\psi}_{j}\|_{L^{1}}\lesssim 1$, Young’s convolution
inequality, and (4.3), we have

$\|\mu_{1}^{x}-\mu_{1}|_{G_{\infty}(x)}\|_{L^{1}}\lesssim\sum_{j\geq
0}\mu_{1}(G_{j}(x)\setminus G_{\infty}(x))\lesssim\sum_{j\geq
0}R_{j}^{-\eta/2}\lesssim R_{0}^{-\eta/2}.$

∎

######  Remark 4.4.

We can make the last step more efficient using Abel summation. Let
$\check{P}_{k}=\sum_{j=1}^{k}\check{\psi}_{j}$. Note that if we chose
$\psi_{j}$ to be Littlewood-Paley functions $\chi(x/R_{j})-\chi(x/R_{j-1})$ as
in Section 4.1, then $\|\check{P}_{k}\|_{L^{1}}\leq C$ for some universal
constant $C$. Now, Abel summation gives

$\mu_{1}^{x}-\mu_{1}|_{G_{\infty}(x)}=\sum_{j=0}^{\infty}\mu_{1}|_{G_{j}(x)\setminus
G_{j+1}(x)}*\check{P}_{j},$

and thus we get a slightly sharper bound
$\|\mu_{1}^{x}-\mu_{1}|_{G_{\infty}(x)}\|_{L^{1}}\lesssim\mu(G_{0}(x)\setminus
G_{\infty}(x))$. This can be useful in some potential applications where
$\mu(G_{0}(x)\setminus G_{\infty}(x))$ is controlled but not
$\mu(G_{j}(x)\setminus G_{\infty}(x))$ as $j\to\infty$.

###  4.5. Tube Geometry and bound of
$\|d_{*}^{x}(\mu_{1}^{x})-d_{*}^{x}(\mu_{1,g}^{x})\|_{L^{1}}$

The goal of this subsection is to establish a bound on
$\|d_{*}^{x}(\mu_{1}^{x})-d_{*}^{x}(\mu_{1,g}^{x})\|_{L^{1}}$ in terms of the
geometry of the tubes (Lemma 4.7). Showing such a bound will allow us to prove
Proposition 2.1, thanks to Lemma 4.3.

Recall that

$\mu_{1}^{x}=\sum_{j\geq 0}\mu_{1}|_{G_{j}(x)}*\check{\psi}_{j}.$

We first give a good approximation to $\mu_{1}^{x}$. The following lemma can
be proved the same way as Lemma 3.4 in [10] and we omit the details. It shows
that problems about the mysterious $\mu_{1}^{x}$ actually reduce to problems
about restricted measures of $\mu_{1}$. We remark that the next few lemmas
will be proved for all $x\in E_{2}$, even though we only need these results
for $x\in E_{2}^{\prime\prime}$ to prove Proposition 2.1. This is a minor
difference: restricting to $x\in E_{2}^{\prime\prime}$ doesn’t make the lemmas
easier to prove.

######  Lemma 4.5.

Let $x\in E_{2}$. Then

$\|\mu_{1}^{x}-M_{0}(\mu_{1}|_{G_{0}(x)})-\sum_{j=1}^{\infty}\sum_{T\in\mathbb{T}_{j}}M_{T}(\mu_{1}|_{G_{j}(x)})\|_{L^{1}}\leq\mathrm{RapDec}(R_{0}).$

We make the following observations relating the geometry of tubes to analytic
estimates.

######  Lemma 4.6.

Let $T\in\mathbb{T}_{j}$ be an $R_{j}^{-1/2+\beta}$-tube and $x\in E_{2}$.

  1. (a)

If $2T$ doesn’t pass through $x$, then
$\|d_{*}^{x}(M_{T}\mu_{1})\|_{L^{1}}\leq\mathrm{RapDec}(R_{j})$ and
$\|d_{*}^{x}(M_{T}(\mu_{1}|_{G_{j}(x)}))\|_{L^{1}}\leq\mathrm{RapDec}(R_{j})$.

  2. (b)

If $2T\cap B(x,C_{\text{sep}}^{-1})^{c}\subset G_{j}(x)$, then
$\|M_{T}\mu_{1}-M_{T}(\mu_{1}|_{G_{j}(x)})\|_{L^{1}}\leq\mathrm{RapDec}(R_{j})$.

  3. (c)

If $2T\cap B(x,C_{\text{sep}}^{-1})^{c}\subset F(x,aH)$ for some
$H\in\mathcal{H}_{\leq\max(j,j_{*})}(x)$, then
$\|M_{T}(\mu_{1}|_{G_{j}(x)})\|_{L^{1}}\leq\mathrm{RapDec}(R_{j})$.

  4. (d)

$\|M_{T}(\mu_{1}|_{G_{j}(x)})\|_{L^{1}}$ and $\|M_{T}\mu_{1}\|_{L^{1}}$ are
both $\lesssim\mu_{1}(2T)+\mathrm{RapDec}(R_{j})$.

###### Proof.

(a) This is Lemma 3.1 of [10] applied to $\mu_{1}$ and $\mu_{1}|_{G_{j}(x)}$;
note that $\|\mu_{1}|_{G_{j}(x)}\|_{L^{1}}\leq\|\mu_{1}\|_{L^{1}}\leq 1$.

(b) By assumption, $\mu_{1}-\mu_{1}|_{G_{j}(x)}$ is supported outside $2T\cap
B(x,C_{\text{sep}}^{-1})^{c}\supset 2T\cap E_{1}$, so we can apply Lemma 3.2
of [10].

(c) By assumption, $\mu_{1}|_{G_{j}(x)}$ is supported outside $2T\cap E_{1}$,
so we can apply Lemma 3.2 of [10].

(d) This is a direct consequence of Lemma 3.2 of [10]. ∎

Using Lemma 4.6, we are able to compare $\mu_{1}^{x}$ with $\mu_{1,g}^{x}$. We
first need some definition. For $x\in E_{2}$, $j\geq 0$, define
$\mathrm{Bad}_{j}(x)$ to be the union of $2T$, where $T\in\mathbb{T}_{j}$ is
an $R_{j}^{-1/2+\beta}$-tube such that $2T$ passes through $x$ and either (1)
$2T\cap B(x,C_{\text{sep}}^{-1})^{c}$ is not contained in $G_{j}(x)$ or any
$F(x,aH)$ for $H\in\mathcal{H}_{\leq\max(j,j_{*})}(x)$; or (2) $2T\cap
B(x,C_{\text{sep}}^{-1})^{c}\subset G_{j}(x)$ and
$\mu_{2}(4T)>R_{j}^{-\alpha/2+\varepsilon_{0}}$. By Lemma 4.1, (2) is morally
the union of the acceptable but not good tubes through $x$, while (1) is the
union of the tubes through $x$ that are “borderline” between acceptable and
non-acceptable.

One may compare the following lemma with Lemma 3.1 of [5].

######  Lemma 4.7.

Let $x\in E_{2}$ and $\text{Bad}_{j}(x)$ be defined as above, $\forall j\geq
0$. Then,

$\|d_{*}^{x}(\mu_{1}^{x})-d_{*}^{x}(\mu_{1,g}^{x})\|_{L^{1}}\lesssim\sum_{j\geq
1}R_{j}^{100\beta d}\mu_{1}(\mathrm{Bad}_{j}(x))+\mathrm{RapDec}(R_{0}).$

###### Proof.

We apply Lemma 4.5 and the definition of $\mu_{1,g}^{x}$. Note that the
$M_{0}(\mu_{1}|_{G_{0}(x)})$ terms cancel in the resulting expression for
$\mu_{1}^{x}-\mu_{1,g}^{x}$ (this is the reason why $\mu_{1,g}^{x}$ needs to
depend on $x$). Let $g(T)=1$ if $T$ is good and $0$ otherwise. Thus, it
suffices to prove for each $j\geq 1$,

$\displaystyle\|\sum_{T\in\mathbb{T}_{j}}[d^{x}_{*}(M_{T}(\mu_{1}|_{G_{j}(x)}))-d^{x}_{*}(M_{T}\mu_{1})g(T)]\|_{L^{1}}$
(4.4) $\displaystyle\lesssim$ $\displaystyle R_{j}^{100\beta
d}\mu_{1}(\mathrm{Bad}_{j}(x))+\mathrm{RapDec}(R_{j}).$

Let $\mathbb{T}_{j,\mathrm{bad}}$ be the set of tubes $T\in\mathbb{T}_{j}$
such that $2T$ passes through $x$ and either condition (1) or (2) in the
definition of $\text{Bad}_{j}(x)$ holds. We claim that if
$T\notin\mathbb{T}_{j,\mathrm{bad}}$, then

(4.5)
$\|d^{x}_{*}(M_{T}(\mu_{1}|_{G_{j}(x)}))-d^{x}_{*}(M_{T}\mu_{1})g(T)\|_{L^{1}}\leq\mathrm{RapDec}(R_{j}),$

while if $T\in\mathbb{T}_{j,\mathrm{bad}}$, then

(4.6)
$\|d^{x}_{*}(M_{T}(\mu_{1}|_{G_{j}(x)}))-d^{x}_{*}(M_{T}\mu_{1})g(T)\|_{L^{1}}\lesssim\mu_{1}(2T)+\mathrm{RapDec}(R_{j}).$

We show how the claim implies (4.4). For any $y\in E_{1}$, $d(x,y)\gtrsim 1$
and so there are $\lesssim R_{j}^{100\beta d}$ many $R_{j}^{-1/2+\beta}$-tubes
in $\mathbb{T}_{j}$ passing through both $x$ and $y$. Thus,

(4.7) $\sum_{T\in\mathbb{T}_{j,\mathrm{bad}}}\mu_{1}(2T)\lesssim
R_{j}^{100\beta d}\mu_{1}(\mathrm{Bad}_{j}(x)).$

Combining (4.5), (4.6), (4.7) proves (4.4).

