# Analytic capacity and dimension
of sets with plenty of big projections

Damian Dąbrowski  Damian Dąbrowski, University of Jyväskylä, P.O. Box 35
(MaD), 40014, Finland<EMAIL_ADDRESS>and  Michele Villa  Michele
Villa, Research Unit of Mathematical Sciences, University of Oulu, P.O. Box
8000, FI-90014, University of Oulu, Finland<EMAIL_ADDRESS>
###### Abstract.

Our main result marks progress on an old conjecture of Vitushkin. We show that
a compact set in the plane with plenty of big projections (PBP) has positive
analytic capacity, along with a quantitative lower bound. A higher dimensional
counterpart is also proved for capacities related to the Riesz kernel,
including the Lipschitz harmonic capacity. The proof uses a construction of a
doubling Frostman measure on a lower content regular set, which may be of
independent interest.

Our second main result is the Analyst’s Traveling Salesman Theorem for sets
with plenty of big projections. As a corollary, we obtain a lower bound for
the Hausdorff dimension of uniformly wiggly sets with PBP. The second
corollary is an estimate for the capacities of subsets of sets with PBP, in
the spirit of the quantitative solution to Denjoy’s conjecture.

###### Key words and phrases:

analytic capacity, orthogonal projections, Vitushkin’s conjecture, Besicovitch
projection theorem, quantitative rectifiability, $\beta$-numbers, Analyst’s
Traveling Salesman Theorem, wiggly sets

###### 1991 Mathematics Subject Classification:

28A75 (primary), 28A80, 42B20 (secondary)

D.D. and M.V. were supported by the Academy of Finland via the project
Incidences on Fractals, grant No. 321896. M.V. was also supported by a
starting grant of the University of Oulu

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Vitushkin’s conjecture
  4. 4 Traveling Saleseman Theorem
  5. 5 Dimension of wiggly sets
  6. 6 Denjoy’s conjecture
  7. A Frostman’s lemma for lower content regular sets

##  1\. Introduction

The aim of this paper is to study sets with plenty of big projections.

######  Definition 1.1.

We say that $E\subset\mathbb{R}^{n}$ has _plenty of ( $d$-dimensional) big
projections_ (abbreviated to PBP, or $d$-PBP) if there exists a constant
$0<\delta\leq 1$ such that the following holds. For all $x\in E$ and
$0<r<\operatorname{diam}(E)$ there exists $V_{x,r}\in{\mathcal{G}}(n,d)$ such
that

$\mathcal{H}^{d}(\pi_{V}(E\cap B(x,r)))\geq\delta r^{d}\quad\text{for all
}V\in B(V_{x,r},\delta).$

Here ${\mathcal{G}}(n,d)$ is the Grassmannian manifold of $d$-dimensional
(linear) planes in $\mathbb{R}^{n}$,
$\pi_{V}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{n}\to V$ is the
orthogonal projection, and the ball $B(V_{x,r},\delta)$ is defined with
respect to the standard metric on ${\mathcal{G}}(n,d)$ (see §2.1).

This definition first appeared in [DS93], although originally David and Semmes
assumed additionally that sets with PBP are Ahlfors regular.

######  Definition 1.2.

A set $E\subset\mathbb{R}^{n}$ is called _Ahlfors $d$-regular_ if there exists
a constant $C\geq 1$ such that for all $x\in E$ and
$0<r<\operatorname{diam}(E)$ we have

$C^{-1}r^{d}\leq\mathcal{H}^{d}(E\cap B(x,r))\leq Cr^{d}.$

The smallest constant $C$ for which the above holds is called the _Ahlfors
regularity constant_ of $E$.

In [DS93] David and Semmes conjectured that Ahlfors regular sets with PBP
satisfy the so-called _big pieces of Lipschitz graphs condition_ , a strong
quantitative rectifiability property (see Definition 2.4). Their conjecture
was recently solved in a breakthrough work of Orponen [Orp21] (see Theorem 2.5
for the precise statement).

In this article we use Orponen’s result to further study sets with PBP. The
main novelty of our work stems from the fact that we do not assume our sets to
be Ahlfors regular. In fact, some of our results remain valid for sets with
non-$\sigma$-finite $\mathcal{H}^{d}$-measure.

###  1.1. Analytic capacity: Vitushkin’s conjecture and quantitative Denjoy’s
conjecture

Recall that a compact set $E\subset{\mathbb{C}}\simeq\mathbb{R}^{2}$ is said
to be _removable for bounded analytic functions_ if all bounded analytic
functions $f$ defined on ${\mathbb{C}}\setminus E$ are constant. In the 40s
Ahlfors [Ahl47] managed to quantify the notion of removability by introducing
_analytic capacity_. Recall that the analytic capacity of a compact set
$E\subset{\mathbb{C}}$ is defined as

$\gamma(E)=\sup|f^{\prime}(\infty)|,$

where the supremum is taken over all analytic functions
$f\mathrel{\mathop{\ordinarycolon}}{\mathbb{C}}\setminus E\to{\mathbb{C}}$
with $\|f\|_{\infty}\leq 1$, and
$f^{\prime}(\infty)=\lim_{z\to\infty}z(f(z)-f(\infty))$. Ahlfors proved that a
set is removable for bounded analytic functions if and only if $\gamma(E)=0$.

In 1967 Vitushkin [Vit67] conjectured a geometric characterization of
removability in terms of orthogonal projections. He asked the following: is it
true that for compact sets $E\subset{\mathbb{C}}$ one has

$\gamma(E)=0\quad\Leftrightarrow\quad\operatorname{Fav}(E)=0,$ (1.1)

where $\operatorname{Fav}(E)$ is the Favard length of $E$, defined as

$\operatorname{Fav}(E)=\int_{0}^{\pi}\mathcal{H}^{1}(\pi_{\theta}(E))\
d\theta.$

In the above
$\pi_{\theta}\mathrel{\mathop{\ordinarycolon}}{\mathbb{C}}\to\ell_{\theta}$ is
the orthogonal projection to
$\ell_{\theta}=\mathop{\mathrm{missing}}{span}((\sin\theta,\cos\theta))$.

It has been known for a very long time that sets with $\mathcal{H}^{1}(E)=0$
are removable, while sets with $\mathop{\mathrm{missing}}{dim}_{H}(E)>1$ are
non-removable, so Vitushkin’s conjecture holds for such sets. The case of sets
$E$ satisfying $0<\mathcal{H}^{1}(E)<\infty$ was much more difficult to
establish, but the answer to Vitushkin’s question is also positive: the
implication $(\Rightarrow)$ in (1.1) is due to Calderón [Cal77], while
$(\Leftarrow)$ was shown by David [Dav98]. Finally, the case of sets with
$\sigma$-finite $\mathcal{H}^{1}$-measure follows from the finite measure case
together with subadditivity of analytic capacity, which is due to Tolsa
[Tol03].

In 1986 Mattila [Mat86] showed that Vitushkin’s conjecture fails for sets with
Hausdorff dimension $1$ and non-$\sigma$-finite $\mathcal{H}^{1}$-measure. It
wasn’t clear from Mattila’s proof which of the implications in (1.1) is false.
Soon thereafter Jones and Murai [JM88] constructed an example of a set with
$\gamma(E)>0$ and $\operatorname{Fav}(E)=0$. A simpler example was found later
on by Joyce and Mörters [JM00]. Hence, the implication $(\Leftarrow)$ in (1.1)
is false. For more information on analytic capacity and Vitushkin’s conjecture
see the books [Tol14, Dud11, Paj02]. See also recent surveys on related topics
[Ver21, Mat21].

After all these extraordinary developments one question remains: what about
the implication $(\Rightarrow)$ in (1.1)? Equivalently, is it true that

$\operatorname{Fav}(E)>0\quad\Rightarrow\quad\gamma(E)>0?$ (1.2)

A quantitative version of this question is the following: is it true that

$\gamma(E)\gtrsim\operatorname{Fav}(E)?$ (1.3)

As far as we know, the only partial result related to these open problems is
due to Chang and Tolsa [CT20] (more on that in Remark 1.8). In this article we
make further progress on (1.2) and (1.3). If the assumption
$\operatorname{Fav}(E)>0$ is replaced by the (significantly stronger) PBP
assumption, then the analytic capacity is positive.

######  Theorem 1.3.

Let $E\subset\mathbb{R}^{2}$ be a compact set with $1$-PBP. Then,

$\gamma(E)\gtrsim\operatorname{diam}(E),$ (1.4)

where the implicit constant depends only on the PBP constant.

######  Remark 1.4.

In the theorem above we do not assume $\mathcal{H}^{1}(E)<\infty$. However,
our result bears new information even in the $\mathcal{H}^{1}(E)<\infty$ case.
Recall that in this regime Vitushkin’s conjecture is true, so in particular
$\operatorname{Fav}(E)>0$ implies $\gamma(E)>0$. However, the existing proof
does not offer any quantitative lower bound on $\gamma(E)$ in terms of
$\operatorname{Fav}(E)$. Roughly speaking, the reason is the following: the
Besicovitch projection theorem states that if $\operatorname{Fav}(E)>0$, then
there exists a $1$-dimensional Lipschitz graph $\Gamma$ such that
$\mathcal{H}^{1}(E\cap\Gamma)>0$, and then
$\gamma(E)\geq\gamma(E\cap\Gamma)>0$ follows from [Cal77]. However, the
Besicovitch projection theorem does not say anything about the size of
$\mathcal{H}^{1}(E\cap\Gamma)$, and so deriving a quantitative lower bound for
$\gamma(E)$ is impossible as long as one follows this strategy. In other
words, the question (1.3) remains wide open even in the
$\mathcal{H}^{1}(E)<\infty$ case. Although Theorem 1.3 is quite far from
establishing (1.3), it is (as far as we know) the first quantitative lower
bound for $\gamma(E)$ which depends only on the projections of $E$.

In the finite measure case, we are able to prove more: we show a quantitative
bound for the analytic capacity of subsets of sets with $1$-PBP. This is a
form of quantitative Denjoy’s conjecture, where rectifiable curves are
substituted with sets with PBP of finite measure. Recall that Denjoy’s
conjecture, which predates Vitushkin’s conjecture by about 60 years, stated
that if $\Gamma\subset\mathbb{R}^{2}$ is a rectifiable curve, and a compact
set $E\subset\Gamma$ satisfies $\mathcal{H}^{1}(E)>0$, then $\gamma(E)>0$.
This statement can be seen as a special case of Vitushkin’s conjecture, if one
restricts attention to the implication (1.2) and sets of finite length.
Denjoy’s conjecture was confirmed by Calderón in [Cal77]. A quantitative
version of Calderón’s result was established by Murai [Mur87], who showed that
if $\Gamma$ is a $1$-rectifiable graph, and $E\subset\Gamma$ is compact, then

$\gamma(E)\gtrsim\frac{\mathcal{H}^{1}_{\infty}(E)^{\frac{3}{2}}}{\mathcal{H}^{1}(\Gamma)^{\frac{1}{2}}}.$

Later on Verdera observed that Murai’s estimate can be obtained for arbitrary
rectifiable curves $\Gamma$ using Menger curvature and Jones’ Travelling
Salesman Theorem, see [Tol14, Theorem 4.31]. The exponent $3/2$ above is
optimal, see [Mur90] and [Tol14, §4.8].

Using an Analyst’s Travelling Salesman Theorem for sets with PBP (see Theorem
1.10 below) we obtain the following.

######  Theorem 1.5.

Let $\Gamma\subset\mathbb{R}^{2}$ be a compact set with $1$-PBP and
$0<\mathcal{H}^{1}(\Gamma)<\infty$. Then, for any compact subset
$E\subset\Gamma$

$\displaystyle\gamma(E)\gtrsim\frac{\mathcal{H}^{1}_{\infty}(E)^{\frac{3}{2}}}{\mathcal{H}^{1}(\Gamma)^{\frac{1}{2}}},$

where the implicit constant depends only on the PBP constant.

Theorem 1.5 is interesting for two reasons: first, as mentioned above, it
applies to subsets of sets with PBP. These subsets could very well have
vanishing density in many places, and have very small projections. Note that a
similar statement cannot be true for sets $\Gamma$ with infinite length: for
example, the unit square $[0,1]^{2}$ has PBP, but the 4-corner Cantor set
$K\subset[0,1]^{2}$ satisfies $\mathcal{H}^{1}(K)\sim 1$ and $\gamma(K)=0$.

The second interesting aspect of Theorem 1.5 is that it suggests that, for
many geometric problems, sets with PBP and finite measure are just as good as
rectifiable curves. Let us however make the following remark: from the
Analyst’s TST for sets with PBP (Theorem 1.10 below) it follows that a set
with PBP and finite $\mathcal{H}^{1}$-measure can be covered by a rectifiable
curve of comparable length. Thus, Theorem 1.5 can be derived directly from
Theorem 1.10 together with the previous result of Murai and Verdera. However,
in higher dimension (see the section below) the result is altogether new,
since no result analogous to that of Murai was known for capacities associated
to vector-valued Riesz kernel.

###  1.2. Higher dimensional variants

Higher dimensional variants of Theorem 1.3 and Theorem 1.5 are also true.
Instead of analytic capacity one has to consider certain capacities associated
to the vector-valued Riesz kernel. Let $0<d<n$ be integers. Given a compact
set $E\subset\mathbb{R}^{n}$ the capacity $\Gamma_{n,d}(E)$ is defined as

$\Gamma_{n,d}(E)=\sup|\langle T,1\rangle|,$

where the supremum is taken over all real distributions $T$ supported in $E$
such that

$\frac{x}{|x|^{d+1}}\ast T\in
L^{\infty}(\mathbb{R}^{n})\quad\text{and}\quad\bigg{\|}\frac{x}{|x|^{d+1}}\ast
T\bigg{\|}_{L^{\infty}(\mathbb{R}^{n})}\leq 1.$

We remark that by [Tol03] one has $\Gamma_{2,1}(E)\sim\gamma(E)$. In the
codimension 1 case the capacity $\Gamma_{d+1,d}(E)$ is also called the
_Lipschitz harmonic capacity_ , and it is often denoted by $\kappa(E)$. It was
introduced by Paramonov, who observed that sets with $\kappa(E)=0$ are
precisely the sets removable for Lipschitz harmonic functions, see [Par90,
Remark 2.4] and [MP95]. The counterpart of Vitushkin’s conjecture for
$\kappa(E)$ and sets with $\mathcal{H}^{d}(E)<\infty$ was established by
Nazarov, Tolsa and Volberg in [NTV14b, NTV14a]. More specifically, they showed
that for a compact set $E\subset\mathbb{R}^{d+1}$ with
$\mathcal{H}^{d}(E)<\infty$ one has $\kappa(E)=0$ if and only if $E$ is purely
$d$-unrectifiable (equivalently, $\mathcal{H}^{d}(\pi_{V}(E))=0$ for a.e.
$V\in\mathcal{G}(n,d)$). Whether an analogous statement is true for
$\Gamma_{n,d}$ with $1<n<d-1$ is an open problem.