Now we prove the claim. Suppose $T\notin\mathbb{T}_{j,\mathrm{bad}}$. By
working through the definition of $\mathbb{T}_{j,\mathrm{bad}}$, we have three
possibilities of $T$: either

  1. (i)

$x\notin 2T$;

  2. (ii)

$x\in 2T$, $2T\cap B(x,C_{\text{sep}}^{-1})^{c}\subset G_{j}(x)$ and
$\mu_{2}(4T)\leq R_{j}^{-\alpha/2+\varepsilon_{0}}$. Then $T$ is acceptable by
Lemma 4.1(a), so it is good since $\mu_{2}(4T)\leq
R_{j}^{-\alpha/2+\varepsilon_{0}}$.

  3. (iii)

$x\in 2T$, $2T\cap B(x,C_{\text{sep}}^{-1})^{c}\subset F(x,aH)$ for some
$H\in\mathcal{H}_{\leq\max(j,j_{*})}(x)$. Then $T$ is non-acceptable by Lemma
4.1(b).

In case (i), we get (4.5) by Lemma 4.6(a), regardless of $g(T)=0$ or $1$. In
case (ii), we use Lemma 4.6(b) and $g(T)=1$, and in case (iii), we use Lemma
4.6(c) and $g(T)=0$. Thus, if $T\notin\mathbb{T}_{j,\mathrm{bad}}$, then (4.5)
holds.

Now suppose $T\in\mathbb{T}_{j,\mathrm{bad}}$. Then we get (4.6) by applying
Lemma 4.6(d), regardless of whether $g(T)=0$ or $1$. This proves the claim. ∎

The crucial estimate about $\mathrm{Bad}_{j}(x)$ is the following lemma, which
will be proved in Section 4.6.

######  Lemma 4.8.

For every $\varepsilon_{0}>0$, there exist
$\eta_{0}(\varepsilon_{0}),\kappa_{0}(\varepsilon_{0})>0$ such that for any
$\eta\in(0,\eta_{0}],\kappa\in(0,\kappa_{0}]$ and sufficiently small $\beta$
depending on $\varepsilon_{0},\eta,\kappa$, the following holds. In the
construction of $\mu_{1,g}^{x}$ in Section 4.2, we can choose some
$a\in[99C_{\text{sep}},100C_{\text{sep}}]$ such that for any $j\geq 1$, if we
define

$\mathrm{Bad}_{j}:=\\{(x,y)\in E_{2}\times
E_{1}:y\in\mathrm{Bad}_{j}(x)\\}\,,$

then $\mu_{2}\times\mu_{1}(\mathrm{Bad}_{j})\lesssim R_{j}^{-200\beta d}.$

Using this, we complete the proof of Proposition 2.1.

###### Proof of Proposition 2.1.

Our goal is to find a large subset $E_{2}^{\prime}\subset E_{2}$ with
$\mu_{2}(E_{2}^{\prime})\geq 1-R_{0}^{-\beta}$ such that for each $x\in
E_{2}^{\prime}$, we have that $B^{d}(0,10)\setminus G(x)$ is contained within
some $(R_{0}^{-\beta},k)$-plate and

(4.8) $\|d_{*}^{x}(\mu_{1}|_{G(x)})-d_{*}^{x}(\mu_{1,g}^{x})\|_{L^{1}}\leq
R_{0}^{-\beta}.$

First, let us determine the auxiliary parameters
$\varepsilon_{0},\kappa,\eta,\beta$. (Recall that $\varepsilon,\alpha,k,d$ are
fixed constants.) We defer the choice of $\varepsilon_{0}(\varepsilon)$ to the
next section, see Lemma 5.1. Then, choose $\kappa=\kappa_{0}(\varepsilon_{0})$
in Lemma 4.8. Next, choose $\eta=\eta(\kappa,\varepsilon_{0})$ to be the
smaller of the two $\eta_{0}$’s in Lemma 4.2 and Lemma 4.8. Finally, choose
$\beta$ to the smaller of the $\beta(\varepsilon_{0},\eta,\kappa)$ in Lemma
4.8 and $\min(\frac{\eta}{3},\frac{\kappa}{2})$.

Now, we shall construct $E_{2}^{\prime}$ by taking the set
$E_{2}^{\prime\prime}$ from Lemma 4.2 and removing some “bad parts” given by
Lemma 4.8. Fix a choice of $a$ in the construction of $\mu_{1,g}^{x}$ in
Section 4.2 such that the conclusion in Lemma 4.8 holds for any $j\geq 1$. By
Lemma 4.8, for each $j\geq 1$, we can find a set $F_{j}\subset E_{2}$ with
$\mu_{2}(F_{j})\leq R_{j}^{-50\beta d}$ such that
$\mu_{1}(\mathrm{Bad}_{j}(x))\lesssim R_{j}^{-150\beta d}$ for $x\in
E_{2}\setminus F_{j}$. Finally, define
$E_{2}^{\prime}:=E_{2}^{\prime\prime}\setminus\bigcup_{j\geq 1}F_{j}$. We now
verify that $E_{2}^{\prime}$ satisfies the desired conditions.

First, observe that $\mu_{2}(E_{2}^{\prime})\geq\mu(E_{2})-\mu(E_{2}\setminus
E_{2}^{\prime\prime})-\sum_{j\geq 1}R_{j}^{-50\beta d}>1-R_{0}^{-\beta}$ if
$R_{0}$ is sufficiently large.

Next, fix $x\in E_{2}^{\prime}$. Since $x\in E_{2}^{\prime\prime}$ and
$\beta<\frac{\kappa}{2}$, we get from the first part of Lemma 4.3 that
$B(0,10)\setminus G(x)$ is contained in some $(R_{0}^{-\beta},k)$-plate. Now
by the second part of Lemma 4.3, since $\beta\leq\frac{\eta}{3}$, we have that

(4.9)
$\|d_{*}^{x}(\mu_{1}|_{G(x)})-d^{x}_{*}(\mu_{1}^{x})\|_{L^{1}}<\frac{1}{2}R_{0}^{-\beta}.$

For each $x\in E_{2}^{\prime}$, Lemma 4.7 tells us (for some constant $C$
depending only on our parameters):

(4.10) $\|d_{*}^{x}(\mu_{1}^{x})-d_{*}^{x}(\mu_{1,g}^{x})\|_{L^{1}}\leq
C\cdot\sum_{j\geq 1}R_{j}^{-50\beta
d}+\mathrm{RapDec}(R_{0})<\frac{1}{2}R_{0}^{-\beta}$

if $R_{0}$ is sufficiently large. Combining (4.9) and (4.10) via triangle
inequality proves the desired equation (4.8). ∎

###  4.6. Control of bad part

The goal of this subsection is to prove Lemma 4.8. To do so, we will use the
new radial projection estimate, Theorem 3.1.

###### Proof of Lemma 4.8.

Let $\mathrm{Bad}_{j}=\\{(x,y)\in E_{2}\times
E_{1}:y\in\mathrm{Bad}_{j}(x)\\}$. By definition of $\mathrm{Bad}_{j}(x)$
above Lemma 4.7, we have
$\mathrm{Bad}_{j}\subset\mathrm{Bad}_{j}^{1}\cup\mathrm{Bad}_{j}^{2}$. Here
$\mathrm{Bad}_{j}^{1}$ is the set of pairs $(x,y)\in E_{2}\times E_{1}$ such
that $x,y$ lie in $2T$ for some $R_{j}^{-1/2+\beta}$-tube $T\in\mathbb{T}_{j}$
with $2T\cap B(x,C_{\text{sep}}^{-1})^{c}\subset G_{j}(x)$ and
$\mu_{2}(4T)\geq R_{j}^{-\alpha/2+\varepsilon_{0}}$. And
$\mathrm{Bad}_{j}^{2}$ is the set of pairs $(x,y)\in E_{2}\times E_{1}$
satisfying that $x,y$ lie in $2T$ for some $R_{j}^{-1/2+\beta}$-tube
$T\in\mathbb{T}_{j}$ such that $2T\cap B(x,C_{\text{sep}}^{-1})^{c}$ is not
contained in $G_{j}(x)$ or any $F(x,aH)$ for
$H\in\mathcal{H}_{\leq\max(j,j_{*})}(x)$.

Our bound of $\mu_{2}\times\mu_{1}(\mathrm{Bad}^{1}_{j})$ will not depend on
the choice of $a$ in the construction of $\mu_{1,g}^{x}$ in Section 4.2, while
$\mu_{2}\times\mu_{1}(\mathrm{Bad}^{2}_{j})$ will. For a given
$H\in\mathcal{H}_{\ell}$ with $\ell\leq\max(j,j_{*})$, let
$\mathrm{Bad}^{2}_{j}(H)$ be the set of pairs $(x,y)\in E_{2}\times E_{1}$
satisfying that $H\in\mathcal{H}_{\leq\max(j,j_{*})}(x)$ and $x,y$ lie in $2T$
for some $R_{j}^{-1/2+\beta}$-tube $T\in\mathbb{T}_{j}$ such that $2T\cap
B(x,C_{\text{sep}}^{-1})^{c}$ is not contained in $\mathbb{R}^{d}\setminus
F(x,aH)$ or $F(x,aH)$. We’ll prove

$\mu_{2}\times\mu_{1}(\mathrm{Bad}^{1}_{j})\lesssim R_{j}^{-200\beta d}$

and that there exists $a$ such that

(4.11)
$\sum_{H\in\mathcal{H}_{\leq\max(j,j_{*})}}\mu_{2}\times\mu_{1}(\mathrm{Bad}^{2}_{j}(H))\lesssim
R_{j}^{-1/8}\,,$

for any $j\geq 0$. Since
$\mathrm{Bad}_{j}^{2}\subset\bigcup_{H\in\mathcal{H}_{\leq\max(j,j_{*})}}\mathrm{Bad}_{j}^{2}(H)$,
these two bounds will prove Lemma 4.8.

_Upper bound of $\mu_{2}\times\mu_{1}(\mathrm{Bad}^{1}_{j})$._ This is a
consequence of Theorem 3.1. By applying Theorem 3.1 with parameters
$(\delta,r,\delta^{\eta},\varepsilon)=(R_{j}^{-1/2+\beta},R_{j}^{-\kappa}/(200C_{\text{sep}}),R_{j}^{-\eta},\varepsilon_{0}/4)$,
we can find $\gamma>0$ such that the following is true. There exists a set
$B\subset E_{2}\times E_{1}$ with $\mu_{2}\times\mu_{1}(B)\leq
R_{j}^{-\gamma}$ such that for each $y\in E_{1}$ and $R_{j}^{-1/2+\beta}$-tube
$T$ with $2T$ containing $y$, we have (assuming $\kappa$ and $\beta$ are
chosen small enough in terms of $\varepsilon_{0}$):

(4.12) $\mu_{2}(2T\setminus(A|_{y}\cup
B|_{y}))\lesssim_{C_{\text{sep}}}\frac{R_{j}^{(-1/2+\beta)\alpha}}{R_{j}^{-\kappa(\alpha-k+1)}}\cdot
R_{j}^{(-1/2+\beta)(-\varepsilon_{0}/4)}\leq
R_{j}^{-\alpha/2+\varepsilon_{0}/2}\,,$

where $A$ is the set of pairs $(x,y)\in E_{2}\times E_{1}$ satisfying that $x$
and $y$ lie in some $R_{j}^{-\eta}$-concentrated
$(R_{j}^{-\kappa}/(200C_{\text{sep}}),k)$-plate on $\mu_{1}+\mu_{2}$. Since
decreasing the values of $\beta,\gamma$ makes the previous statement weaker,
we may assume $200\beta d=\gamma$.

Now, observe that since $d(y,E_{2})\geq C_{\text{sep}}^{-1}\gtrsim 1$ for
$y\in E_{1}$ and $\mu_{2}$ is a probability measure, there are at most
$\lesssim R_{j}^{\alpha/2-\varepsilon_{0}+O(\beta)}$ many tubes
$T\in\mathbb{T}_{j}$ with $2T$ containing $y$ satisfying $\mu_{2}(4T)\geq
R_{j}^{-\alpha/2+\varepsilon_{0}}$. Moreover, we claim the following.

Claim 1.. Let $y\in E_{1}$. Suppose there exist $x\in E_{2}$ and
$T\in\mathbb{T}_{j}$ such that $x,y$ lie in $2T$ with $2T\cap
B(x,C_{\text{sep}}^{-1})^{c}\subset G_{j}(x)$. Then $2T\cap A|_{y}=\emptyset$.

Assuming Claim 1, by (4.12) we get

$\mu_{2}(\mathrm{Bad}^{1}_{j}|_{y}\setminus B|_{y})\lesssim
R_{j}^{\alpha/2-\varepsilon_{0}+O(\beta)}\cdot
R_{j}^{-\alpha/2+\varepsilon_{0}/2}\leq R_{j}^{-200\beta d}\,.$

Thus, $\mu_{2}\times\mu_{1}(\mathrm{Bad}^{1}_{j}\setminus B)\lesssim
R_{j}^{-200\beta d}$, and so
$\mu_{2}\times\mu_{1}(\mathrm{Bad}^{1}_{j})\lesssim R_{j}^{-200\beta d}$.

It remains to prove Claim 1. Let $y\in E_{1}$, $x\in E_{2}$, and
$T\in\mathbb{T}_{j}$ be such that $x,y$ lie in $2T$ with $2T\cap
B(x,C_{\text{sep}}^{-1})^{c}\subset G_{j}(x)$. Suppose $2T\cap
A|_{y}\neq\emptyset$. Pick a point $x^{\prime}\in 2T\cap A|_{y}$; by
definition, we have $x^{\prime}\in E_{2}\cap 2T$ and there exists a
$R_{j}^{-\eta}$-concentrated $(R_{j}^{-\kappa}/(200C_{\text{sep}}),k)$-plate
$H^{\prime}$ such that $x^{\prime}$ and $y$ lie in $H^{\prime}$. We also know
$x^{\prime},y$ both belong to $2T$. Since $d(x^{\prime},y)\geq
C_{\text{sep}}^{-1}$, we have that $2T$ is contained in
$100C_{\text{sep}}H^{\prime}$. This in turn is a $R_{j}^{-\eta}$-concentrated
$(R_{j}^{-\kappa}/2,k)$-plate, so it must be contained in some
$H\in\mathcal{H}_{j}$. Hence, $2T\subset H$.

By assumption, we know $2T\cap B(x,C_{\text{sep}}^{-1})^{c}\not\subset
F(x,aH)$ (where $a=99C_{\text{sep}}$), so there exists $z\in 2T\cap
B(x,C_{\text{sep}}^{-1})^{c}$ such that $\ell(x,z)$ intersects
$\mathrm{Surf}(aH)$ at some point $w$. Let $P$ be the $k$-plane through $x$
parallel to the central plane $P_{H}$ of $H$. Since $x,w,z$ are collinear, we
have

$\frac{d(w,P)}{d(z,P)}=\frac{d(w,x)}{d(z,x)}.$

However, we know that $w\in\mathrm{Surf}(aH)$, so
$d(w,P_{H})=aR_{j}^{-\kappa}$. Also $d(P,P_{H})\leq R_{j}^{-\kappa}$ since
$x\in 2T\subset H$. Thus by triangle inequality, we get
$d(w,P)\geq(a-1)R_{j}^{-\kappa}$. While $d(z,P)\leq 2R_{j}^{-\kappa}$ (since
$z\in 2T\subset H$), so $\frac{d(w,P)}{d(z,P)}\geq\frac{a-1}{2}$. On the other
hand, $d(w,x)\leq 20$ and $d(z,x)\geq C_{\text{sep}}^{-1}$, so
$\frac{d(w,x)}{d(z,x)}\leq 20C_{\text{sep}}$. Hence, we get $\frac{a-1}{2}\leq
20C_{\text{sep}}$, contradiction to $a=99C_{\text{sep}}$ and
$C_{\text{sep}}\geq 1$.