The higher dimensional variants of Theorem 1.3 and Theorem 1.5 are the
following.

######  Theorem 1.6.

Let $E\subset\mathbb{R}^{n}$ be a compact set with $d$-PBP. Then,

$\Gamma_{n,d}(E)\gtrsim\operatorname{diam}(E)^{d},$ (1.5)

where the implicit constant depends only on $n,d,$ and the PBP constant.

######  Theorem 1.7.

Let $\Sigma\subset\mathbb{R}^{n}$ be a compact set with $d$-PBP and
$0<\mathcal{H}^{d}(\Sigma)<\infty$. Then for any compact subset
$E\subset\Sigma$,

$\displaystyle\Gamma_{n,d}(E)\gtrsim\frac{\mathcal{H}^{d}_{\infty}(E)^{\frac{3}{2}}}{\mathcal{H}^{d}(\Sigma)^{\frac{1}{2}}},$

where the implicit constant depends only on $n,d$, and the PBP constant.

######  Remark 1.8.

Theorems 1.3 and 1.6 should be compared with the results of Chang and Tolsa
[CT20], who proved the following. If $I\subset[0,\pi)$ is an interval and
$E\subset\mathbb{C}$ is a compact set supporting a probability measure $\mu$
such that $\pi_{\theta}\mu\in L^{2}(\ell_{\theta})$ for a.e. $\theta\in I$,
then

$\gamma(E)\gtrsim\frac{1}{\int_{I}\|\pi_{\theta}\mu\|^{2}_{L^{2}}\,d\theta},$

with the implicit constant depending on $\mathcal{H}^{1}(I)$. They also prove
a higher dimensional analogue of this estimate for capacities $\Gamma_{n,d}$.

The main advantage of [CT20] over our results is that their
“$L^{2}$-projections” assumption is single-scale, while the PBP condition is
multi-scale. On the other hand, our result may be easier to apply as it
derives a lower bound for $\gamma(E)$ directly from the information on the
projections of $E$, without the need of constructing a measure supported on
$E$ and studying its projections.

Even more importantly, note that the $L^{2}$-projections assumption
$\pi_{\theta}\mu\in L^{2}(\ell_{\theta})$ implies a big projection
$\mathcal{H}^{1}(\pi_{\theta}(E))\geq\|\pi_{\theta}\mu\|^{-2}_{2}$ using the
Cauchy-Schwarz inequality

$1=\mu(E)=\|\pi_{\theta}\mu\|_{1}\leq\|\pi_{\theta}\mu\|_{2}\,\mathcal{H}^{1}(\pi_{\theta}(E))^{1/2}.$

The difference between $L^{2}$-projections and big projections is fundamental.
A characterization of Ahlfors regular sets with big pieces of Lipschitz graphs
in terms of $L^{2}$-projections has been achieved by Martikainen and Orponen
in [MO18], but it took another major breakthrough [Orp21] to find an analogous
characterization in terms of PBP. Moreover, a “single-scale version” of
[Orp21] is an open problem, see [Orp21, §1.3], whereas [MO18] contains the
relevant single-scale result, see [MO18, Theorem 1.7].

######  Remark 1.9.

The proof of Theorem 1.6 is robust, and our techniques are likely to be useful
in the future work on problems (1.2) and (1.3).

In particular, suppose that the following strengthening of Orponen’s result
(Theorem 2.5) was available: “if a set $E\subset\mathbb{R}^{n}$ is Ahlfors
$d$-regular and it has _uniformly large Favard length_ (there exists $C>0$
such that for all $x\in E$ and all $0<r<\operatorname{diam}(E)$ we have
$\operatorname{Fav}(E\cap B(x,r))\geq Cr^{d}$), then $E$ has big pieces of
Lipschitz graphs.” With such result at hand, the proof of Theorem 1.6 would
immediately yield that any set $E\subset\mathbb{R}^{d}$ with uniformly large
Favard length satisfies

$\Gamma_{n,d}(E)\gtrsim\operatorname{diam}(E)^{d}.$

A related question is of course the following: suppose that
$E\subset[0,1]^{2}$ is a $1$-Ahlfors regular set with ${\rm
Fav}(E)\geq\delta$. Is it then true that there exists a Lipschitz graph
$\Gamma$ so that $\mathcal{H}^{1}(E\cap\Gamma)\gtrsim\delta$? A recent related
result by A. Chang, T. Orponen and the authors [CDOV22] shows that if $E$ has
almost maximal Favard length, then it is contained in a Lipschitz graph with
small constant, save for a tiny subset. A natural question is whether this
result can be used to obtain a new estimate for analytic capacity.

###  1.3. Analyst’s Traveling Salesman Theorem

Another main result of this article is the Analyst’s Travelling Salesman
Theorem (TST) for sets with PBP. Recall that the original Analyst’s TST is due
to Jones [Jon90], and it is a characterization of sets
$E\subset\mathbb{R}^{2}$ such that there exists a rectifiable curve $\Gamma$
containing $E$, along with a quite precise estimate on
$\mathcal{H}^{1}(\Gamma)$. Much work has been put into proving analogs of this
result in other spaces; for example, Okikiolu [Oki92] proved an Analyst’s TST
for curves in $\mathbb{R}^{d}$, and in [Sch07] Schul further generalized it to
the Hilbert space setting. Another research direction consists in finding
statements analogous to Jones’ TST but for higher dimensional sets rather than
curves. While we are still lacking a theorem as complete as that of Jones, in
recent years there has been much progress in this direction. Two major
difficulties were: first, the coefficients used by Jones simply do not work
for higher dimensional set; second, it is not obvious what to use as an
analogue to curves (e.g. topological sphere do not work, see [Vil19,
Introduction]). These issues have largely been overcome in the series of
papers [AS18, AV21, Vil19, Hyd20, Hyd21]. We refer the reader to Sections 1
and 3 of [Vil19] for a more in-depth discussion of Analyst’s TST and its
relevance. Let us mention that while these results are related to the
rectifiability of sets, they do not say much about whether these sets can be
parameterised. This is a major open problem in the area.

Our present result continues the line of work of Azzam, Schul, and the second
author [AS18, AV21]. Let $\mathcal{D}$ be a system of Christ-David cubes (see
Lemma 2.1), and set

$\displaystyle\beta(Q_{0})=\beta_{E,C_{0},p,d}(Q_{0})\mathrel{\mathop{\ordinarycolon}}=\sum_{Q\in\mathcal{D}(Q_{0})}\beta_{E}^{d,p}(C_{0}B_{Q})^{2}\ell(Q)^{d},$
(1.6)

where the coefficients $\beta_{E}^{d,p}$ are a variant of Jones’
$\beta$-numbers (1.9) introduced by Azzam and Schul in [AS18], see Definition
2.12.

######  Theorem 1.10.

Let $E\subset\mathbb{R}^{n}$ be a set with $d$-PBP with constant $\delta>0$,
$\mathcal{D}$ be a system of Christ-David cubes, $Q_{0}\in\mathcal{D}$ and
$C_{0}\geq 3$. Let $1\leq p<p(d)$, where $p(d)=\tfrac{2d}{d-2}$ for $d>2$ and
$p(d)=\infty$ if $d\leq 2$. Then

$\displaystyle\operatorname{diam}(Q_{0})^{d}+\beta(Q_{0})\sim\mathcal{H}^{d}(Q_{0}),$
(1.7)

where the implicit constant depends on $\delta$, $C_{0}$, $p$, $n,d$ and on
the constants from the Azzam-Schul TST (see Theorem A.1 in [AV21]).

######  Remark 1.11.

What we really prove here is one direction of the inequality (bound on the
$\beta$ sum with the measure), as the other one follows from [AS18]. See
Proposition 4.4.

Estimates similar to (1.7) have been proven in [AS18] and [AV21] for general
_lower content regular sets_ (see §2.3 for the definition; sets with PBP are
lower content regular), with the following crucial caveat: in general, an
additional error term needs to be added to the right hand side of (1.7). In
[Vil19] the second author proved that the error term may be omitted if one
assumes that $E$ is a _topologically stable $d$-surface_ (a condition
satisfied e.g. by Reifenberg flat sets, or Semmes surfaces), see [Vil19,
Theorem 3.6]. In Theorem 1.10 we prove that the error term disappears also in
the case of sets with PBP. In other words, both PBP and topological stability
give the set enough rigidity so that one can estimate its $\beta$-numbers
using the Hausdorff measure.

To see why the estimate (1.7) may be useful, let us mention that in [AS18,
Theorem II] it was shown that a lower content regular set with
$\beta(E)<\infty$ is rectifiable. Thus, any class of lower content regular
sets for which (1.7) holds satisfies

$\mathcal{H}^{d}(E)<\infty\,\implies\,\mbox{ rectifiability.}$ (1.8)

In particular, (1.8) holds for topologically stable surfaces, and for sets
with PBP (of course, for sets with PBP (1.8) easily follows from the
Besicovitch projection theorem, without the need to refer to $\beta$-numbers
and TST). Identifying classes of sets for which (1.8) holds is an interesting
problem. For example, while (1.8) is true for connected one dimensional sets,
there exist $2$-dimensional sets, homeomorphic to the $2$-sphere, with finite
$\mathcal{H}^{2}$-measure, and containing a purely 2-unrectifiable set of
positive measure (see [Vil19, Figure 1]) See also [DK19] and [DLD20] in
connection with (1.8) and quantitative topological conditions.

Theorem 1.10 is used in the proofs of Theorems 1.5 and 1.7. Another
application is an estimate for the dimension of wiggly sets.

###  1.4. Dimension of wiggly sets

A set $E$ is said to be uniformly wiggly of dimension $d$ and with parameter
$\beta_{0}$ if for all balls $B$ centered on $E$ and with
$0<r(B)<\operatorname{diam}(E)$ it holds that

$\displaystyle\beta_{E,\infty}^{d}(B)>\beta_{0}.$ (1.9)

Here $\beta_{E,\infty}^{d}$ is the $d$-dimensional version of the so-called
$L^{\infty}$ $\beta$-number of Peter Jones [Jon90] (see Definition 2.6).
Wiggly sets appear naturally in various contexts where some form of self-
similarity is present. Examples include limit sets of certain Kleinian groups,
some Julia sets of polynomials, and random sets (see [BP17], pp. 340-341).
They were first studied via the Analyst’s TST by Bishop and Jones in [BJ97],
were they proved a lower bound on the Hausdorff dimension of wiggly connected
sets in the plane, assuming $d=1$. Their result was generalised to continua in
metric spaces by Azzam [Azz15]. A result in this vein was later proved by
David [Dav04] (and quantified in [Vil19]), this time for uniformly non flat
sets of any integer dimension, satisfying a topological condition. David’s
work was motivated by a question of L. Potyagailo, concerning higher
dimensional limit sets (see the introduction of [Dav04]). The result below can
be seen as a version of David’s theorem with the topological condition
replaced by the PBP assumption.

######  Theorem 1.12.

Let $n\geq 2$ and $1\leq d\leq n-1$. Let $E\subset\mathbb{R}^{n}$ be a closed
set with $d$-PBP (with parameter $\delta>0$) and which is uniformly wiggly of
dimension $d$ and constant $\beta_{0}$. Then

${\rm dim}_{H}(E)\geq d+c\beta_{0}^{2(d+1)},$ (1.10)

where $c$ depends on $\delta,n,d,p$.

######  Remark 1.13.

We defined uniformly wiggly sets in terms of $\beta_{E,\infty}^{d}$, but, in
principle, we could have used the $L^{p}$ versions of the coefficients
instead. Indeed, if we take a set $E$ that satisfies the hypotheses of Theorem
1.12 with respect to the Azzam-Schul $\beta_{E}^{d,p}$-numbers, assuming
$1\leq p<p(d)$ (where $p(d)$ is defined in Theorem 1.10), then we obtain a
somewhat sharper dimension estimate

${\rm dim}_{H}(E)\geq d+c\beta_{0}^{2},$ (1.11)

see Proposition 5.1. This is not unexpected, since the correct
$\beta$-coefficients to use when working with higher dimensional sets are the
averaged ones.

Another comment on definitions is in order: many interesting sets satisfy non-
flatness hypotheses weaker than uniform wiggly. For example, attractors of
many dynamical systems are only mean wiggly, in the sense that the $\beta$
coefficients are large in many (but not all) scales and locations. See [GJM12,
GM22] for a definition. See also [KR97] for a result similar in spirit
concerning mean porous sets.

######  Remark 1.14.

The dependence of $\mathop{\mathrm{missing}}{dim}_{H}(E)$ on the PBP
parameters is not a proof artifact: suppose we had a lower bound for Hausdorff
diemnsion depending only on $\beta_{0}$ and not on $\delta$. Consider the four
corners Cantor set $E$, constructed in the usual way, except that we dilate by
constant $\eta>1$ the squares in the construction. In the limit, we will
obtain that a) $\mathop{\mathrm{missing}}{dim}_{H}(E)=1+C(\eta)$ with
$C(\eta)\to 0$ as $\eta\to 1$, b) $\beta_{E}^{1,\infty}(B)\gtrsim
1\geq\beta_{0}>0$ for any ball $B$ centered on $E$, and c) $E$ will have PBP
with parameters depending on $\eta$. But then, our hypothetical (and false)
theorem would tell us that $\mathop{\mathrm{missing}}{dim}_{H}(E)\geq
1+c\beta_{0}^{4}$ independently of $\eta$. This example was pointed out by T.
Orponen in a discussion on a first draft, where the exact dependence of the
dimension estimate on the PBP constant had been overlooked.