_Upper bound of $\mu_{2}\times\mu_{1}(\mathrm{Bad}^{2}_{j})$._ We will prove
there exists a choice for $a$ such that for all $j,\ell$ satisfying
$\ell\leq\max(j,j_{*})$, we have

(4.13)
$F_{j,\ell}(a)=\sum_{H\in\mathcal{H}_{\ell}}\mu_{2}\times\mu_{1}(\mathrm{Bad}_{j}^{2}(H))\leq
R_{j}^{-1/4}.$

Then given $j$, summing (4.13) over $\ell\leq\max(j,j_{*})$ gives (4.11) (use
$\max(j,j_{*})\lesssim\log R_{j}$). To prove (4.13), we fix $j,\ell$ and upper
bound the measure of the set of $a$’s for which (4.13) fails. We would like to
apply Markov’s inequality, so we compute the expectation of $F_{j,\ell}(a)$
over $a$. Let
$P(a)=\frac{1}{C_{\text{sep}}}\mathbbm{1}_{[99C_{\text{sep}},100C_{\text{sep}}]}\,da$
be a probability measure of $a\in[99C_{\text{sep}},100C_{\text{sep}}]$. Then
we have

$\displaystyle\int_{I}F_{j,\ell}(a)\,dP(a)=$
$\displaystyle\int_{E_{1}}\int_{E_{2}}\int_{I}\sum_{H\in\mathcal{H}_{\ell}}1_{\mathrm{Bad}^{2}_{j}(H)}(x,y)\,dP(a)d\mu_{2}(x)d\mu_{1}(y)\,$
$\displaystyle\leq$ $\displaystyle\sup_{x\in E_{2},y\in
E_{1}}\int_{I}\sum_{H\in\mathcal{H}_{\ell}}1_{\mathrm{Bad}^{2}_{j}(H)}(x,y)\,dP(a).$

The following claim shows that it is unlikely that
$(x,y)\in\mathrm{Bad}_{j}^{2}(H)$ for a given $H$.

Claim 2. Suppose $(x,y)\in\mathrm{Bad}_{j}^{2}(H)$, where
$H\in\mathcal{H}_{\ell}(x)$ with $\ell\leq\max(j,j_{*})$. Let $z_{1},z_{2}$ be
the intersections of the line through $x,y$ with $B(0,10)$. Then for one of
$i=1,2$, we have $|d(z_{i},P_{H})-aR_{\ell}^{-\kappa}|\lesssim
R_{j}^{-1/2+\beta}$, where $P_{H}$ is the central plane of $H$.

Proof. By definition of $\mathrm{Bad}_{j}^{2}(H)$, there exists
$T\in\mathbb{T}_{j}$ with $2T$ containing both $x,y$ such that $2T\cap
B(x,C_{\text{sep}}^{-1})^{c}$ is not contained in $\mathbb{R}^{d}\setminus
F(x,aH)$ or $F(x,aH)$.

Fix a large constant $C>0$. If
$d(z_{i},P_{H})-aR_{\ell}^{-\kappa}>CR_{j}^{-1/2+\beta}$ for $i=1$ or $2$,
then we claim $2T\cap
B(x,C_{\text{sep}}^{-1})^{c}\subset\mathbb{R}^{d}\setminus F(x,aH)$. Note that
for any $y^{\prime}\in 2T\cap B(x,C_{\text{sep}}^{-1})^{c}$, one of the
intersections $z^{\prime}$ of the line through $x,y^{\prime}$ with $B(0,10)$
is contained in $B(z_{i},CR_{j}^{-1/2+\beta})$, and so by triangle inequality,
we get $d(z^{\prime},P_{H})>aR_{\ell}^{-\kappa}$. (This is why the
$B(x,C_{\text{sep}}^{-1})^{c}$ is important: it is not true that for all
$y^{\prime}\in 2T$, one of the intersections $z^{\prime}$ of the line through
$x,y^{\prime}$ with $B(0,10)$ is contained in $B(z_{i},CR_{j}^{-1/2+\beta})$.
Take $y^{\prime}$ such that $(y-x)\perp(y^{\prime}-x)$, for instance.) This
shows that $2T\cap B(x,C_{\text{sep}}^{-1})^{c}\subset\mathbb{R}^{d}\setminus
F(x,aH)$.

If on the other hand $d(z_{i},P_{H})<aR_{\ell}^{-\kappa}-CR_{j}^{-1/2+\beta}$
for both $i=1,2$, then a similar argument shows that $2T\cap
B(x,C_{\text{sep}}^{-1})^{c}\subset F(x,aH)$. Thus, we have established the
contrapositive of the claim. ∎

Using Claim 2, observe that, for a fixed pair $(x,y)\in E_{2}\times E_{1}$ and
a fixed $H\in\mathcal{H}_{\ell}(x)$ with $\ell\leq\max(j,j_{*})$ (so
$R_{\ell}\leq R_{j}^{2}$), we have $(x,y)\in\mathrm{Bad}_{j}^{2}(H)$ for $a$
lying in two intervals each of length $\lesssim
R_{j}^{-1/2+\beta}R_{\ell}^{\kappa}\lesssim R_{j}^{-1/2+\beta+2\kappa}$.
Recall that by Lemma 3.3 and $R_{\ell}\leq R_{j}^{2}$ we have
$|\mathcal{H}_{\ell}|\lesssim R_{j}^{2N\eta}$. Therefore,

$\displaystyle\sum_{H\in\mathcal{H}_{\ell}}\int_{I}1_{\mathrm{Bad}^{2}_{j}(H)}(x,y)\,dP(a)\lesssim$
$\displaystyle\frac{R_{j}^{2N\eta}R_{j}^{-1/2+\beta+2\kappa}}{|I|}$
$\displaystyle\sim$ $\displaystyle R_{j}^{-1/2+\beta+2N\eta+2\kappa},$

Thus, assuming $\beta,\eta,\kappa$ are small enough and $R_{0}$ is large
enough, by Markov’s inequality we have
$|\\{a:F_{j,\ell}(a)>R_{j}^{-1/4}\\}|\leq R_{j}^{-1/8}$. By the union bound,
(4.13) fails for some $j,\ell$ satisfying $\ell\leq\max(j,j_{*})$ only in a
set of measure

$\displaystyle\sum_{j=0}^{\infty}\sum_{\ell\leq\max(j,j_{*})}R_{j}^{-1/8}\leq\sum_{j=0}^{j_{*}}\sum_{\ell=0}^{j_{*}}R_{0}^{-1/8}+\sum_{\ell=0}^{\infty}\sum_{j=\ell}^{\infty}R_{j}^{-1/8}$
$\displaystyle\lesssim$ $\displaystyle
R_{0}^{-1/8}\cdot(\log_{2}R_{0})^{2}+\sum_{\ell\geq 0}R_{\ell}^{-1/8}\lesssim
R_{0}^{-1/8}\cdot\left[(\log_{2}R_{0})^{2}+1\right]<1,$

if $R_{0}$ is chosen sufficiently large. ∎

##  5\. Refined decoupling and Proposition 2.2

In this section, we prove Proposition 2.2, which will complete the proof of
Theorem 1.2. This part of the argument proceeds very similarly as [5, Section
4] and [10, Section 5].

Let $\sigma_{r}$ be the normalized surface measure on the sphere of radius
$r$. The main estimates in the proof of Proposition 2.2 are the following.

######  Lemma 5.1.

For any $\alpha>0$, $r>10R_{0}$, and $\varepsilon_{0}$ sufficiently small
depending on $\alpha,\varepsilon$:

$\int_{E_{2}}|\mu_{1,g}^{x}*\widehat{\sigma}_{r}(x)|^{2}d\mu_{2}(x)\lesssim_{\varepsilon}r^{-\frac{\alpha
d}{d+1}+\varepsilon}r^{-(d-1)}\int|\widehat{\mu}_{1}|^{2}\psi_{r}d\xi+{\rm
RapDec}(r),$

where $\psi_{r}$ is a weight function which is $\sim 1$ on the annulus
$r-1\leq|\xi|\leq r+1$ and decays off of it. To be precise, we could take

$\psi_{r}(\xi)=\left(1+|r-|\xi||\right)^{-100}.$

######  Lemma 5.2.

For any $\alpha>0$, $r>0$, we have

$\int_{E_{2}}|\mu_{1,g}^{x}*\widehat{\sigma}_{r}(x)|^{2}\,d\mu_{2}(x)\lesssim(r+1)^{d-1}r^{-(d-1)}\,.$

###### Proof of Proposition 2.2, given Lemmas 5.1 and 5.2.

Note that

$d_{*}^{x}(\mu_{1,g}^{x})(t)=t^{d-1}\mu_{1,g}^{x}*\sigma_{t}(x).$

Since $\mu_{1,g}^{x}$ is essentially supported in the $R_{0}^{-1/2+\beta}$
neighborhood of $E_{1}$, for $x\in E_{2}$, we only need to consider $t\sim 1$.
Hence, up to a loss of $\text{RapDec}(R_{0})$ which is negligible in our
argument, we have

$\displaystyle\int\|d_{*}^{x}(\mu_{1,g}^{x})\|^{2}_{L^{2}}\,d\mu_{2}(x)$
$\displaystyle\lesssim\int_{0}^{\infty}\int|\mu_{1,g}^{x}*\sigma_{t}(x)|^{2}\,d\mu_{2}(x)t^{d-1}dt$
$\displaystyle=\int_{0}^{\infty}\int|\mu_{1,g}^{x}*\widehat{\sigma}_{r}(x)|^{2}\,d\mu_{2}(x)r^{d-1}dr,$

where in the second step, we used a limiting process and an $L^{2}$-identity
proved by Liu [14, Theorem 1.9]: for any Schwartz function $f$ on
$\mathbb{R}^{d},d\geq 2$, and any $x\in\mathbb{R}^{d}$,