In fact, after a minute’s thinking, the dependence on $\delta$ of (1.10)
appears altogether natural: the parameter $\delta$ regulates the “rigidity” of
a set with PBP, where one should think of connected set as having ‘rigidity’
1.

###  1.5. Outline of the paper and proof ideas

In Section 2 we establish notation and state some quantitative rectifiability
results used in the paper.

In Section 3 we prove Theorem 1.6. Since $\Gamma_{2,1}(E)\sim\gamma(E)$ by
[Tol03], Theorem 1.3 follows from Theorem 1.6 by taking $n=2,\,d=1$. To prove
Theorem 1.6 we use a lower bound on $\Gamma_{n,d}$ due to Prat [Pra12] and
Girela-Sarrión [GS19], formulated in terms of measures satisfying certain
flatness condition, see Theorem 2.10. We show that if $E$ has PBP, then the
Frostman measure $\mu$ on $E$ satisfies the flatness condition, and so Theorem
2.10 yields the desired lower bound. To show that $\mu$ satisfies the flatness
condition, we conduct a stopping time argument to construct a corona
decomposition of the David-Christ lattice $\mathcal{D}$ into trees such that
on each tree the measure $\mu$ is approximately Ahlfors regular. For each
tree, we construct an approximating measure $\eta$ such that it is Ahlfors
regular and $\operatorname{supp}\eta$ satisfies the PBP condition. A recent
result of Orponen [Orp21] implies that $\eta$ satisfies the desired flatness
condition. Finally, we transfer the flatness condition back to the original
Frostman measure $\mu$.

In Section 4 we prove the TST for sets with PBP, Theorem 1.10. The idea is the
following: we start off with $E$ which has PBP. Then it is easy to see that
$E$ is lower content regular. We can then use a coronisation of lower content
regular sets by Ahlfors regular sets proved in [AV21], which gives a
decomposition of the cubes of $E$ into trees, and for each tree $\mathcal{T}$
we have an Ahlfors regular set $E_{\mathcal{T}}$. The family of root cubes of
these trees satisfy a packing condition. Each $E_{\mathcal{T}}$ inherits the
PBP from $E$, and therefore it is uniformly rectifiable by Orponen’s result
mentioned above. By the David-Semmes theory, each $E_{\mathcal{T}}$ satisfies
the strong geometric lemma. Theorem 1.10 is obtained by carefully transferring
these good estimated on $\beta$-numbers from $E_{\mathcal{T}}$ to $E$.

The proofs of Theorems 1.6 and 1.10 bear many similarities. In fact, the proof
of Theorem 1.6 hinges on what can be seen as an “Analyst’s TST for measures”
(Proposition 3.1 below). In turn, the proof of Proposition 3.1 is based on a
coronisation of Frostman measures of lower content regular sets by Ahlfors
regular measures. It is, in spirit at least, a version of the coronisation in
[AV21], where instead of working directly with a set $E$, we work with a
Frostman measure $\mu$ on $E$. This allows us to deal with sets with
non-$\sigma$-finite Hausdorff measure, a setting where the techniques from
[AV21] become useless. Another advantage of this approach, which will be
explored in future work, is that it allows us to study rectifiability of lower
content regular sets (e.g. Analyst’s TST) without the need for Azzam-Schul
$\beta$-numbers defined in terms of Hausdorff content. These object are very
useful in some respects, but difficult to work with at times.

The last two sections are dedicated to Theorem 1.12 and Theorem 1.7 (note that
Theorem 1.5 follows from Theorem 1.7 since $\gamma(E)\sim\Gamma_{2,1}(E)$).
The strategy of the proof of Theorem 1.12 is that of Bishop and Jones [BJ97],
while the proof of Theorem 1.7 follows Verdera’s [Tol14, Theorem 4.31].

In Appendix A we construct a doubling Frostman measure supported on a lower
content regular set. We use this modified Frostman measure in the proof of
Theorem 1.6. The idea of the proof is rather simple: the doubling condition is
broken if two neighbouring cubes have very unequal amount of mass. We fix this
by recursively redistributing “the wealth”, that is, the mass, form the cubes
which are too wealthy, to the poorer ones. We are grateful to Tuomas Orponen
for helping us with the construction.

### Acknowledgments

We thank Alan Chang, Matthew Hyde, Sebastiano Nicolussi Golo, Tuomas Orponen,
Polina Perstneva, and Xavier Tolsa for useful comments and discussions on the
manuscript and related topics. In particular, T. Orponen suggested us to work
on the problem (1.2), he pointed out that a previous version of Theorem 1.12
was incorrectly stated, suggesting some counterexamples, and finally he helped
us with the construction of a doubling Frostman measure from the appendix. We
are grateful to A. Chang for helping us with the proof of Lemma 3.10.

Parts of this work were completed during the conference “Rajchman, Zygmund,
and Marcinkiewicz” at IMPAN (Warsaw), and during the Trimester Programme
“Interactions between Geometric measure theory, Singular integrals, and PDE”
at HIM (Bonn). We thank the organisers of both events for their kind
hospitality.

Finally, the second author would like to thank the Mathematical Research unit
at the University of Oulu for giving him full support to pursue his research
during the first year spent at the university.

##  2\. Preliminaries

###  2.1. Notation

We gather here some notation and some results which will be used later on. We
write $a\lesssim b$ if there exists an constant $C$ such that $a\leq Cb$. If
the constant $C$ depends on a parameter $t$, we write $a\lesssim_{t}b$. By
$a\sim b$ we mean $a\lesssim b\lesssim a$.

For two subsets $A,B\subset\mathbb{R}^{n}$, we let
$\mathop{\mathrm{missing}}{dist}(A,B)\mathrel{\mathop{\ordinarycolon}}=\inf_{a\in
A,b\in B}|a-b|.$ For a point $x\in\mathbb{R}^{n}$ and a set
$A\subset\mathbb{R}^{n}$,
$\mathop{\mathrm{missing}}{dist}(x,A)\mathrel{\mathop{\ordinarycolon}}=\mathop{\mathrm{missing}}{dist}(\\{x\\},A)=\inf_{a\in
A}|x-a|.$ The cardinality of a set $A$ is denoted by $\\#A$.

We write
$B(x,r)\mathrel{\mathop{\ordinarycolon}}=\\{y\in\mathbb{R}^{n}\,\mathrel{\mathop{\ordinarycolon}}\,|x-y|<r\\},$
and, for $\lambda>0$, $\lambda
B(x,r)\mathrel{\mathop{\ordinarycolon}}=B(x,\lambda r).$ At times, we may
write $\mathbb{B}$ to denote $B(0,1)$. When necessary we write $B_{n}(x,r)$ to
distinguish a ball in $\mathbb{R}^{n}$ from one in $\mathbb{R}^{d}$, which we
may denote by $B_{d}(x,r)$. Given a ball $B$, we denote by $r(B)$ its radius.

If $\mu$ is a Radon measure on $\mathbb{R}^{n}$, then the $d$-dimensional
density of $\mu$ in the ball $B=B(x,r)$ is

$\theta_{\mu}(B)=\theta_{\mu}(x,r)=\frac{\mu(B(x,r))}{r^{d}}.$

We denote by ${\mathcal{G}}(n,d)$ the Grassmannian, that is, the manifold of
all $d$-dimensional linear subspaces of $\mathbb{R}^{n}$. A ball in
${\mathcal{G}}(n,d)$ is defined with respect to the standard metric

$\displaystyle d_{{\mathcal{G}}}(V,W)=\|\pi_{V}-\pi_{W}\|_{{\rm op}}.$

Recall that $\pi_{V}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{n}\to V$ is
the standard orthogonal projection onto $V$. With ${\mathcal{A}}(n,d)$ we
denote the affine Grassmannian, the manifold of all affine $d$-planes in
$\mathbb{R}^{n}$.

###  2.2. Dyadic lattice and Christ-David cubes

The family of dyadic cubes in $\mathbb{R}^{n}$ will be denoted by $\Delta$,
and the family of dyadic cubes with sidelength $\ell(I)=2^{-k}$ by
$\Delta_{k}$. The $d$-dimensional skeleton of $I\in\Delta$ (i.e. the union of
the $d$-dimensional faces of $I$) will be denoted by ${\partial}_{d}I$.

The idea to consider _generalized dyadic cubes_ , that is, nested partitions
of sets with nice properties, goes back to David [Dav88] and Christ [Chr90].
In many contexts it is important that these generalized cubes have thin
boundaries with respect to a given measure. Since we will not need this
property, we may use for example the cubes from [KRS12]. A special case of
their construction gives the following.

######  Lemma 2.1 ([KRS12]).

Let $E\subset\mathbb{R}^{n}$, $\rho=1/1000$ and $c_{0}=1/500.$ Then, for each
$k\in\mathbb{Z}$, there is a collection $\mathcal{D}_{k}$ of generalized cubes
on $E$ such that the following hold.

  1. (1)

For each $k\in\mathbb{Z},\ E=\bigcup_{Q\in\mathcal{D}_{k}}Q,$ and the union is
disjoint.

  2. (2)

If $Q_{1},Q_{2}\in\bigcup_{k}\mathcal{D}_{k}$ and $Q_{1}\cap
Q_{2}\neq\varnothing$, then either $Q_{1}\subset Q_{2}$ or $Q_{2}\subset
Q_{1}$.

  3. (3)

For $Q\in\bigcup_{k}\mathcal{D}_{k}$, let $k(Q)$ be the unique integer so that
$Q\in\mathcal{D}_{k}$ and set $\ell(Q)=5\rho^{k}$. Then there is $x_{Q}\in Q$
such that

$\displaystyle B(x_{Q},c_{0}\ell(Q))\cap E\subseteq Q\subseteq
B(x_{Q},\ell(Q)).$

We introduce some notation related to cubes. Given two integers $k<l$, we set
$\mathcal{D}_{k}^{l}=\bigcup_{i=k}^{l}\mathcal{D}_{i}$. If $R\in\mathcal{D}$,
we will denote the descendants of $R$ by

$\mathcal{D}(R)=\\{Q\in\mathcal{D}\ \mathrel{\mathop{\ordinarycolon}}\
Q\subset R\\},$

and $\mathcal{D}_{k}(R)=\mathcal{D}(R)\cap\mathcal{D}_{k}$. On the other hand,
$Q^{1}$ will denote the _parent_ of $Q$, i.e. the unique cube such that
$\ell(Q)=\rho\ell(Q^{1})$ and $Q\subset Q^{1}$.

The child-parent relation endows $\mathcal{D}$ with a natural tree structure.
We will say that a collection $\mathcal{T}\subset\mathcal{D}$ is a _tree_ if

  * •

$\mathcal{T}\subset\mathcal{D}(R)$ for some $R\in\mathcal{T}$. This maximal
cube $R$ will be called the _root_ of $\mathcal{T}$.

  * •

for any $Q\in\mathcal{T}$ we also have $P\in\mathcal{T}$ for all cubes
$P\in\mathcal{D}$ with $Q\subset P\subset R,$ where $R$ is the root of
$\mathcal{T}$.

The minimal cubes of $\mathcal{T}$ will be called its _stopping cubes_.

For every $Q\in\mathcal{D}$ we set
$B(Q)\mathrel{\mathop{\ordinarycolon}}=B(x_{Q},c_{0}\ell(Q))$ and
$B_{Q}=B(x_{Q},\ell(Q))$, so that

$B(Q)\cap E\subset Q\subset B_{Q}.$

Note that if $P\subset Q$, then $2B_{P}\subset 2B_{Q}$.

Given a Radon measure $\mu$ and a cube $Q\in\mathcal{D}$, we define the
$d$-dimensional density as

$\theta_{\mu}(Q)=\frac{\mu(Q)}{\ell(Q)^{d}}.$

###  2.3. Lower content regular sets

Recall that a set $E\subset\mathbb{R}^{n}$ is lower content
$(d,c_{1})$-regular if, for all balls $B$ centered on $E$,

$\displaystyle\mathcal{H}^{d}_{\infty}(E\cap B)\geq c_{1}r(B)^{d}.$

We show below that sets with $d$-PBP are lower content regular.

######  Lemma 2.2.

Let $E\subset\mathbb{R}^{n}$ be a set with $d$-PBP with constants $\delta>0$.
Then $E$ is lower content $d$-regular with constant $c\sim_{d}\delta$.

###### Proof.

Without loss of generality, we identify $V$ with $\mathbb{R}^{d}$. For an
arbitrary $\epsilon_{1}>0$, let ${\mathcal{B}}$ be a family of balls in
$\mathbb{R}^{d}$ so that
$\sum_{B^{\prime}\in{\mathcal{B}}}r(B^{\prime})^{d}\leq\mathcal{H}^{d}(\pi_{V}(E\cap
B))-\epsilon_{1}$. Note that, since these are balls in a $d$-plane,
$\mathcal{H}^{d}(B^{\prime}\cap\pi_{V}(B\cap
E))\lesssim_{d}r(B^{\prime})^{d}$. Let $\delta$ be the parameter with which
$E$ satisfied $d$-PBP. Fix a ball $B$ centered on $E$, with
$r(B)\leq\operatorname{diam}(E)$, and a plane $V$ in $B(V_{B},\delta)$. Then,

$\displaystyle\delta r(B)^{d}\leq$ $\displaystyle\mathcal{H}^{d}(\pi_{V}(E\cap
B))$
$\displaystyle\leq\sum_{B^{\prime}\in{\mathcal{B}}}\mathcal{H}^{d}(\pi_{V}(E\cap
B)\cap
B^{\prime})\lesssim_{d}\sum_{B^{\prime}\in{\mathcal{B}}}r(B^{\prime})^{d}\leq
C\mathcal{H}^{d}(\pi_{V}(E\cap B))+C\epsilon_{1}$ $\displaystyle\quad\quad\leq
C^{\prime}\mathcal{H}^{d}_{\infty}(\pi_{V}(E\cap B))+C\epsilon_{1}.$

Now, since $\pi_{V}$ is $1$-Lipschitz and $\epsilon_{1}$ was arbitrary, we
obtain the lemma. The lower content regularity constant $c$ depends only on
$\delta$ and $d$, since $C$ in the above display only depends on $d$. ∎

###  2.4. Frostman measure associated to $E$

In Appendix A we construct a particularly nice Frostman measure supported on a
lower content regular set $E$. Its properties are listed in the lemma below.