$\int_{0}^{\infty}|f*\sigma_{t}(x)|^{2}\,t^{d-1}\,dt=\int_{0}^{\infty}|f*\widehat{\sigma}_{r}(x)|^{2}\,r^{d-1}\,dr.$

For $r\leq 10R_{0}$ we use Lemma 5.2, and for $r>10R_{0}$ we use Lemma 5.1.
The small $r$ contribution to
$\int\|d_{*}^{x}(\mu_{1,g}^{x})\|^{2}_{L^{2}}d\mu_{2}(x)$ is

$\displaystyle\int_{0}^{10R_{0}}(r+1)^{d-1}\,dr\lesssim R_{0}^{d}.$

The large $r$ contribution is (dropping the negligible $\mathrm{RapDec}(r)$
term)

$\displaystyle\int_{10R_{0}}^{\infty}\int_{\mathbb{R}^{d}}r^{-\frac{\alpha
d}{d+1}+\varepsilon}\psi_{r}(\xi)|\widehat{\mu}_{1}(\xi)|^{2}\,d\xi dr$
$\displaystyle\lesssim\int_{\mathbb{R}^{d}}|\xi|^{-\frac{\alpha
d}{d+1}+\varepsilon}|\widehat{\mu}_{1}(\xi)|^{2}\,d\xi.$

The proof of Proposition 2.2 is thus complete upon verification of Lemmas 5.1
and 5.2. ∎

###### Proof of Lemma 5.2.

We follow the proof of Proposition 5.3 in [10], case $r<10R_{0}$. Since
$\mu_{2}$ is a probability measure, it suffices to upper bound
$\sup_{x}|\mu_{1,g}^{x}*\widehat{\sigma}_{r}(x)|$. Fix $x$ and note that

$|\mu_{1,g}^{x}*\widehat{\sigma}_{r}(x)|^{2}\leq\|\widehat{\mu_{1,g}^{x}}\|^{2}_{L^{1}(d\sigma_{r})}\leq\|\widehat{\mu_{1,g}^{x}}\|^{2}_{L^{2}(d\sigma_{r})}\,.$

Then by the approximately orthogonal argument in [10, Proof of Proposition
5.3], we have

$\|\widehat{\mu_{1,g}^{x}}\|^{2}_{L^{2}(d\sigma_{r})}\lesssim
r^{-(d-1)}\int(|\widehat{\mu_{1}|_{G_{0}(x)}}|^{2}+|\widehat{\mu_{1}}|^{2})\psi_{r}d\xi.$

Finally, since $\|\widehat{\mu}_{1}\|_{L^{\infty}}\leq\|\mu_{1}\|_{L^{1}}=1$,
$\|\widehat{\mu_{1}|_{G_{0}(x)}}\|_{L^{\infty}}\leq\|\mu_{1}|_{G_{0}(x)}\|_{L^{1}}\leq
1$, and $\int\psi_{r}\,d\xi\lesssim(r+1)^{d-1}$, we get the desired result. ∎

###  5.1. Refined decoupling estimates

The key ingredient in the proof of Lemma 5.1 is the following refined
decoupling theorem, which is derived by applying the $l^{2}$ decoupling
theorem of Bourgain and Demeter [3] at many different scales.

Here is the setup. Suppose that $S\subset\mathbb{R}^{d}$ is a compact and
strictly convex $C^{2}$ hypersurface with Gaussian curvature $\sim 1$. For any
$\varepsilon>0$, suppose there exists $0<\beta\ll\varepsilon$ satisfying the
following. Suppose that the 1-neighborhood of $RS$ is partitioned into
$R^{1/2}\times...\times R^{1/2}\times 1$ blocks $\theta$. For each $\theta$,
let $\mathbb{T}_{\theta}$ be a set of finitely overlapping tubes of dimensions
$R^{-1/2+\beta}\times\cdots\times R^{-1/2+\beta}\times 1$ with long axis
perpendicular to $\theta$, and let
$\mathbb{T}=\cup_{\theta}\mathbb{T}_{\theta}$. Each $T\in\mathbb{T}$ belongs
to $\mathbb{T}_{\theta}$ for a single $\theta$, and we let $\theta(T)$ denote
this $\theta$. We say that $f$ is microlocalized to $(T,\theta(T))$ if $f$ is
essentially supported in $2T$ and $\widehat{f}$ is essentially supported in
$2\theta(T)$.

######  Theorem 5.3.

[10, Corollary 4.3] Let $p$ be in the range $2\leq p\leq\frac{2(d+1)}{d-1}$.
For any $\varepsilon>0$, suppose there exists $0<\beta\ll\varepsilon$
satisfying the following. Let $\mathbb{W}\subset\mathbb{T}$ and suppose that
each $T\in\mathbb{W}$ lies in the unit ball. Let $W=|\mathbb{W}|$. Suppose
that $f=\sum_{T\in\mathbb{W}}f_{T}$, where $f_{T}$ is microlocalized to
$(T,\theta(T))$. Suppose that $\|f_{T}\|_{L^{p}}$ is $\sim$ constant for each
$T\in\mathbb{W}$. Let $Y$ be a union of $R^{-1/2}$-cubes in the unit ball each
of which intersects at most $M$ tubes $T\in\mathbb{W}$. Then

$\|f\|_{L^{p}(Y)}\lesssim_{\varepsilon}R^{\varepsilon}\left(\frac{M}{W}\right)^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in\mathbb{W}}\|f_{T}\|_{L^{p}}^{2}\right)^{1/2}.$

The proof of Lemma 5.1 using Theorem 5.3 proceeds almost identically as in [5,
Lemma 4.1], as the $x$ dependence of the good measure doesn’t exist in the
regime $r>10R_{0}$. We include the proof below for the sake of completeness.

###  5.2. Proof of Lemma 5.1

Assume $r>10R_{0}$. By definition,

$\mu_{1,g}^{x}*\widehat{\sigma}_{r}=\sum_{j:R_{j}\sim
r}\sum_{T\in\mathbb{T}_{j}:T\textrm{
good}}M_{T}\mu_{1}*\widehat{\sigma}_{r}+{\rm RapDec}(r).$

The key point to notice is that $\mu_{1,g}^{x}*\widehat{\sigma}_{r}$ is
independent of $x$.

The contribution of ${\rm RapDec}(r)$ is already taken into account in the
statement of Lemma 5.1. Hence without loss of generality we may ignore the
tail ${\rm RapDec}(r)$ in the argument below.

Let $\eta_{1}$ be a bump function adapted to the unit ball and define

$f_{T}=\eta_{1}\left(M_{T}\mu_{1}*\widehat{\sigma}_{r}\right).$

One can easily verify that $f_{T}$ is microlocalized to $(T,\theta(T))$.

Let $p=\frac{2(d+1)}{d-1}$. By dyadic pigeonholing, there exists $\lambda>0$
such that

$\int|\mu_{1,g}^{x}*\widehat{\sigma}_{r}(x)|^{2}\,d\mu_{2}(x)\lesssim\log
r\int|f_{\lambda}(x)|^{2}d\mu_{2}(x),$

where

$f_{\lambda}=\sum_{T\in\mathbb{W}_{\lambda}}f_{T},\quad\mathbb{W}_{\lambda}:=\bigcup_{j:R_{j}\sim
r}\Big{\\{}T\in\mathbb{T}_{j}:T\text{ good
},\|f_{T}\|_{L^{p}}\sim\lambda\Big{\\}}.$

Next, we divide the unit ball into $r^{-1/2}$-cubes $q$ and sort them. Denote

$\mathcal{Q}_{M}:=\\{r^{-1/2}\textrm{-cubes }q:q\textrm{ intersects }\sim
M\textrm{ tubes }T\in\mathbb{W}_{\lambda}\\}.$

Let $Y_{M}:=\bigcup_{q\in\mathcal{Q}_{M}}q$. Since there are only $\sim\log r$
many choices of $M$, there exists $M$ such that

$\int|\mu_{1,g}^{x}*\widehat{\sigma}_{r}(x)|^{2}\,d\mu_{2}(x)\lesssim(\log
r)^{2}\int_{Y_{M}}|f_{\lambda}(x)|^{2}d\mu_{2}(x)\,.$

Since $f_{\lambda}$ only involves good wave packets, by considering the
quantity

$\sum_{q\in\mathcal{Q}_{M}}\sum_{T\in\mathbb{W}_{\lambda}:T\cap
q\neq\emptyset}\mu_{2}(q),$

we get

(5.1)
$M\mu_{2}(\mathcal{N}_{r^{-1/2}}(Y_{M}))\lesssim|\mathbb{W}_{\lambda}|r^{-\frac{\alpha}{2}+\varepsilon_{0}},$

where $\mathcal{N}_{r^{-1/2}}(Y_{M})$ is the $r^{-1/2}$-neighborhood of
$Y_{M}$.

The rest of the proof of Lemma 5.1 will follow from Theorem 5.3 and estimate
(5.1).