######  Lemma 2.3.

Let $E\subset\mathbb{R}^{n}$ be a compact lower content $(d,c_{1})$-regular
set. Then, there exists a measure $\mu$ with $\operatorname{supp}\mu\subset E$
satisfying the following properties:

  1. (1)

$\mu(E)=\mathcal{H}^{d}_{\infty}(E)\gtrsim c_{1}\operatorname{diam}(E)^{d}$,

  2. (2)

$\mu$ has polynomial growth, that is, there exists a constant $C_{1}\geq 1$
such that for all $x\in E$ and $0<r<\operatorname{diam}(E)$ we have

$\mu(B(x,r))\leq C_{1}r,$

  3. (3)

$\mu$ is doubling, that is, there exists a constant $C_{db}>1$ such that for
all $x\in E$ and $0<r<\operatorname{diam}(E)$ we have

$\mu(B(x,2r))\leq C_{db}\,\mu(B(x,r))$ (2.1)

  4. (4)

the $d$-dimensional density of $\mu$ is almost monotone, that is, there exists
a constant $A\geq 1$ such that if $P,Q\in\mathcal{D}$, and $P\subset Q$, then

$\theta_{\mu}(P)\leq A\,\theta_{\mu}(Q).$

In the above, $C_{1}$ may depend only on $d,n$, while $C_{db}$ and $A$ may
also depend on the LCR-constant $c_{1}$.

Observe that thanks to (2.1), if $E$ is lower content regular and
$\mathcal{D}$ is the associated David-Christ lattice, then for all
$Q\in\mathcal{D}$

$\mu(Q)\leq\mu(2B_{Q})\lesssim_{c_{1}}\mu(B(Q))\leq\mu(Q).$ (2.2)

###  2.5. Quantitative rectifiability and $\beta$-numbers

Recall that one of the quantitative notions of rectifiability introduced by
David and Semmes in [DS91] is given by the big pieces of Lipschitz graphs
condition.

######  Definition 2.4.

An Ahlfors $d$-regular set $E\subset\mathbb{R}^{n}$ has _big pieces of
Lipschitz graphs (BPLG)_ if there exist constants $C_{0},L>0$ such that for
any $x\in E$ and $0<r<\operatorname{diam}(E)$ there exists a Lipschitz graph
$\Gamma\subset\mathbb{R}^{n}$ with $\operatorname{Lip}(\Gamma)\leq L$ and

$\mathcal{H}^{d}(E\cap B(x,r)\cap\Gamma)\geq C_{0}r^{d}.$ (2.3)

In [DS93] David and Semmes conjectured that for Ahlfors regular sets, the PBP
and BPLG conditions are equivalent. This was confirmed in a recent
breakthrough result of Orponen [Orp21].

######  Theorem 2.5 ([Orp21]).

Suppose that a set $E\subset\mathbb{R}^{n}$ is Ahlfors $d$-regular. Then, it
has $d$-PBP if and only if it has BPLG.

We recall different variants of $\beta$-numbers that we will use.

######  Definition 2.6 (Jones).

Let $E\subseteq\mathbb{R}^{n}$ and $B$ a ball. Define

$\displaystyle\beta_{E,\infty}^{d}(B)=\frac{1}{r(B)}\inf_{L\in\mathcal{A}(n,d)}\sup\\{\text{dist}(y,L)\mathrel{\mathop{\ordinarycolon}}y\in
E\cap B\\}.$

For $x\in\mathbb{R}^{n}$ and $r>0$ we set also
$\beta_{E,\infty}^{d}(x,r)=\beta_{E,\infty}^{d}(B(x,r))$, and we will use the
same notation for other types of $\beta$-numbers, defined below. We will also
usually omit the superscript $d$.

######  Definition 2.7 (David-Semmes).

Let $\mu$ be a Radon measure on $\mathbb{R}^{n}$, $B\subset\mathbb{R}^{n}$ a
ball, and $1\leq p<\infty$. The $L^{p}$ variant of Jones’ $\beta$-numbers is
defined as

$\beta_{\mu,p}^{d}(B)=\inf_{L\in\mathcal{A}(n,d)}\left(\frac{1}{r(B)^{d}}\int_{B}\left(\frac{\mathop{\mathrm{missing}}{dist}(y,L)}{r(B)}\right)^{p}\,d\mu(y)\right)^{\frac{1}{p}}.$

The following is a special case of a classical result of David and Semmes.

######  Theorem 2.8 ([DS91]).

Let $E\subset\mathbb{R}^{n}$ be a bounded Ahlfors regular set with BPLG, and
let $\mu=\mathcal{H}^{d}|_{E}$. Then, there exists a constant $C$, depending
only on $n,d,$ and the BPLG and Ahlfors regularity constants of $E$, such that

$\int_{E}\int_{0}^{\operatorname{diam}(E)}\beta_{\mu,2}(x,r)^{2}\,\frac{dr}{r}d\mu(x)\leq
C\mu(E).$

Together with Theorem 2.5 this gives the following.

######  Corollary 2.9.

If an Ahlfors regular set $E$ has PBP, then the surface measure
$\mu=\mathcal{H}^{d}|_{E}$ satisfies

$\int_{E}\int_{0}^{\operatorname{diam}(E)}\beta_{\mu,2}(x,r)^{2}\,\frac{dr}{r}d\mu(x)\leq
C\mu(E),$ (2.4)

where $C>0$ depends only on $n,d,$ and the PBP and Ahlfors regularity
constants of $E$.

The estimate (2.4) is extremely useful for estimating $\Gamma_{n,d}(E)$ due to
the following result.

######  Theorem 2.10 ([Pra12, GS19]).

Let $E\subset\mathbb{R}^{n}$ be compact. Then,

$\Gamma_{n,d}(E)\gtrsim\sup\\{\mu(E)\,\mathrel{\mathop{\ordinarycolon}}\,\mu\in\mathcal{F}(E)\\},$
(2.5)

where $\mathcal{F}(E)$ is the set of Radon measures with
$\operatorname{supp}\mu\subset E$ satisfying the polynomial growth condition

$\mu(B(x,r))\leq r^{d}\quad\text{for all $x\in\operatorname{supp}\mu$ and
$r>0$},$

and the flatness condition

$\iint_{0}^{\infty}\beta_{\mu,2}(x,r)^{2}\,\theta_{\mu}(x,r)\,\frac{dr}{r}d\mu(x)\leq\mu(E).$

The theorem above is a combination of two results. Prat [Pra12] related the
capacity $\Gamma_{n,d}$ with the supremum over measures whose Riesz transform
is in $L^{2}$, whereas Girela-Sarrión [GS19] proved that this Riesz transform
condition is true for measures $\mu\in\mathcal{F}(E)$. Prat’s result was first
proved by Tolsa [Tol03] for $d=1,\,n=2$, and by Volberg [Vol03] in the case
$d=n-1$. The result of Girela-Sarrión was first shown for $d=1,\,n=2,$ by
Azzam and Tolsa [AT15]. Finally, let us mention that while (2.5) holds for all
$1\leq d<n$, in the codimension-1 case $d=n-1$ an estimate converse to (2.5)
is also known to be true. This was shown for $d=1$ by Azzam and Tolsa [Tol05,
AT15] and for general $d\in\mathbb{N}$ by Tolsa and the first author [DT21,
Tol21]. Whether the same is true in codimension larger than 1 is an open
problem.

In the statement of Theorem 1.10 we used the content $\beta$-numbers of Azzam
and Schul, which we recall below. For $1\leq p<\infty$ and
$A\subset\mathbb{R}^{n}$ Borel, we define the $p$-Choquet integral as

$\displaystyle\int_{A}f(x)^{p}\,d\mathcal{H}^{d}_{\infty}(x)\mathrel{\mathop{\ordinarycolon}}=\int_{0}^{\infty}\mathcal{H}^{d}_{\infty}(\\{x\in
A\,\mathrel{\mathop{\ordinarycolon}}\,f(x)>t\\})\,t^{p-1}\,dt.$

We refer the reader to [Mat95] for more details on Hausdorff measures and
content and to Section 2 and the Appendix of [AS18] for more details on
Choquet integration.

######  Lemma 2.11 ([AS18, Lemma 2.3]).

Let $E\subseteq\mathbb{R}^{n}$ be either compact or bounded and open so that
$\mathcal{H}^{d}(E)>0,$ and let $f\geq 0$ be continuous on $E$. Then for
$1<p\leq\infty,$

$\frac{1}{\mathcal{H}^{d}_{\infty}(E)}\int_{E}f\,d\mathcal{H}^{d}_{\infty}\lesssim_{n}\left(\frac{1}{\mathcal{H}^{d}_{\infty}(E)}\int_{E}f^{p}\,d\mathcal{H}^{d}_{\infty}\right)^{\frac{1}{p}}$

######  Definition 2.12 (Azzam-Schul).

Let $1\leq p<\infty,$ $E\subseteq\mathbb{R}^{n}$ and $B$ a ball. For a
$d$-dimensional plane $L$ define

$\displaystyle\beta^{d,p}_{E}(B,L)=\left(\frac{1}{r(B)^{d}}\int_{E\cap
B}\left(\frac{\mathop{\mathrm{missing}}{dist}(y,L)}{r(B)}\right)^{p}\,d\mathcal{H}^{d}_{\infty}(y)\right)^{\frac{1}{p}}.$
(2.6)

Then $\beta_{E}^{d,p}(B)=\inf_{L\in{\mathcal{A}}(n,d)}\beta_{E}^{d,p}(B,L)$.

We will need the following lemma.

######  Lemma 2.13 ([AS18], Lemma 2.21).

Let $1\leq p<\infty$ and $E_{1},E_{2}\subset\mathbb{R}^{n}$. Let $x\in E_{1}$
and fix $r>0$. Take some $y\in E_{2}$ so that $B(x,r)\subset B(y,2r)$. Assume
that $E_{1},E_{2}$ are both lower content $d$-regular with constant $c$. Then

$\displaystyle\beta_{E_{1}}^{d,p}(x,r)\lesssim_{c}\beta_{E_{2}}^{d,p}(y,2r)+\left(\frac{1}{r^{d}}\int_{E_{1}\cap
B(x,2r)}\left(\frac{\mathop{\mathrm{missing}}{dist}(y,E_{2})}{r}\right)^{p}\,d\mathcal{H}^{d}_{\infty}(y)\right)^{\frac{1}{p}}.$

##  3\. Vitushkin’s conjecture

In this section we prove Theorem 1.6. Without loss of generality, we may
assume that $\operatorname{diam}(E)=1$ (this follows from scaling
$\Gamma_{n,d}(\lambda E)=\lambda^{d}\,\Gamma_{n,d}(E)$ for $\lambda>0$). By
Theorem 2.10, in order to show that $\Gamma_{n,d}(E)\gtrsim 1$, it suffices to
construct a measure with polynomial growth supported on $E$ such that
$\mu(E)\sim 1$ and

$\iint_{0}^{\infty}\beta_{\mu,2}(x,r)^{2}\,\theta_{\mu}(x,r)\
\frac{dr}{r}d\mu(x)\lesssim\mu(E).$ (3.1)

Let $\mu$ be the measure from Lemma 2.3. Then it has polynomial growth,
$\operatorname{supp}\mu\subset E$, and $\mu(E)\sim 1$. We are going to show
that the estimate (3.1) is satisfied by $\mu$. In fact, we will prove a
somewhat stronger estimate.

######  Proposition 3.1.

Let $E\subset\mathbb{R}^{n}$ be a compact set with $d$-PBP. If $\mu$ is the
measure from Lemma 2.3, then

$\iint_{0}^{\infty}\beta_{\mu,2}(x,r)^{2}\ \frac{dr}{r}d\mu(x)\lesssim\mu(E),$
(3.2)

where the implicit constant depends only on $n,d$ and the PBP constants.

To see that (3.2) implies (3.1), recall that $\theta_{\mu}(x,r)\lesssim 1$ by
the polynomial growth condition.

Observe that since we assume that $\operatorname{diam}(E)=1$, we have that
$\mathcal{D}_{0}$ consists of a single cube, namely $\mathcal{D}_{0}=\\{E\\}$.
By standard techniques, (3.2) is equivalent to

$\sum_{Q\in\mathcal{D}}\beta_{\mu,2}(2B_{Q})^{2}\,\mu(Q)\lesssim\mu(E).$ (3.3)

For reader’s convenience, we sketch the proof of $\eqref{eq:main prop
sum}\Rightarrow\eqref{eq:main prop}$. Observe that if $B_{1}\subset B_{2}$ are
balls, and $r(B_{1})\sim r(B_{2})$, then it follows from the definition of
$\beta_{\mu,2}$-numbers that

$\beta_{\mu,2}(B_{1})\lesssim\beta_{\mu,2}(B_{2}).$

Given $x\in E$ and $0<r<\infty$ let $Q\in\mathcal{D}$ be the unique cube with
$x\in Q$ and $\rho\ell(Q)<r\leq\ell(Q)$, so that $B(x,r)\subset 2B_{Q}$ and
$r(2B_{Q})\sim r$. Then,

$\beta_{\mu,2}(x,r)\lesssim\beta_{\mu,2}(2B_{Q}),$

and it follows easily that

$\iint_{0}^{\infty}\beta_{\mu,2}(x,r)^{2}\
\frac{dr}{r}d\mu(x)\lesssim\sum_{Q\in\mathcal{D}}\beta_{\mu,2}(2B_{Q})^{2}\,\mu(Q).$

Thus, our goal is to prove (3.3). Recalling that $\mathcal{D}_{k}=\\{E\\}$ for
$k\leq 0$ and $\mu(E)\sim 1$, it follows immediately that

$\sum_{k\leq
0}\sum_{Q\in\mathcal{D}_{k}}\beta_{\mu,2}(2B_{Q})^{2}\,\mu(Q)\lesssim\sum_{k\leq
0}\sum_{Q\in\mathcal{D}_{k}}\ell(Q)^{-d}\,\mu(E)^{3}\sim\sum_{k\leq
0}\rho^{-kd}\,\mu(E)^{3}\lesssim\mu(E).$

So when proving (3.3), we may concentrate on $Q\in\mathcal{D}_{k}$ for $k\geq
0$. Since working with infinite sums would entail certain technicalities later
on, we prefer to work with finite sums. Fix some large integer $N>1$. We will
show that

$\sum_{Q\in\mathcal{D}_{0}^{N}}\beta_{\mu,2}(2B_{Q})^{2}\,\mu(Q)\lesssim\mu(E),$
(3.4)

with the estimate independent of $N$. Letting $N\to\infty$ gives the desired
bound (3.3).