By Hölder’s inequality and the observation that $f_{\lambda}$ has Fourier
support in the $1$-neighborhood of the sphere of radius $r$, one has

$\int_{Y_{M}}|f_{\lambda}(x)|^{2}\,d\mu_{2}(x)\lesssim\left(\int_{Y_{M}}|f_{\lambda}|^{p}\right)^{2/p}\left(\int_{Y_{M}}|\mu_{2}*\eta_{1/r}|^{p/(p-2)}\right)^{1-2/p},$

where $\eta_{1/r}$ is a bump function with integral $1$ that is essentially
supported on the ball of radius $1/r$.

To bound the second factor, we note that $\eta_{1/r}\sim r^{d}$ on the ball of
radius $1/r$ and rapidly decaying off it. Using the fact that
$\mu_{2}(B(x,t))\lesssim t^{\alpha},\forall x\in\mathbb{R}^{d},\forall t>0$,
we have

$\|\mu_{2}*\eta_{1/r}\|_{\infty}\lesssim r^{d-\alpha}\,.$

Therefore,

$\begin{split}\int_{Y_{M}}|\mu_{2}*\eta_{1/r}|^{p/(p-2)}\lesssim&\|\mu_{2}*\eta_{1/r}\|_{\infty}^{2/(p-2)}\int_{Y_{M}}d\mu_{2}*\eta_{1/r}\\\
\lesssim&r^{2(d-\alpha)/(p-2)}\mu_{2}(\mathcal{N}_{r^{-1/2}}(Y_{M})).\end{split}$

By Theorem 5.3, the first factor can be bounded as follows:

$\begin{split}\left(\int_{Y_{M}}|f_{\lambda}|^{p}\right)^{2/p}\lessapprox&\left(\frac{M}{\mathbb{W}_{\lambda}}\right)^{1-2/p}\sum_{T\in\mathbb{W}_{\lambda}}\|f_{T}\|_{L^{p}}^{2}\\\
\lesssim&\left(\frac{r^{-\frac{\alpha}{2}+\varepsilon_{0}}}{\mu_{2}(\mathcal{N}_{r^{-1/2}}(Y_{M}))}\right)^{1-2/p}\sum_{T\in\mathbb{W}_{\lambda}}\|f_{T}\|_{L^{p}}^{2},\end{split}$

where the second step follows from (5.1).

Combining the two estimates together, one obtains

$\int_{Y_{M}}|f_{\lambda}(x)|^{2}\,d\mu_{2}(x)\lesssim
r^{\frac{2d}{p}-\alpha(\frac{1}{2}+\frac{1}{p})+O(\varepsilon_{0})}\sum_{T\in\mathbb{W}_{\lambda}}\|f_{T}\|_{L^{p}}^{2}.$

Observe that $\|f_{T}\|_{L^{p}}$ has the following simple bound:

$\begin{split}\|f_{T}\|_{L^{p}}\lesssim&\|f_{T}\|_{L^{\infty}}|T|^{1/p}\lesssim\sigma_{r}(\theta(T))^{1/2}|T|^{1/p}\|\widehat{M_{T}\mu_{1}}\|_{L^{2}(d\sigma_{r})}\\\
=&r^{-(\frac{1}{2p}+\frac{1}{4})(d-1)+O(\beta)}\|\widehat{M_{T}\mu_{1}}\|_{L^{2}(d\sigma_{r})}.\end{split}$

Plugging this back into the above formula, one obtains

$\begin{split}\int_{Y_{M}}|f_{\lambda}(x)|^{2}\,d\mu_{2}(x)\lesssim&r^{\frac{2d}{p}-(\alpha+d-1)(\frac{1}{2}+\frac{1}{p})+O(\varepsilon_{0})}\sum_{T\in\mathbb{W}_{\lambda}}\|\widehat{M_{T}\mu_{1}}\|_{L^{2}(d\sigma_{r})}^{2}\\\
\lesssim&r^{-\frac{\alpha
d}{d+1}+\varepsilon}r^{-(d-1)}\int|\widehat{\mu}_{1}|^{2}\psi_{r}\,d\xi,\end{split}$

where $p=2(d+1)/(d-1)$ and we have used orthogonality and chosen
$\beta\ll\varepsilon_{0}\ll\varepsilon$. The proof of Lemma 5.1 and hence
Proposition 2.2 is complete.

##  6\. Proof of Theorem 1.3

We will prove Theorem 1.3 following the approach in [15]. We shall use the
following criteria to determine the Hausdorff dimension of pinned distance
sets.

######  Lemma 6.1.

[15, Lemma 3.1] Given a compact set $E\subset\mathbb{R}^{d}$,
$x\in\mathbb{R}^{d}$ and a probability measure $\mu_{E}$ on $E$. Suppose there
exist $\tau\in(0,1]$, $K\in\mathbb{Z}_{+}$, $\beta>0$ such that

$\mu_{E}(\\{y:|y-x|\in D_{k}\\})<2^{-k\beta}$

for any

$D_{k}=\bigcup_{j=1}^{M}I_{j},$

where $k>K$, $M\leq 2^{k\tau}$ are arbitrary integers and each $I_{j}$ is an
arbitrary interval of length $\approx 2^{-k}$. Then

$\dim_{H}(\Delta_{x}(E))\geq\tau.$

The next proposition is a key step in the proof of Theorem 1.3, which can be
viewed as a discretized variant of it.

######  Proposition 6.2.

Let $d\geq 2,1\leq k\leq d-1$, $k-1<\alpha\leq k$, and
$\tau<\min(f(\alpha),1)$. There exists $\beta>0$ depending on $\tau,\alpha,k$
such that the following holds for sufficiently small
$\delta<\delta_{0}(\tau,\alpha,k)$. Let $\mu_{1},\mu_{2}$ be
$\alpha$-dimensional measures with $\sim 1$ separation and constant
$C_{\alpha}$ supported on $E_{1},E_{2}$ respectively. Then there exists a set
$F_{\delta}\subset E_{2}$ with $\mu_{2}(F_{\delta})\lesssim\delta^{\beta^{2}}$
such that for all $x\in E_{2}\setminus F_{\delta}$, there exists a set $W(x)$
that is contained within some $(2\delta^{\beta^{2}},k)$-plate such that

$\mu_{1}(\\{y:|x-y|\in J\\}\setminus W(x))\leq\delta^{\beta^{2}/2},$

where $J$ is any union of $\leq\delta^{-\tau}$ many intervals each of length
$\sim\delta$.

Next we prove Proposition 6.2. The proof reproduces the argument in [15,
Section 4] with some minor simplifications.

###### Proof.

First, instead of working with $\mu_{1}$, we will use a mollified version that
removes the high frequency contributions. Let $\phi\in
C_{0}^{\infty}(\mathbb{R}^{d})$ be supported on $B(0,1)$ and satisfy $\phi\geq
0$, $\int\phi=1$, and $\phi\geq 1$ on $B(0,\frac{1}{2})$. This $\phi$ will be
fixed for the rest of the proof (in particular, it does not depend on
$\delta$, and subsequent implicit constants may depend on $\phi$). Let
$\phi_{\delta}(\cdot)=\delta^{-d}\phi(\delta^{-1}\cdot)$ and
$\mu_{1}^{\delta}=\mu_{1}*\phi_{\delta}$. The crucial point is that
$\mu_{1}^{\delta}$ is supported in a $\delta$-neighborhood of the support of
$\mu_{1}$ and in fact serves as a good approximation for $\mu_{1}$ down to
scale $\delta$, but $\mu_{1}^{\delta}$ is rapidly decaying at frequencies much
larger than $\delta^{-1}$.

Fix a small $\varepsilon>0$. We apply Proposition 2.1 with
$R_{0}=\delta^{-\beta}$ and the measure $\mu_{1}^{\delta}$, which is still an
$\alpha$-dimensional measure with constant comparable to $C_{\alpha}$
(independent of $\delta$). (We make $\delta$ sufficiently small to ensure that
$R_{0}$ is sufficiently large.) Then there is a subset $E_{2}^{\prime}\subset
E_{2}$ so that $\mu_{2}(E_{2}^{\prime})\geq 1-\delta^{\beta^{2}}$ and for each
$x\in E_{2}^{\prime}$, there exists a set $G(x)\subset\mathbb{R}^{d}$ where
$B^{d}(0,10)\setminus G(x)$ is contained within some
$(\delta^{\beta^{2}},k)$-plate $H(x)$ such that

$\|d_{*}^{x}(\mu^{\delta}_{1}|_{G(x)})-d_{*}^{x}(\mu_{1,g}^{\delta,x})\|_{L^{1}}\leq\delta^{\beta^{2}}.$

We will define $W(x)=H(x)^{(\delta)}$, which satisfies the condition for
$W(x)$. Let

$\mathcal{J}^{\tau}_{\delta}=\left\\{\bigcup_{j=1}^{M}I_{j}:M\leq\delta^{-\tau},\text{
each }I_{j}\text{ is an open interval of length}\sim\delta\right\\}.$

Let $F^{\prime}$ be the set of points $x\in E_{2}^{\prime}$ such that

$\sup_{J\in\mathcal{J}^{\tau}_{\delta}}\int_{J^{(\delta)}}d_{*}^{x}(\mu_{1}^{\delta}|_{G(x)})(t)\,dt\geq\delta^{\beta^{2}/2}.$

Now, define $F_{\delta}:=F^{\prime}\cup(E_{2}\setminus E_{2}^{\prime})$. Then
for any $x\in E_{2}\setminus F_{\delta}=E_{2}^{\prime}\setminus F^{\prime}$,
we can claim the following.