We are going to prove (3.4) using an appropriate corona decomposition.

###  3.1. Stopping time argument

We begin by conducting a stopping time argument.

Let $R\in\mathcal{D}_{0}^{N-1}$. We will say that $Q\in\mathsf{LD}_{0}(R)$
(here $\mathsf{LD}$ stands for “low density”) if $Q\in\mathcal{D}_{0}^{N},\
Q\subset R$, and

$\theta_{\mu}(Q)\leq\tau\,\theta_{\mu}(R),$

where $\tau=0.1$. The refined family $\mathsf{LD}(R)$ consists of the maximal
cubes from $\mathsf{LD}_{0}(R)$ (i.e., cubes $Q\in\mathsf{LD}_{0}(R)$ which
are not properly contained in any other cube from $\mathsf{LD}_{0}(R)$). We
define also

$\mathsf{End}(R)=\big{\\{}Q\in\mathcal{D}_{N}\,\mathrel{\mathop{\ordinarycolon}}\,Q\subset
R,\ Q\cap\bigcup_{P\in\mathsf{LD}(R)}P=\varnothing\big{\\}}$

Finally, we set

$\mathsf{Stop}(R)=\mathsf{LD}(R)\cup\mathsf{End}(R),$

and

$\mathrm{Tree}(R)=\\{Q\in\mathcal{D}_{0}^{N}\,\mathrel{\mathop{\ordinarycolon}}\,Q\subset
R,\ \text{there exists $P\in\mathsf{Stop}(R)$ such that }P\subset Q\\}$

It follows immediately from the definition that $\mathsf{Stop}(R)$ is a family
of pairwise disjoint cubes covering $R$. Moreover, $R\notin\mathsf{Stop}(R)$
(we have $R\notin\mathsf{LD}(R)$ by the definition of $\mathsf{LD}_{0}(R)$,
and $R\notin\mathsf{End}(R)$ because $R\notin\mathcal{D}_{N}$). Observe also
that $\mathsf{Stop}(R)\subset\mathrm{Tree}(R)$.

In the lemma below we show that $\mu$ is $d$-ADR at the scales and locations
of $\mathrm{Tree}(R)$.

######  Lemma 3.2.

Let $R\in\mathcal{D}_{0}^{N-1}$. Then, for all $Q\in\mathrm{Tree}(R)$

$\tau\theta_{\mu}(R)\lesssim\theta_{\mu}(Q)\lesssim\theta_{\mu}(R).$

###### Proof.

The upper estimate $\theta_{\mu}(Q)\lesssim\theta_{\mu}(R)$ follows
immediately from property (4) in Lemma 2.3, and the fact that $Q\subset R$.

To see the lower bound $\tau\theta_{\mu}(R)\lesssim\theta_{\mu}(Q)$, note that
this is obvious for $Q\in\mathrm{Tree}(R)\setminus\mathsf{LD}(R)$: for such
cubes we have

$\theta_{\mu}(Q)>\tau\theta_{\mu}(R),$

by the stopping time condition of $\mathsf{LD}_{0}(R)$.

Assume now that $Q\in\mathsf{LD}(R)$, and denote the parent of $Q$ by $Q^{1}$.
Then, $Q^{1}\in\mathrm{Tree}(R)\setminus\mathsf{LD}(R)$, so that
$\theta_{\mu}(Q^{1})>\tau\theta_{\mu}(R)$. But now we see by the doubling
property of $\mu$ (2.1) that $\theta_{\mu}(Q)\gtrsim\theta_{\mu}(Q^{1})$, so
that

$\theta_{\mu}(Q)\gtrsim\theta_{\mu}(Q^{1})\geq\tau\theta_{\mu}(R).$

∎

The following key lemma is the only point in the proof where we use the PBP
propety of $E$ (except for the fact that PBP implies lower content regularity
of $E$, which we also use multiple times throughout the proof).

######  Lemma 3.3.

For any $R\in\mathcal{D}_{0}^{N-1}$ we have

$\sum_{Q\in\mathrm{Tree}(R)}\beta_{\mu,2}(2B_{Q})^{2}\,\mu(Q)\lesssim\theta_{\mu}(R)\,\mu(R),$
(3.5)

with the implicit constant depending only on $n,d$ and the PBP constants of
$E$.

We defer the proof of this lemma to Subsections 3.3–3.5.

###  3.2. Corona decomposition

We are ready to perform the corona decomposition of $\mathcal{D}$ using the
stopping time argument of the previous section.

Set $\mathsf{Top}_{0}=\mathcal{D}_{0}=\\{E\\}$ and
$\mathsf{Top}_{1}=\mathsf{LD}(E)\setminus\mathcal{D}_{N}$. We proceed by
induction: assume that $\mathsf{Top}_{k}$ has already been defined, and that
$\mathsf{Top}_{k}\subset\mathcal{D}_{0}^{N-1}$. Then, we set

$\mathsf{Top}_{k+1}=\bigcup_{P\in\mathsf{Top}_{k}}\mathsf{LD}(P)\setminus\mathcal{D}_{N}.$

Note that for each $k$ the family $\mathsf{Top}_{k}$ consists of pairwise
disjoint cubes. Observe also that there exists $1\leq k_{0}\leq N$ such that
$\mathsf{Top}_{k_{0}}\neq\varnothing$, and then $\mathsf{Top}_{k}=\varnothing$
for all $k>k_{0}$. We define

$\mathsf{Top}=\bigcup_{k=0}^{k_{0}}\mathsf{Top}_{k}.$

Remark that

$\mathcal{D}_{0}^{N}=\bigcup_{R\in\mathsf{Top}}\mathrm{Tree}(R),$

and the sum above is disjoint.

Recall that our aim is to prove (3.4), that is,

$\sum_{Q\in\mathcal{D}_{0}^{N}}\beta_{\mu,2}(2B_{Q})^{2}\,\mu(Q)=\sum_{R\in\mathsf{Top}}\sum_{Q\in\mathrm{Tree}(R)}\beta_{\mu,2}(2B_{Q})^{2}\,\mu(Q)\lesssim\mu(E).$

Let us show how the key estimate from Lemma 3.3 together with the corona
decomposition constructed above implies (3.4).

###### Proof of (3.4) using Lemma 3.3.

It immediately follows from (3.5) that

$\sum_{Q\in\mathcal{D}_{0}^{N}}\beta_{\mu,2}(2B_{Q})^{2}\,\mu(Q)=\sum_{R\in\mathsf{Top}}\sum_{Q\in\mathrm{Tree}(R)}\beta_{\mu,2}(2B_{Q})^{2}\,\mu(Q)\\\
\lesssim\sum_{R\in\mathsf{Top}}\theta_{\mu}(R)\mu(R)=\sum_{k=0}^{k_{0}}\sum_{R\in\mathsf{Top}_{k}}\theta_{\mu}(R)\mu(R).$

Now observe that for $R\in\mathsf{Top}_{k},\ 0\leq k\leq k_{0},$ we have
$\theta_{\mu}(R)\leq C\tau^{k}$ for some constant $C$ depending on $\mu(E)$
(and $\mu(E)\sim 1$, with the implicit constant depending on the PBP
constant). Indeed, this follows from a simple induction argument: it is true
for $k=0$ because $\theta_{\mu}(E)\sim\mu(E)$. If $R\in\mathsf{Top}_{k+1}$,
then $R\in\mathsf{LD}(R^{\prime})$ for some $R^{\prime}\in\mathsf{Top}_{k}$.
Since $\theta_{\mu}(R^{\prime})\leq C\tau^{k}$ by the inductive assumption, we
get from the definition of $\mathsf{LD}(R)$ that

$\theta_{\mu}(R)\leq\tau\theta_{\mu}(R^{\prime})\leq C\tau^{k+1},$

which closes the induction.

It follows that

$\sum_{k=0}^{k_{0}}\sum_{R\in\mathsf{Top}_{k}}\theta_{\mu}(R)\mu(R)\lesssim\sum_{k=0}^{k_{0}}\sum_{R\in\mathsf{Top}_{k}}\tau^{k}\mu(R)\leq\sum_{k=0}^{k_{0}}\tau^{k}\mu(E),$

where in the last estimate we used the fact that for each $k$ the cubes in
$\mathsf{Top}_{k}$ are pairwise disjoint. Recalling that $\tau=0.1$, we get
that

$\sum_{Q\in\mathcal{D}_{0}^{N}}\beta_{\mu,2}(2B_{Q})^{2}\,\mu(Q)\lesssim\sum_{k=0}^{k_{0}}\sum_{R\in\mathsf{Top}_{k}}\theta_{\mu}(R)\mu(R)\lesssim\sum_{k=0}^{k_{0}}\tau^{k}\mu(E)\leq\sum_{k=0}^{\infty}\tau^{k}\mu(E)\leq
2\mu(E).$

This finishes the proof of (3.4). We stress that in the estimates above the
implicit constants do not depend on $N$. ∎

In the light of the above, all that remains to do is prove Lemma 3.3. We do
this in the remaining three subsections.

###  3.3. Regularizing trees

Fix $R\in\mathcal{D}_{0}^{N-1}$. In what follows, it will be more convenient
to work with _regular_ trees, in the sense that nearby stopping cubes have
comparable sidelengths. In order to regularize $\mathrm{Tree}(R)$ we will use
a technique from [DS91] which is nowadays considered fairly standard. It will
also be useful to work with certain cubes neighboring with $R$. We provide the
details below.

We define the set of cubes neighboring with $R$ as

$\mathcal{N}(R)=\\{R^{\prime}\in\mathcal{D}\
\mathrel{\mathop{\ordinarycolon}}\ \ell(R^{\prime})=\ell(R),\ R^{\prime}\cap
2B_{R}\neq\varnothing\\}.$ (3.6)

Note that $R\in\mathcal{N}(R)$. We define also an extended version of
$\mathcal{D}(R)$:

$\mathcal{D}_{*}(R)=\bigcup_{R^{\prime}\in\mathcal{N}(R)}\mathcal{D}(R^{\prime}).$

Now we perform the regularization algorithm. Given $x\in\mathbb{R}^{n}$ set

$d_{R}(x)=\inf_{Q\in\mathrm{Tree}(R)}\mathop{\mathrm{missing}}{dist}(x,Q)+\ell(Q)$

and for $Q\in\mathcal{D}$ set

$d_{R}(Q)=\max\big{(}\frac{1}{20}\inf_{x\in Q}d_{R}(x),5\rho^{N}\big{)},$

where the parameter $\rho=1/1000$ comes from the definition of $\mathcal{D}$.
Observe that the quantity $d_{R}(Q)$ is “monotone” in the sense that if
$P\subset Q$, then $d_{R}(P)\geq d_{R}(Q)$.

We define $\mathsf{Reg}_{*}(R)$ to be the family of maximal cubes
$Q\in\mathcal{D}_{*}(R)$ satisfying

$\ell(Q)\leq d_{R}(Q).$ (3.7)

Note that the "$5\rho^{N}$" term in the definition of $d_{R}(Q)$ ensures that
the inequality above is satisfied by all $Q\in\mathcal{D}_{N}$, so that
$\mathsf{Reg}_{*}(R)\subset\mathcal{D}_{*}(R)\cap\mathcal{D}_{0}^{N}$. Observe
that the cubes in $\mathsf{Reg}_{*}(R)$ are pairwise disjoint, by maximality,
and also

$\bigcup_{Q\in\mathsf{Reg}_{*}(R)}Q=\bigcup_{R^{\prime}\in\mathcal{N}({R})}R^{\prime}\supset
2B_{R}\cap E.$ (3.8)

######  Lemma 3.4.

If $Q\in\mathsf{Reg}_{*}(R)$ and $x\in 5B_{Q}$, then $d_{R}(x)\sim\ell(Q)$.
Consequently, if $Q,P\in\mathsf{Reg}_{*}(R)$, and $5B_{Q}\cap
5B_{P}\neq\varnothing$, then $\ell(Q)\sim\ell(P)$.

###### Proof.

Let $Q\in\mathsf{Reg}_{*}(R)$ and $x\in 5B_{Q}$. First we show that
$d_{R}(x)\geq\ell(Q)$. By the definition of $\mathsf{Reg}_{*}(R)$, we have
$\ell(Q)\leq d_{R}(Q)$, so it suffices to show that $d_{R}(x)\geq d_{R}(Q)$,
that is

$d_{R}(x)\geq\max\big{(}\frac{1}{20}\inf_{y\in Q}d_{R}(y),5\rho^{N}\big{)}.$

Suppose the maximum above is achieved by $5\rho^{N}$, i.e.
$d_{R}(Q)=5\rho^{N}$. Then, the estimate $d_{R}(x)\geq 5\rho^{N}$ is clear by
the definition of $d_{R}$, since $\mathrm{Tree}(R)\subset\mathcal{D}_{0}^{N}$.

Assume now that $d_{R}(Q)=\frac{1}{20}\inf_{y\in Q}d_{R}(y)$. Since the
function $d_{R}$ is $1$-Lipschitz, we have

$\frac{1}{20}d_{R}(x)\geq\frac{1}{20}d_{R}(x_{Q})-\frac{1}{20}|d_{R}(x)-d_{R}(x_{Q})|\geq\frac{1}{20}d_{R}(x_{Q})-\frac{5}{20}\ell(Q)\\\
\geq\frac{1}{20}\inf_{y\in
Q}d_{R}(y)-\frac{1}{4}\ell(Q)=d_{R}(Q)-\frac{1}{4}\ell(Q)\geq\frac{3}{4}\ell(Q).$

Hence, $d_{R}(x)\geq 15\,\ell(Q)$.