Claim. For all $x\in E_{2}^{\prime}\setminus F^{\prime}$ and
$J\in\mathcal{J}_{\delta}^{\tau}$, we have

$\mu_{1}(\\{y:|x-y|\in J\\}\setminus W(x))\leq\delta^{\beta^{2}/2}.$

Proof of Claim. Note that if $y\notin W(x)$, then $B(y,\delta)\subset G(x)$.
For $x\in E_{2}^{\prime}\setminus F^{\prime}$ and
$J\in\mathcal{J}_{\delta}^{\tau}$, we have

$\displaystyle\delta^{\beta^{2}/2}$ $\displaystyle\geq\int_{|x-z|\in
J^{(\delta)}}\mu_{1}^{\delta}|_{G(x)}(z)\,dz$
$\displaystyle=\delta^{-d}\iint_{|x-z|\in J^{(\delta)},z\in
G(x)}\phi(\delta^{-1}(z-y))d\mu_{1}(y)dz$
$\displaystyle\geq\delta^{-d}\iint_{|x-y|\in J,|y-z|\leq\delta,y\notin
W(x)}\phi(\delta^{-1}(z-y))d\mu_{1}(y)dz$ $\displaystyle\geq\int_{|x-y|\in
J,y\notin W(x)}d\mu_{1}(y)\int_{B(0,1)}\phi(u)\,du$
$\displaystyle=\mu_{1}(\\{y:|x-y|\in J\\}\setminus W(x)).\qed$

Recall that $\mu_{2}(E_{2}\setminus E_{2}^{\prime})\leq\delta^{\beta^{2}}$. So
it remains to show $\mu_{2}(F^{\prime})\lesssim\delta^{\beta^{2}}$ (assuming
good choice for $\beta,\varepsilon$). For $x\in F^{\prime}$, we have

$\displaystyle\sup_{J\in\mathcal{J}^{\tau}_{\delta}}\int_{J^{(\delta)}}$
$\displaystyle|d_{*}^{x}(\mu_{1,g}^{\delta,x})(t)|\,dt$
$\displaystyle\geq\sup_{J\in\mathcal{J}^{\tau}_{\delta}}\int_{J^{(\delta)}}d_{*}^{x}(\mu_{1}^{\delta}|_{G(x)})(t)\,dt-\|d_{*}^{x}(\mu_{1}^{\delta}|_{G(x)})-d_{*}^{x}(\mu_{1,g}^{\delta,x})\|_{L^{1}}$
$\displaystyle\geq\delta^{\beta^{2}/2}-\delta^{\beta^{2}}\geq\delta^{\beta^{2}}.$

Then by Cauchy-Schwarz, we have for $x\in F^{\prime}$,

$\left(\sup_{J\in\mathcal{J}_{\delta}^{\tau}}|J^{\delta}|^{\frac{1}{2}}\right)\left(\int|d_{*}^{x}(\mu_{1,g}^{\delta,x})(t)|^{2}dt\right)^{\frac{1}{2}}\geq\sup_{J\in\mathcal{J}^{\tau}_{\delta}}\int_{J^{(\delta)}}|d_{*}^{x}(\mu_{1,g}^{\delta,x})(t)|\,dt\geq\delta^{\beta^{2}}.$

For $J\in\mathcal{J}_{\delta}^{\tau}$, $J$ and $J^{(\delta)}$ can both be
covered by $\lesssim\delta^{-\tau}$ many intervals each of length
$\sim\delta$, so
$\sup_{J\in\mathcal{J}_{\delta}^{\tau}}|J^{\delta}|^{1/2}\lesssim\delta^{(1-\tau)/2}$.
Thus, for $x\in F^{\prime}$,

$\|d_{*}^{x}(\mu_{1,g}^{\delta,x})\|^{2}_{L^{2}}\geq\delta^{2\beta^{2}-(1-\tau)}q.$

Integrate over $F^{\prime}$ and apply Proposition 2.2 to get

$\displaystyle\delta^{2\beta^{2}-(1-\tau)}\mu_{2}(F^{\prime})$
$\displaystyle\leq\int\|d_{*}^{x}(\mu_{1,g}^{\delta,x})\|^{2}_{L^{2}}d\mu_{2}(x)$
$\displaystyle\leq\int|\widehat{\mu_{1}^{\delta}}(\xi)|^{2}|\xi|^{-\frac{\alpha
d}{d+1}+\varepsilon}\,d\xi+\delta^{-d\beta}+\mathrm{RapDec}(\delta)$
$\displaystyle\leq\int|\widehat{\mu_{1}}(\xi)|^{2}|\widehat{\phi}(\delta\xi)|^{2}|\xi|^{-\frac{\alpha
d}{d+1}+\varepsilon}\,d\xi+\delta^{-d\beta}+\mathrm{RapDec}(\delta)$
$\displaystyle\leq\int_{|\xi|\leq\delta^{-1-\beta}}|\widehat{\mu_{1}}(\xi)|^{2}|\xi|^{-\frac{\alpha
d}{d+1}+\varepsilon}\,d\xi+\delta^{-d\beta}+\mathrm{RapDec}(\delta)\,.$

Since $\tau<1$, the condition $|\xi|\leq\delta^{-1-\beta}$, which implies
$|\xi|^{1-\beta}\leq\delta^{-(1+\beta)(1-\beta)}\leq\delta^{-1}$, gives us

$\delta^{-3\beta^{2}+(1-\tau)}\leq|\xi|^{(3\beta^{2}-(1-\tau))(1-\beta)}=|\xi|^{-(1-\tau)+O(\beta)}.$

Thus,

$\displaystyle\mu_{2}(F^{\prime})$ $\displaystyle\leq$
$\displaystyle\delta^{-2\beta^{2}+(1-\tau)}\int_{|\xi|\leq\delta^{-1-\beta}}|\widehat{\mu_{1}}(\xi)|^{2}|\xi|^{-\frac{\alpha
d}{d+1}+\varepsilon}\,d\xi+\delta^{-d\beta-2\beta^{2}+(1-\tau)}+\mathrm{RapDec}(\delta)$
$\displaystyle\leq$
$\displaystyle\delta^{\beta^{2}}\int|\widehat{\mu_{1}}(\xi)|^{2}|\xi|^{-\frac{\alpha
d}{d+1}+\varepsilon-(1-\tau)+O(\beta)}\,d\xi+\delta^{-d\beta-2\beta^{2}+(1-\tau)}+\mathrm{RapDec}(\delta)\,.$

Finally, since $\tau<f(\alpha)=\alpha\cdot\frac{2d+1}{d+1}-(d-1)$ and
$\tau<1$, we may choose small $\varepsilon$ and $\beta$ such that

$\displaystyle-d\beta-2\beta^{2}+(1-\tau)>\beta^{2},$
$\displaystyle-\frac{\alpha d}{d+1}+\varepsilon-(1-\tau)+O(\beta)<-d+\alpha.$

This guarantees the energy integral

$\int|\widehat{\mu_{1}}(\xi)|^{2}|\xi|^{-\frac{\alpha
d}{d+1}+\varepsilon-(1-\tau)+O(\beta)}\,d\xi$

to be finite (in fact, $\lesssim C_{\mu}$) and so
$\mu_{2}(F^{\prime})\lesssim\delta^{\beta^{2}}$. ∎

We will also need a result of Shmerkin to prove Theorem 1.3.

######  Theorem 6.3.

[25, Theorem 6.3 and Theorem B.1] Fix $1\leq k\leq d-1$, $c>0$. Given
$\kappa_{1},\kappa_{2}>0$, there is $\gamma>0$ (depending continuously on
$\kappa_{1},\kappa_{2}$) such that the following holds.

Let $\mu,\nu$ be probability measures on $B^{d}(0,1)$ satisfying decay
conditions

$\displaystyle\mu(V)$ $\displaystyle\leq C_{\mu}r^{\kappa_{1}},$
$\displaystyle\nu(V)$ $\displaystyle\leq C_{\nu}r^{\kappa_{2}},$

for any $(r,k-1)$-plate $V$, and $0<r\leq 1$. Suppose $\nu$ gives zero mass to
every $k$-dimensional affine plane. Then for all $x$ in a set $E$ of
$\mu$-measure $\geq 1-c$ there is a set $K(x)$ with $\nu(K(x))\geq 1-c$ such
that

(6.2) $\nu(W\cap K(x))\leq r^{\gamma},$

for any $r\in(0,r_{0}]$ and any $(r,k)$-plate $W$ passing through $x$, where
$r_{0}>0$ depends only on $d,\mu,C_{\nu},\kappa_{2},c$.