We move on to the estimate $d_{R}(x)\lesssim\ell(Q)$. Recall that, by the
definition of $\mathsf{Reg}_{*}(R)$, $Q$ is a maximal cube satisfying
$\ell(Q)\leq d_{R}(Q)$. In particular, $Q^{1}$ satisfies

$\ell(Q^{1})>d_{R}(Q^{1})=\max\big{(}\frac{1}{20}\inf_{y\in
Q^{1}}d_{R}(y),5\rho^{N}\big{)}\geq\frac{1}{20}\inf_{y\in Q^{1}}d_{R}(y).$

Let $y\in Q^{1}$ be such that $\ell(Q^{1})\geq 1/20\,d_{R}(y)$. Since $x,y\in
2B_{Q^{1}}$, we may use the $1$-Lipschitz property of $d_{R}$ to conclude that

$\ell(Q^{1})\geq\frac{1}{20}d_{R}(y)\geq\frac{1}{20}d_{R}(x)-\frac{1}{20}|d_{R}(y)-d_{R}(x)|\geq\frac{1}{20}d_{R}(x)-\frac{4}{20}\ell(Q^{1}).$

Thus, $\ell(Q)=\rho^{-1}\ell(Q^{1})\gtrsim d_{R}(x)$. ∎

We define the extended, regularized “tree” as

$\mathrm{Tree}_{*}(R)=\\{Q\in\mathcal{D}_{*}(R)\,\mathrel{\mathop{\ordinarycolon}}\,\text{there
exists $P\in\mathsf{Reg}_{*}(R)$ such that }P\subset Q\\}.$

The family $\mathrm{Tree}_{*}(R)$ might not be a “true” tree since we cannot
guarantee that all $Q\in\mathrm{Tree}_{*}(R)$ are contained in $R$.
Nevertheless, it is a union of a bounded number of trees
$\mathrm{Tree}_{*}(R)\cap\mathcal{D}(R^{\prime}),\
R^{\prime}\in\mathcal{N}(R)$.

Remark that since $\mathsf{Reg}_{*}(R)\subset\mathcal{D}_{0}^{N}$, we also
have $\mathrm{Tree}_{*}(R)\subset\mathcal{D}_{0}^{N}$. Below we prove that
$\mathrm{Tree}_{*}(R)$ is larger than the original tree $\mathrm{Tree}(R)$.

######  Lemma 3.5.

We have $\mathrm{Tree}(R)\subset\mathrm{Tree}_{*}(R)$.

###### Proof.

It suffices to show that each $Q\in\mathsf{Stop}(R)$ contains some
$P\in\mathsf{Reg}_{*}(R)$. To this end, observe that if $x\in Q$, then from
the definition of $d_{R}$ we have $d_{R}(x)\leq\ell(Q)$. It follows that

$\frac{1}{20}\inf_{x\in Q}d_{R}(x)\leq\frac{\ell(Q)}{20}.$

There are two cases to consider.

_Case_ $Q\in\mathsf{Stop}(R)\cap\mathcal{D}_{N}$. Then
$d_{R}(Q)=5\rho^{N}=\ell(Q)$, so $Q$ satisfies (3.7), while
$\ell(Q^{1})>5\rho^{N}=d_{R}(Q^{1})$, so that $Q^{1}$ does not satisfy (3.7).
Consequently, $Q\in\mathsf{Reg}_{*}(R)$.

_Case_ $Q\in\mathsf{Stop}(R)\cap\mathcal{D}_{0}^{N-1}$. Then we have

$\ell(Q)>\max\bigg{(}\frac{\ell(Q)}{20},5\rho^{N}\bigg{)}\geq d_{R}(Q),$

so that $Q$ does not satisfy (3.7). Taking into account (3.8) we get that
there exists $P\in\mathsf{Reg}_{*}(R)$ such that $P\subset Q$. ∎

Recall that in Lemma 3.2 we proved that $\mu$ is AD-regular at the scales and
locations of $\mathrm{Tree}(R)$. In the lemma below we show that despite
enlarging $\mathrm{Tree}(R)$ to its regularized version $\mathrm{Tree}_{*}(R)$
we did not lose this property.

######  Lemma 3.6.

For each $Q\in\mathrm{Tree}_{*}(R)$ we have

$\tau\,\theta_{\mu}(R)\lesssim\theta_{\mu}(Q)\lesssim\theta_{\mu}(R).$ (3.9)

###### Proof.

If $Q\in\mathrm{Tree}(R)$, then this was already shown in Lemma 3.2. Suppose
that $Q\in\mathrm{Tree}_{*}(R)\setminus\mathrm{Tree}(R)$. Let
$P\in\mathsf{Reg}_{*}(R)$ be such that $P\subset Q$. By Lemma 3.4 we have
$d_{R}(x_{P})\sim\ell(P)$. By the definition of $d_{R}(x_{P})$ there exists
$P^{\prime}\in\mathrm{Tree}(R)$ such that
$d_{R}(x_{P})\sim\mathop{\mathrm{missing}}{dist}(x_{P},P^{\prime})+\ell(P^{\prime})$.
Hence,

$\ell(P)\sim\mathop{\mathrm{missing}}{dist}(x_{P},P^{\prime})+\ell(P^{\prime}).$
(3.10)

If $\ell(P^{\prime})<\ell(Q)$ let $Q^{\prime}\in\mathrm{Tree}(R)$ be the
ancestor of $P^{\prime}$ with $\ell(Q^{\prime})=\ell(Q)$, otherwise set
$Q^{\prime}=P^{\prime}$. We claim that

$\ell(Q^{\prime})\sim\ell(Q).$

This is clearly the case if $\ell(P^{\prime})<\ell(Q)$. On the other hand, if
$Q^{\prime}=P^{\prime}$, then $\ell(Q^{\prime})=\ell(P^{\prime})\geq\ell(Q)$,
and at the same time $\ell(P^{\prime})\lesssim\ell(P)\leq\ell(Q)$. This shows
that $\ell(Q^{\prime})\sim\ell(Q)$.

Recalling (3.10) we get that

$\mathop{\mathrm{missing}}{dist}(Q,Q^{\prime})\lesssim\ell(Q)\sim\ell(Q^{\prime}).$

Hence, $B_{Q}\subset CB_{Q^{\prime}}\subset 2CB_{Q}$ for some $C\sim 1$. The
estimate (3.9) then follows from the fact that it is satisfied by $Q^{\prime}$
(which was shown in Lemma 3.2) and from the doubling property of $\mu$.

∎

The following auxiliary result will be useful later on.

######  Lemma 3.7.

There exists $C_{0}>1$, with $C_{0}\sim 1$, such that for any
$Q\in\mathrm{Tree}_{*}(R)$ and $P\in\mathsf{Reg}_{*}(R)$ satisfying
$2B_{P}\cap 2B_{Q}\neq\varnothing$ we have

$P\subset 2B_{P}\subset C_{0}B_{Q}.$

###### Proof.

To prove the lemma it suffices to show that $\ell(P)\lesssim\ell(Q)$. If
$\ell(Q)\geq\ell(P),$ we are done. Assume $\ell(Q)\leq\rho\,\ell(P)$, so that
$Q\subset 3B_{P}$.

By the definition of $\mathrm{Tree}_{*}(R)$, there exists
$Q^{\prime}\in\mathsf{Reg}_{*}(R)$ such that $Q^{\prime}\subset Q$. In
particular, we have $B_{Q^{\prime}}\cap 3B_{P}\neq\varnothing$, and since
$Q^{\prime},P\in\mathsf{Reg}_{*}(R)$, we get from Lemma 3.4 that
$\ell(P)\sim\ell(Q^{\prime})\leq\ell(Q)$. ∎

###  3.4. Approximating measure

In this subsection we construct a $d$-AD-regular set $\Gamma$ with $d$-PBP
that approximates $E$ at the level of $\mathrm{Tree}_{*}(R)$.

Recall that $\Delta$ denotes the family of usual half-open dyadic cubes in
$\mathbb{R}^{n}$. For each $Q\in\mathrm{Tree}_{*}(R)$ set

$\Delta_{Q}=\\{I\in\Delta\ \mathrel{\mathop{\ordinarycolon}}\
\ell(Q)/2\leq\ell(I)<\ell(Q),\,I\cap Q\neq\varnothing\\}.$

In particular,

$Q\subset\bigcup_{I\in\Delta_{Q}}I\subset 2B_{Q}.$ (3.11)

We define also

$Q_{\Gamma}=\bigcup_{I\in\Delta_{Q}}{\partial}_{d}I,$

where ${\partial}_{d}I$ denotes the $d$-dimensional skeleton of
$I\in\Delta_{Q}$. Observe that

$Q_{\Gamma}\subset 2B_{Q},$ (3.12)

and

$\mathcal{H}^{d}(Q_{\Gamma})\sim\ell(Q)^{d}$ (3.13)

because $\mathcal{H}^{d}({\partial}_{d}I)\sim\ell(Q)^{d}$ for each
$I\in\Delta_{Q}$, and $\\#\Delta_{Q}\lesssim 1$.

Finally, we set

$\Gamma=\bigcup_{Q\in\mathsf{Reg}_{*}(R)}Q_{\Gamma}.$

Note that

$\Gamma\subset\bigcup_{R^{\prime}\in\mathcal{N}(R)}2B_{R^{\prime}}\subset
6B_{R},$ (3.14)

by (3.12) and the definition of $\mathcal{N}(R)$ (3.6).

######  Lemma 3.8.

We have

$\sum_{Q\in\mathsf{Reg}_{*}(R)}\mathds{1}_{Q_{\Gamma}}\sim\mathds{1}_{\Gamma}.$
(3.15)

###### Proof.

Observe that if $Q,P\in\mathsf{Reg}_{*}(R)$ are such that $2B_{Q}\cap
2B_{P}=\varnothing$, then $Q_{\Gamma}\cap P_{\Gamma}=\varnothing$, by (3.12).

It may happen that $Q_{\Gamma}\cap P_{\Gamma}\neq\varnothing$ for
$Q,P\in\mathsf{Reg}_{*}(R)$ such that $2B_{Q}\cap 2B_{P}\neq\varnothing$.
However, for any fixed $Q\in\mathsf{Reg}_{*}(R)$ there is only a bounded
number of $P\in\mathsf{Reg}_{*}(R)$ where this can happen, by Lemma 3.4. ∎

The set $\Gamma$ approximates $E$ at the scales and locations of
$\mathrm{Tree}_{*}(R)$. In order to get a measure $\nu$ approximating $\mu$,
we also define a density
$g\mathrel{\mathop{\ordinarycolon}}\Gamma\to\mathbb{R}$, so that
$\nu=g\,\mathcal{H}^{d}|_{\Gamma}$.

For each $Q\in\mathsf{Reg}_{*}(R)$ we define
$g_{Q}\mathrel{\mathop{\ordinarycolon}}\Gamma\to\mathbb{R}$ as

$g_{Q}=\frac{\mu(Q)}{\mathcal{H}^{d}(Q_{\Gamma})}\mathds{1}_{Q_{\Gamma}},$

so that

$\int_{\Gamma}g_{Q}\ d\mathcal{H}^{d}=\mu(Q).$ (3.16)

Note also that

$g_{Q}\sim\theta_{\mu}(Q)\mathds{1}_{Q_{\Gamma}}\sim\theta_{\mu}(R)\mathds{1}_{Q_{\Gamma}},$
(3.17)

by (3.13) and Lemma 3.6.

Set

$g=\sum_{Q\in\mathsf{Reg}_{*}(R)}g_{Q}$

and

$\nu=g\,\mathcal{H}^{d}|_{\Gamma}.$

Remark that for any $x\in\Gamma$

$g(x)=\sum_{Q\in\mathsf{Reg}_{*}(R)}g_{Q}(x)\overset{\eqref{eq:gQsimtheta}}{\sim}\theta_{\mu}(R)\sum_{Q\in\mathsf{Reg}_{*}(R)}\mathds{1}_{Q_{\Gamma}}(x)\sim\theta_{\mu}(R),$

where in the last estimate we used (3.15). It follows that

$\nu=\theta_{\mu}(R)\cdot\,h\mathcal{H}^{d}|_{\Gamma},$ (3.18)

where $h(x)=g(x)/\theta_{\mu}(R)\sim\mathds{1}_{\Gamma}$.

In the next two lemmas we show that $\Gamma$ is $d$-AD-regular, and that it
satisfies $d$-PBP, with constants depending only on $n,d$ and the PBP
constants of $E$. This will allow us to use Corollary 2.9 and get estimates
for the $\beta_{\nu,2}$-numbers in the following subsection.

######  Lemma 3.9.

Measure $\nu$ is $d$-AD-regular. More precisely, for $x\in\Gamma$ and
$0<r<\operatorname{diam}(\Gamma)$ we have

$\nu(B(x,r))\sim\theta_{\mu}(R)\,r^{d}.$ (3.19)

In consequence, $\Gamma$ is a $d$-AD-regular set, with ADR constants depending
only on $n,d,$ and the $PBP$ constants of $E$.

###### Proof.

Fix $x\in\Gamma$, and let $Q\in\mathsf{Reg}_{*}(R)$ be a cube such that $x\in
Q_{\Gamma}$; if $Q$ is non-unique, just choose one.

First we check that $\nu$ is upper ADR. Assume that $0<r\leq\ell(Q)$. Then,

$\nu(B(x,r))=\sum_{P\in\mathsf{Reg}_{*}(R)}\int_{B(x,r)}g_{P}\
d\mathcal{H}^{d}\overset{\eqref{eq:gQsimtheta}}{\lesssim}\sum_{P\in\mathsf{Reg}_{*}(R),\,P_{\Gamma}\cap
B(x,r)\neq\varnothing}\int_{B(x,r)}\theta_{\mu}(R)\mathds{1}_{P_{\Gamma}}\
d\mathcal{H}^{d}\\\
=\theta_{\mu}(R)\sum_{P\in\mathsf{Reg}_{*}(R),\,P_{\Gamma}\cap
B(x,r)\neq\varnothing}\mathcal{H}^{d}({P_{\Gamma}\cap
B(x,r)})\lesssim\theta_{\mu}(R)\sum_{P\in\mathsf{Reg}_{*}(R),\,P_{\Gamma}\cap
B(x,r)\neq\varnothing}r^{d},$

where in the last estimate we used the fact that each $P_{\Gamma}$ is upper
AD-regular. To estimate the last term above, note that $B(x,r)\subset 2B_{Q}$
and so $B(x,r)$ may intersect only a bounded number $P_{\Gamma},\
P\in\mathsf{Reg}_{*}(R),$ by (3.12) and Lemma 3.4. Hence,

$\nu(B(x,r))\lesssim\theta_{\mu}(R)\sum_{P\in\mathsf{Reg}_{*}(R),\,P_{\Gamma}\cap
B(x,r)\neq\varnothing}r^{d}\sim\theta_{\mu}(R)\,r^{d}.$

Now suppose that $\ell(Q)<r<\operatorname{diam}(\Gamma)$.

$\nu(B(x,r))=\sum_{P\in\mathsf{Reg}_{*}(R)}\int_{B(x,r)}g_{P}\
d\mathcal{H}^{d}\leq\sum_{P\in\mathsf{Reg}_{*}(R),\,P_{\Gamma}\cap
B(x,r)\neq\varnothing}\int g_{P}\ d\mathcal{H}^{d}\\\
\overset{\eqref{eq:gQmeas}}{=}\sum_{P\in\mathsf{Reg}_{*}(R),\,P_{\Gamma}\cap
B(x,r)\neq\varnothing}\mu(P).$

Let $Q^{\prime}\in\mathrm{Tree}_{*}(R)$ be the minimal cube satisfying
$Q\subset Q^{\prime}$ and $\ell(Q^{\prime})>r$. Note that
$\ell(Q^{\prime})\sim r$, and $B(x,r)\subset 2B_{Q^{\prime}}$. If
$P\in\mathsf{Reg}_{*}(R)$ satisfies $P_{\Gamma}\cap B(x,r)\neq\varnothing$,
then in particular we have $2B_{P}\cap 2B_{Q^{\prime}}\neq\varnothing$. Thus,
by Lemma 3.7, we have $P\subset C_{0}B_{Q^{\prime}}$. In consequence,

$\nu(B(x,r))\leq\sum_{P\in\mathsf{Reg}_{*}(R),\,P_{\Gamma}\cap
B(x,r)\neq\varnothing}\mu(P)\leq\mu(C_{0}B_{Q^{\prime}})\sim\mu(Q^{\prime}),$

where in the last estimate we used the doubling property of $\mu$. We have
$\mu(Q^{\prime})\sim\theta_{\mu}(R)\ell(Q^{\prime})^{d}$ by Lemma 3.6. Since
$\ell(Q^{\prime})\sim r$, we get the desired upper bound

$\nu(B(x,r))\lesssim\theta_{\mu}(R)\,r^{d}.$

We turn to the lower regularity of $\nu$. The proof at scales $0<r\leq
20\,\ell(Q)$ is very similar to the one before. We have

$\nu(B(x,r))=\sum_{P\in\mathsf{Reg}_{*}(R)}\int_{B(x,r)}g_{P}\
d\mathcal{H}^{d}\overset{\eqref{eq:gQsimtheta}}{\gtrsim}\int_{B(x,r)}\theta_{\mu}(R)\mathds{1}_{Q_{\Gamma}}\
d\mathcal{H}^{d}\\\ =\theta_{\mu}(R)\ \mathcal{H}^{d}({Q_{\Gamma}\cap
B(x,r)})\sim\theta_{\mu}(R)\,r^{d},$

where in the last line we used the AD-regularity of $Q_{\Gamma}$.

Concerning scales $20\,\ell(Q)\leq r<\operatorname{diam}(\Gamma)$, we proceed
as follows:

$\nu(B(x,r))=\sum_{P\in\mathsf{Reg}_{*}(R)}\int_{B(x,r)}g_{P}\
d\mathcal{H}^{d}\geq\sum_{P\in\mathsf{Reg}_{*}(R),\,P_{\Gamma}\subset
B(x,r)}\int g_{P}\ d\mathcal{H}^{d}\\\
\overset{\eqref{eq:gQmeas}}{=}\sum_{P\in\mathsf{Reg}_{*}(R),\,P_{\Gamma}\subset
B(x,r)}\mu(P).$

Let $Q^{\prime}\in\mathrm{Tree}_{*}(R)$ be the maximal cube satisfying
$Q\subset Q^{\prime}$ and $2B_{Q^{\prime}}\subset B(x,r)$ (such cube exists
because $2B_{Q}\subset B(x,r)$). Note that $\ell(Q^{\prime})\sim r$.

We claim that if $P\in\mathsf{Reg}_{*}(R)$ and $P\subset Q^{\prime}$, then
$P_{\Gamma}\subset B(x,r)$ (so in particular, $P$ appears in the sum on the
right hand side above). Indeed, we have $P_{\Gamma}\subset 2B_{P}\subset
2B_{Q^{\prime}}\subset B(x,r)$, by the definition of $Q^{\prime}$. Thus,

$\nu(B(x,r))\geq\sum_{P\in\mathsf{Reg}_{*}(R),\,P_{\Gamma}\subset
B(x,r)}\mu(P)\\\ \geq\sum_{P\in\mathsf{Reg}_{*}(R),\,P\subset
Q^{\prime}}\mu(P)=\mu(Q^{\prime})\sim\theta_{\mu}(R)\,\ell(Q^{\prime})^{d}\sim\theta_{\mu}(R)\,r^{d},$

where we have used Lemma 3.6 and the fact that $\ell(Q^{\prime})\sim r$. This
finishes the proof of the AD-regularity of $\nu$.

As a consequence of (3.18) and (3.19) we get that $\Gamma$ is a $d$-ADR set,
with ADR constants depending only on $n,d,$ and the PBP constants. ∎

In order to show that $\Gamma$ has plenty of big projections, we need the
following result on $d$-dimensional skeletons of dyadic cubes.

######  Lemma 3.10.

Let $I\in\Delta$, and $V\in{\mathcal{G}}(n,d)$. Then,

$\pi_{V}({I})\subset\pi_{V}({\partial}_{d}I).$ (3.20)

###### Proof.

Let $p\in\pi_{V}(I)$. Then $W\coloneqq\pi^{-1}(\\{p\\})$ is an
$(n-d)$-dimensional plane intersecting $I$. We are going to show that

$W\cap\partial_{d}I\neq\varnothing,$ (3.21)

so that $p\in\pi_{V}({\partial}_{d}I)$, and in consequence (3.20) holds.

By rotating and translating, we may assume that

$W=\mathop{\mathrm{missing}}{span}(e_{1},\dots,e_{n-d})=\\{(x_{1},\dots,x_{n-d},0,\dots,0)\,\mathrel{\mathop{\ordinarycolon}}\,(x_{1},\dots,x_{n-d})\in\mathbb{R}^{n-d}\\}.$

In this coordinates, the cube $I$ can be described by a system of inequalities

$I=\\{x\in\mathbb{R}^{n}\,\mathrel{\mathop{\ordinarycolon}}\,a\leq Lx\leq
b\\},$

for some orthogonal matrix $L\in SO(n)$ and some vectors
$a=(a_{i})_{i=1,\dots,n}\in\mathbb{R}^{n}$ and
$b=(a_{i}+\ell(I))_{i=1,\dots,n}\in\mathbb{R}^{n}$ (note that before the
rotation and translation, we would have $L=\mathop{\mathrm{missing}}{id}_{n}$,
and $a$ a vector consisting of dyadic numbers).

Recall that all $d$-dimensional faces of $I$ are of the following form: for
any choice of indices $A,B\subset\\{1,\dots,n\\}$ with $A\cap B=\varnothing$
and $\\#(A\cup B)=n-d$ we set

$F(A,B)=\\{x\in\mathbb{R}^{n}\,\mathrel{\mathop{\ordinarycolon}}\,a\leq Lx\leq
b,\,a_{i}=L_{i}\cdot x\text{ for $i\in A$, }L_{i}\cdot x=b_{i}\text{ for $i\in
B$}\\}.$

Above, $L_{i}$ denotes the $i$-th row of $L$. This is a one-to-one
representation, in the sense that for any such $A,B$ we get a $d$-dimensional
face of $I$, and each $d$-dimensional face $F$ satisfies $F=F(A,B)$ for a
unique choice of $A$ and $B$.

Recalling $W=\mathop{\mathrm{missing}}{span}(e_{1},\dots,e_{n-d})$, we get
that

$I\cap
W=\\{x=(x^{\prime},\textbf{0})\in\mathbb{R}^{n-d}\times\\{0\\}^{d}\,\mathrel{\mathop{\ordinarycolon}}\,a\leq
Lx\leq b\\}\neq\varnothing.$

Let $L^{\prime}$ be a rectangular $(n-d)\times d$-matrix obtained by taking
the first $(n-d)$ columns of $L$. Then,

$I\cap
W=\\{x=(x^{\prime},\textbf{0})\in\mathbb{R}^{n-d}\times\\{0\\}^{d}\,\mathrel{\mathop{\ordinarycolon}}\,a\leq
L^{\prime}x^{\prime}\leq b\\},$

and so $I\cap W$ might be identified with the (non-empty) polytope

$P=\\{x^{\prime}\in\mathbb{R}^{n-d}\,\mathrel{\mathop{\ordinarycolon}}\,a\leq
L^{\prime}x^{\prime}\leq b\\}.$

Let $v^{\prime}\in P$ be a vertex of $P$, i.e., a $0$-dimensional face of $P$.
We claim that the point
$v=(v^{\prime},\textbf{0})\in\mathbb{R}^{n-d}\times\\{0\\}^{d}$ belongs to
some $d$-dimensional face of $I$. Since it clearly lies on $W$, this will give
(3.21).

Since $v^{\prime}$ is a vertex of $P$, we get that there exist sets of indices
$A,B\subset\\{1,\dots,n\\}$ with $A\cap B=\varnothing$ and $\\#(A\cup B)=n-d$
such that

$a_{i}=L^{\prime}_{i}\cdot v^{\prime}\text{ for $i\in A$,}\quad
L^{\prime}_{i}\cdot v^{\prime}=b_{i}\text{ for $i\in B$},$

where $L_{i}^{\prime}$ denotes the $i$-th row of $L^{\prime}$. Note that the
property $\\#(A\cup B)=n-d$ follows from the fact that $v^{\prime}$ should be
a unique solution to the equations above (a $0$-dimensional face is a single
point), and we are in $\mathbb{R}^{n-d}$. We do not claim that the choice of
$A$ and $B$ is unique, but we do not care. Finally, observe that the remaining
inequalities $a_{i}\leq L^{\prime}_{i}\cdot v^{\prime}\leq b_{i},\
i\in\\{1,\dots,n\\}\setminus(A\cup B)$ must also be satisfied since $v\in P$.

It remains to observe that $v=(v^{\prime},\textbf{0})\in F(A,B)\cap W$ because
$Lv=L^{\prime}v^{\prime}$. In particular, $v\in{\partial}_{d}I\cap W$, which
gives (3.21).

∎

######  Lemma 3.11.

The set $\Gamma$ has $d$-PBP, with PBP constants depending only on $n,d$, and
the PBP constants of $E$.

###### Proof.

Fix $x\in\Gamma$, and let $Q\in\mathsf{Reg}_{*}(R)$ be such that $x\in
Q_{\Gamma}$ (if there are many such $Q$, just choose one). We will show that
for any $0<r<\operatorname{diam}(\Gamma)$ there exists
$V_{x,r}^{\Gamma}\in{\mathcal{G}}(n,d)$ such that for all $V\in
B(V_{x,r}^{\Gamma},\delta)$ we have

$\mathcal{H}^{d}(\pi_{V}(\Gamma\cap B(x,r)))\gtrsim r^{d}.$ (3.22)

Let $0<r<10\,\ell(Q)$. Since $x\in Q_{\Gamma}$, we have $x\in{\partial}_{d}I$
for some $I\in\Delta_{Q}$, and so it is immediate to see that for _any_
$V\in{\mathcal{G}}(n,d)$ we have

$\mathcal{H}^{d}(\pi_{V}(\Gamma\cap
B(x,r)))\geq\mathcal{H}^{d}(\pi_{V}({\partial}_{d}I\cap B(x,r)))\gtrsim
r^{d}.$

So the PBP property holds trivially at small scales.

Assume now that $10\,\ell(Q)<r<\operatorname{diam}(\Gamma)\sim\ell(R)$. Let
$Q^{\prime}\in\mathcal{D}(R)$ be the maximal cube satisfying $Q\subset
Q^{\prime}$ and $2B_{Q^{\prime}}\subset B(x,r)$. Clearly,
$\ell(Q^{\prime})\sim r$. Recall that there exists a ball $B(Q^{\prime})$
centered at $x_{Q^{\prime}}\in E$ with $r(B(Q^{\prime}))\sim\ell(Q^{\prime})$
and such that $B(Q^{\prime})\cap E\subset Q^{\prime}$ (see Lemma 2.1). Set

$V^{\Gamma}_{x,r}\coloneqq V^{E}_{B(Q^{\prime})},$

where $V^{E}_{B(Q^{\prime})}$ is the $d$-plane coming from the PBP property of
$E$ in the ball $B(Q^{\prime})$. Fix $V\in B(V^{\Gamma}_{x,r},\delta)$. We
will show that (3.22) holds.

Note that the PBP property of $E$ in the ball $B(Q^{\prime})$ gives us

$\mathcal{H}^{d}(\pi_{V}(Q^{\prime}))\geq\mathcal{H}^{d}(\pi_{V}(E\cap
B(Q^{\prime})))\gtrsim\ell(Q^{\prime})^{d}\sim r^{d}.$ (3.23)

We claim that

$\pi_{V}(\Gamma\cap B(x,r))\supset\pi_{V}(Q^{\prime}).$ (3.24)

Together with (3.23), this will give us the desired estimate (3.22).