Finally, the set $\\{(x,y):x\in E,y\in K(x)\\}$ is compact.

We now prove the following theorem that implies the second claim of Theorem
1.3 (that $\sup_{x\in E}\dim_{H}(\Delta_{x}(E))\geq\min(f(\alpha),1)$).
(Specifically, apply Theorem 6.4 with $\alpha-\varepsilon$ for any
$\varepsilon>0$.) The first and third claims then follow by applying the
second claim of Theorem 1.3 to the set $\\{x\in
E:\dim_{H}(\Delta_{x}(E))\leq\min(f(\alpha),1)-\varepsilon\\}$ and taking a
sequence of $\varepsilon_{n}\to 0$.

######  Theorem 6.4.

Let $0<\alpha\leq d-1$. Suppose $E_{1},E_{2}\subset B^{d}(0,1)$ are separated
by $\sim 1$, and each of them has positive $\alpha$-dimensional Hausdorff
measure. Then there exists $x\in E_{1}\cup E_{2}$ with
$\dim_{H}(\Delta_{x}(E_{1}))\geq\min(f(\alpha),1)$.

###### Proof.

Let $1\leq k\leq d-1$, $k-1<\alpha\leq k$. Let $\mu_{1},\mu_{2}$ be
$\alpha$-dimensional measures supported on $E_{1},E_{2}$ respectively. Suppose
$\mu_{1}$ gives nonzero mass to some $k$-dimensional affine plane $H$. We have
three possible cases:

  * •

If $k\geq 3$, then $\alpha>k-1\geq\frac{k+1}{2}$, so by [21], there exists
$x\in E_{1}$ such that $|\Delta_{x}(E_{1})|>0$.

  * •

If $k=2$, then by [15, Theorem 1.1] we have
$\dim_{H}(\Delta_{x}(E_{1}))\geq\min(f(\alpha),1)$ for some $x\in E_{1}$.

  * •

If $k=1$, then for all $x\in E_{1}\cap H$, we have
$\dim_{H}(\Delta_{x}(E_{1}))\geq\dim_{H}(E_{1}\cap
H)\geq\alpha>f(\alpha)=\min(f(\alpha),1)$.

Now assume $\mu_{1}$ gives zero mass to every $k$-dimensional affine plane.
Also, note that $\mu_{1}$ and $\mu_{2}$ satisfy decay conditions

$\mu_{i}(V)\lesssim C_{\mu_{i}}r^{\alpha-(k-1)},\quad i=1,2,$

for any $(r,k-1)$-plate $V$, and $0<r\leq 1$. Then we can use Theorem 6.3 to
find $r_{0},\gamma>0$ and a set $E_{2}^{\prime}\subset E_{2}$ with
$\mu_{2}(E_{2}^{\prime})\geq\frac{1}{2}$ such that for any $x\in
E_{2}^{\prime}$, there exists $K(x)\subset E_{1}$ with
$\mu_{1}(K(x))\geq\frac{1}{2}$ such that for any $(r,k)$-plate $H$ containing
$x$ with $r\leq r_{0}$,

$\mu_{1}(K(x)\cap H)\leq r^{\gamma}.$

Additionally, the set $\\{(x,y):x\in E_{2}^{\prime},y\in K(x)\\}$ is compact,
which in particular means that $K(x)$ is compact for all $x\in
E_{2}^{\prime}$.

Fix $0<\tau<\min(f(\alpha),1)$. We apply Proposition 6.2 at all sufficiently
small dyadic scales $\delta$. By the Borel-Cantelli lemma, a.e. $x\in
E_{2}^{\prime}$ lie in finitely many of the $F_{\delta}$. For such $x$, we
have for all sufficiently small $\delta>0$ and any $J$ which is a union of
$\leq\delta^{-\tau}$ many intervals each of length $\sim\delta$,

$\displaystyle\mu_{1}(\\{y:|x-y|$ $\displaystyle\in J\\}\cap K(x))$
$\displaystyle\leq\mu_{1}(\\{y:|x-y|\in J\\}\setminus W(x))+\mu_{1}(K(x)\cap
W(x))$ $\displaystyle\lesssim\delta^{\beta^{2}/2}+\delta^{\gamma\beta^{2}}.$

Hence, by Lemma 6.1 applied to the restricted measure $\mu_{1}|_{K(x)}$, we
see that $\dim_{H}(\Delta_{x}(E_{1}))\geq\tau$ for a.e. $x\in E_{2}^{\prime}$.
Taking a sequence of $\tau\to\min(f(\alpha),1)$, we see that
$\dim_{H}(\Delta_{x}(E_{1}))\geq\min(f(\alpha),1)$ for a.e. $x\in
E_{2}^{\prime}$. ∎

##  7\. Other norms and connections with Erdős distance problem

Theorem 1.2 also extends to more general norms. For a symmetric convex body
$K$ in $\mathbb{R}^{d}$, let $\|\cdot\|_{K}$ be the norm with unit ball $K$.

######  Theorem 7.1.

Let $d\geq 3$. Let $K$ be a symmetric convex body in $\mathbb{R}^{d}$ whose
boundary is $C^{\infty}$ smooth and has strictly positive curvature. Let
$E\subset\mathbb{R}^{d}$ be a compact set. Suppose that
$\dim_{H}(E)>\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$. Then there is a point
$x\in E$ such that the pinned distance set $\Delta_{K,x}(E)$ has positive
Lebesgue measure, where

$\Delta_{K,x}(E):=\\{\|x-y\|_{K}:\,y\in E\\}.$

The argument for this generalization is similar to that in [10]. Indeed, the
definition of bad tubes and heavy plates depends only on the geometry of
$\mathbb{R}^{d}$ and not the specific norm involved (note that the
$r$-neighborhood of a set $A$ is still defined via Euclidean metric, not the
new norm’s metric). The change of norm only affects the conversion from
geometry to analysis, as manifested in Lemma 4.6(a) and our use of Liu’s
$L^{2}$-identity [14, Theorem 1.9] in the proof of Proposition 2.2. These
considerations were already done in [10] (see also [5]); we omit the details.

As discussed in [10, 5], one can go from Falconer-type results to Erdős-type
results.

######  Definition 7.2.

Let $P$ be a set of $N$ points contained in ${[0,1]}^{d}$. Define the measure

(7.3) $d\mu^{s}_{P}(x)=N^{-1}\cdot N^{\frac{d}{s}}\cdot\sum_{p\in
P}\chi_{B}(N^{\frac{1}{s}}(x-p))\,dx,$

where $\chi_{B}$ is the indicator function of the ball of radius $1$ centered
at the origin. We say that $P$ is _$s$ -adaptable_ if there exists $C$
independent of $N$ such that

(7.4)
$I_{s}(\mu_{P})=\int\int{|x-y|}^{-s}\,d\mu^{s}_{P}(x)\,d\mu^{s}_{P}(y)\leq C.$

It is not difficult to check that if the points in set $P$ are separated by
distance $cN^{-1/s}$, then (7.4) is equivalent to the condition

(7.5) $\frac{1}{N^{2}}\sum_{p\not=p^{\prime}}{|p-p^{\prime}|}^{-s}\leq C,$

where the exact value of $C$ may be different from line to line. In dimension
$d$, it is also easy to check that if the distance between any two points of
$P$ is $\gtrsim N^{-1/d}$, then (7.5) holds for any $s\in[0,d)$, and hence $P$
is $s$-adaptable.

Using the same argument as in [5], from Theorem 7.1 we get the following
Erdős-type result.

######  Proposition 7.3.

Let $d\geq 3$. Let $K$ be a symmetric convex body in $\mathbb{R}^{d}$ whose
boundary is $C^{\infty}$ smooth and has strictly positive curvature. Let $P$
be a set of $N$ points contained in $[0,1]^{d}$.

(a). If the distance between any two points of $P$ is $\gtrsim N^{-1/d}$, then
there exists $x\in P$ such that

$\left|\Delta_{K,x}(P)\right|\gtrapprox
N^{\frac{1}{\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}}}=N^{\frac{2d+1}{d(d+1)}}.$

(b). More generally, if $P$ is $s_{n}$-adaptable for a decreasing sequence
$(s_{n})_{n=1}^{\infty}$ converging to
$\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, then there exists $x\in P$ such that

$\left|\Delta_{K,x}(P)\right|\gtrapprox N^{\frac{2d+1}{d(d+1)}}.$

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Xiumin Du, Northwestern University<EMAIL_ADDRESS>

Yumeng Ou, University of Pennsylvania<EMAIL_ADDRESS>

Kevin Ren, Princeton University<EMAIL_ADDRESS>

Ruixiang Zhang, UC Berkeley<EMAIL_ADDRESS>