Recall that

$Q^{\prime}=\bigcup_{P\in\mathsf{Reg}_{*}(R),\,P\subset Q^{\prime}}P,$

and for each $P\in\mathsf{Reg}_{*}(R)$ we have
$P\subset\bigcup_{I\in\Delta_{P}}I$. In consequence,

$Q^{\prime}\subset\bigcup_{P\in\mathsf{Reg}_{*}(R),\,P\subset
Q^{\prime}}\bigcup_{I\in\Delta_{P}}I.$

Using the above and Lemma 3.10 we get that

$\pi_{V}(Q^{\prime})\subset\bigcup_{P\in\mathsf{Reg}_{*}(R),\,P\subset
Q^{\prime}}\bigcup_{I\in\Delta_{P}}\pi_{V}(I)\subset\bigcup_{P\in\mathsf{Reg}_{*}(R),\,P\subset
Q^{\prime}}\bigcup_{I\in\Delta_{P}}\pi_{V}({\partial}_{d}I)\\\
=\bigcup_{P\in\mathsf{Reg}_{*}(R),\,P\subset Q^{\prime}}\pi_{V}(P_{\Gamma}).$

Recalling (3.12) we see that for all $P\in\mathsf{Reg}_{*}(R),\,P\subset
Q^{\prime},$ we have

$P_{\Gamma}\subset 2B_{P}\cap\Gamma\subset 2B_{Q^{\prime}}\cap\Gamma\subset
B(x,r)\cap\Gamma,$

and so

$\pi_{V}(Q^{\prime})\subset\bigcup_{P\in\mathsf{Reg}_{*}(R),\,P\subset
Q^{\prime}}\pi_{V}(P_{\Gamma})\subset\pi_{V}(\Gamma\cap B(x,r)).$

This finishes the proof (3.24), which together with (3.23) gives (3.22). ∎

###  3.5. Estimating $\beta$-numbers

In this subsection we finally prove our key estimate, Lemma 3.3. We recall it
for reader’s convenience.

######  Lemma.

For any $R\in\mathcal{D}_{0}^{N-1}$ we have

$\sum_{Q\in\mathrm{Tree}(R)}\beta_{\mu,2}(2B_{Q})^{2}\,\mu(Q)\lesssim\theta_{\mu}(R)\,\mu(R),$
(3.25)

with the implicit constant depending only on $n,d$ and the PBP constants of
$E$.

In Lemmas 3.9 and 3.11 we showed that $\Gamma$ is a $d$-AD-regular set
satisfying $d$-PBP, with ADR and PBP constants depending only on $n,d,$ and
the PBP constants of $E$. Thus, we get from Corollary 2.9 that $\Gamma$
satisfies

$\int_{\Gamma}\int_{0}^{\ell(R)}\beta_{\Gamma,2}(x,r)^{2}\,\frac{dr}{r}d\mathcal{H}^{d}(x)\lesssim\ell(R)^{d}.$

In the above we also use that $\Gamma\subset 6B_{R}$, by (3.14). Recalling
(3.18), we get that

$\int_{\Gamma}\int_{0}^{\ell(R)}\beta_{\nu,2}(x,r)^{2}\,\frac{dr}{r}d\mathcal{H}^{d}(x)\lesssim\theta_{\mu}(R)\,\ell(R)^{d}.$

By standard arguments (similar to the ones below (3.3)) this implies

$\sum_{Q\in\mathrm{Tree}(R)}\beta_{\nu,2}(C_{0}B_{Q})^{2}\ell(Q)^{d}\lesssim\theta_{\mu}(R)\,\ell(R)^{d},$

and since by Lemma 3.6 we have $\mu(Q)\sim\theta_{\mu}(R)\ell(Q)^{d}$ for
$Q\in\mathrm{Tree}(R)$, we arrive at

$\sum_{Q\in\mathrm{Tree}(R)}\beta_{\nu,2}(C_{0}B_{Q})^{2}\mu(Q)\lesssim\theta_{\mu}(R)\,\mu(R).$
(3.26)

This is very similar to (3.25), which we are aiming to prove. The last step
consists of transferring the estimate from $\beta_{\nu,2}$ to $\beta_{\mu,2}$.

######  Lemma 3.12.

For any $Q\in\mathrm{Tree}(R)$ we have

$\beta_{\mu,2}(2B_{Q})^{2}\lesssim\beta_{\nu,2}(C_{0}B_{Q})^{2}+\ell(Q)^{-d}\sum_{P\in\mathsf{Reg}_{*}(R),\,2B_{P}\subset
C_{0}B_{Q}}\frac{\ell(P)}{\ell(Q)}\mu(P),$ (3.27)

where $C_{0}\sim 1$ is the constant from Lemma 3.7.

###### Proof.

Fix $Q\in\mathrm{Tree}(R)$, and let $L_{Q}$ be a $d$-plane minimizing
$\beta_{\nu,2}(C_{0}B_{Q})$.

Observe that for any $Q\in\mathrm{Tree}(R)$, the set $2B_{Q}\cap E$ is fully
covered by $P\in\mathsf{Reg}_{*}(R)$ satisfying $P\cap 2B_{Q}\neq\varnothing$.
Indeed, this is true for $R$ because by the definition of $\mathcal{N}(R)$ and
$\mathsf{Reg}_{*}(R)$ we have

$2B_{R}\cap E\subset\bigcup_{P\in\mathsf{Reg}_{*}(R),\,P\cap
2B_{R}\neq\varnothing}P.$

Since $2B_{Q}\subset 2B_{R}$ for any $Q\in\mathrm{Tree}(R)$, the claim
follows.

The observation above gives

$\beta_{\mu,2}(2B_{Q})^{2}\ell(Q)^{d}\lesssim\int_{2B_{Q}}\bigg{(}\frac{\mathop{\mathrm{missing}}{dist}(x,L_{Q})}{\ell(Q)}\bigg{)}^{2}\,d\mu(x)\\\
\leq\sum_{P\in\mathsf{Reg}_{*}(R),\,P\cap
2B_{Q}\neq\varnothing}\int_{P}\bigg{(}\frac{\mathop{\mathrm{missing}}{dist}(x,L_{Q})}{\ell(Q)}\bigg{)}^{2}\,d\mu(x)\\\
\leq\sum_{P\in\mathsf{Reg}_{*}(R),\,2B_{P}\subset
C_{0}B_{Q}}\int_{P}\bigg{(}\frac{\mathop{\mathrm{missing}}{dist}(x,L_{Q})}{\ell(Q)}\bigg{)}^{2}\,d\mu(x),$

where in the last line we used Lemma 3.7.

Let $P\in\mathsf{Reg}_{*}(R),\,2B_{P}\subset C_{0}B_{Q}$. Then,

$\int_{P}\bigg{(}\frac{\mathop{\mathrm{missing}}{dist}(x,L_{Q})}{\ell(Q)}\bigg{)}^{2}\,d\mu(x)=\int_{\Gamma}\bigg{(}\frac{\mathop{\mathrm{missing}}{dist}(x,L_{Q})}{\ell(Q)}\bigg{)}^{2}g_{P}(x)\,d\mathcal{H}^{d}(x)\\\
+\int\bigg{(}\frac{\mathop{\mathrm{missing}}{dist}(x,L_{Q})}{\ell(Q)}\bigg{)}^{2}\,(\mathds{1}_{P}(x)d\mu(x)-g_{P}(x)d\mathcal{H}^{d}|_{\Gamma}(x))\eqqcolon
I_{1}(P)+I_{2}(P).$

It follows that

$\beta_{\mu,2}(2B_{Q})^{2}\ell(Q)^{d}\lesssim\sum_{P\in\mathsf{Reg}_{*}(R),\,2B_{P}\subset
C_{0}B_{Q}}I_{1}(P)+\sum_{P\in\mathsf{Reg}_{*}(R),\,2B_{P}\subset
C_{0}B_{Q}}I_{2}(P)\eqqcolon S_{1}+S_{2}.$

Estimating $S_{1}$ is straightforward. Recall that for
$P\in\mathsf{Reg}_{*}(R)$ we have $\operatorname{supp}g_{P}\subset
P_{\Gamma}\subset 2B_{P}$. Hence,

$S_{1}=\sum_{P\in\mathsf{Reg}_{*}(R),\,2B_{P}\subset
C_{0}B_{Q}}\int_{\Gamma}\bigg{(}\frac{\mathop{\mathrm{missing}}{dist}(x,L_{Q})}{\ell(Q)}\bigg{)}^{2}g_{P}(x)\,d\mathcal{H}^{d}(x)\\\
\leq\sum_{P\in\mathsf{Reg}_{*}(R)}\int_{\Gamma\cap
C_{0}B_{Q}}\bigg{(}\frac{\mathop{\mathrm{missing}}{dist}(x,L_{Q})}{\ell(Q)}\bigg{)}^{2}g_{P}(x)\,d\mathcal{H}^{d}(x)\\\
=\int_{\Gamma\cap
C_{0}B_{Q}}\bigg{(}\frac{\mathop{\mathrm{missing}}{dist}(x,L_{Q})}{\ell(Q)}\bigg{)}^{2}g(x)\,d\mathcal{H}^{d}(x)\\\
=\int_{C_{0}B_{Q}}\bigg{(}\frac{\mathop{\mathrm{missing}}{dist}(x,L_{Q})}{\ell(Q)}\bigg{)}^{2}\,d\nu(x)\sim\beta_{\nu,2}(C_{0}B_{Q})^{2}\ell(Q)^{d},$

where in the last part we used our choice of $L_{Q}$. This gives us the first
term from the right hand side of (3.27).

We turn to estimating $S_{2}$. Let $P\in\mathsf{Reg}_{*}(R)$. Using the fact
that $\int_{\Gamma}g_{P}\,d\mathcal{H}^{d}=\mu(P)$ we get

$I_{2}(P)=\int\bigg{(}\frac{\mathop{\mathrm{missing}}{dist}(x,L_{Q})}{\ell(Q)}\bigg{)}^{2}\,(\mathds{1}_{P}(x)d\mu(x)-g_{P}(x)d\mathcal{H}^{d}|_{\Gamma}(x))\\\
=\int\bigg{(}\bigg{(}\frac{\mathop{\mathrm{missing}}{dist}(x,L_{Q})}{\ell(Q)}\bigg{)}^{2}-\bigg{(}\frac{\mathop{\mathrm{missing}}{dist}(x_{P},L_{Q})}{\ell(Q)}\bigg{)}^{2}\bigg{)}\,(\mathds{1}_{P}(x)d\mu(x)-g_{P}(x)d\mathcal{H}^{d}|_{\Gamma}(x)).$

For $x\in P\cup\operatorname{supp}g_{P}\subset 2B_{P}\subset C_{0}B_{Q}$ we
have

$\bigg{|}\bigg{(}\frac{\mathop{\mathrm{missing}}{dist}(x,L_{Q})}{\ell(Q)}\bigg{)}^{2}-\bigg{(}\frac{\mathop{\mathrm{missing}}{dist}(x_{P},L_{Q})}{\ell(Q)}\bigg{)}^{2}\bigg{|}\leq\frac{|x-x_{P}|}{\ell(Q)}\cdot\frac{\mathop{\mathrm{missing}}{dist}(x,L_{Q})+\mathop{\mathrm{missing}}{dist}(x_{P},L_{Q})}{\ell(Q)}\\\
\lesssim\frac{\ell(P)}{\ell(Q)}\cdot\frac{\ell(Q)}{\ell(Q)}=\frac{\ell(P)}{\ell(Q)}.$

Thus,

$I_{2}(P)\lesssim\frac{\ell(P)}{\ell(Q)}\mu(P).$

Summing over $P\in\mathsf{Reg}_{*}(R)$ with $2B_{P}\subset C_{0}B_{Q}$ yields

$S_{2}\lesssim\sum_{P\in\mathsf{Reg}_{*}(R),\,2B_{P}\subset
C_{0}B_{Q}}\frac{\ell(P)}{\ell(Q)}\mu(P).$

This corresponds to the second term from the right hand side of (3.27). ∎

######  Lemma 3.13.

We have

$\sum_{Q\in\mathrm{Tree}(R)}\beta_{\mu,2}(2B_{Q})^{2}\mu(Q)\lesssim\sum_{Q\in\mathrm{Tree}(R)}\beta_{\nu,2}(C_{0}B_{Q})^{2}\mu(Q)+\theta_{\mu}(R)\,\mu(R).$
(3.28)

###### Proof.

In the previous lemma we showed that for any $Q\in\mathrm{Tree}(R)$ we have
(3.27). Multliplying by $\mu(Q)$ and summing over $Q\in\mathrm{Tree}(R)$
yields

$\sum_{Q\in\mathrm{Tree}(R)}\beta_{\mu,2}(2B_{Q})^{2}\mu(Q)\lesssim\sum_{Q\in\mathrm{Tree}(R)}\beta_{\nu,2}(C_{0}B_{Q})^{2}\mu(Q)\\\
+\sum_{Q\in\mathrm{Tree}(R)}\theta_{\mu}(Q)\sum_{P\in\mathsf{Reg}_{*}(R),\,2B_{P}\subset
C_{0}B_{Q}}\frac{\ell(P)}{\ell(Q)}\mu(P).$

Thus, in order to obtain (3.28) it suffices to prove that

$\sum_{Q\in\mathrm{Tree}(R)}\theta_{\mu}(Q)\sum_{P\in\mathsf{Reg}_{*}(R),\,2B_{P}\subset
C_{0}B_{Q}}\frac{\ell(P)}{\ell(Q)}\mu(P)\lesssim\theta_{\mu}(R)\mu(R).$

First, recall that by Lemma 3.2 we have $\theta_{\mu}(Q)\sim\theta_{\mu}(R)$
for all $Q\in\mathrm{Tree}(R)$. So we only need to show

$\sum_{Q\in\mathrm{Tree}(R)}\sum_{P\in\mathsf{Reg}_{*}(R),\,2B_{P}\subset
C_{0}B_{Q}}\frac{\ell(P)}{\ell(Q)}\mu(P)\lesssim\mu(R).$ (3.29)

Changing the order of summation transforms the left hand side to

$\sum_{P\in\mathsf{Reg}_{*}(R)}\mu(P)\sum_{Q\in\mathrm{Tree}(R),\,2B_{P}\subset
C_{0}B_{Q}}\frac{\ell(P)}{\ell(Q)}.$

The inner sum is essentially a geometric series: if $P\in\mathsf{Reg}_{*}(R)$
is fixed, then for any generation $k\in[0,N]$, there is at most a bounded
number of $Q\in\mathrm{Tree}(R)\cap\mathcal{D}_{k}$ such that $2B_{P}\subset
C_{0}B_{Q}$ (the bound depends on $C_{0}\sim 1$). Moreover, all such $Q$
necessarily satisfy $C_{0}\ell(Q)\geq\ell(P)$. Hence,

$\sum_{P\in\mathsf{Reg}_{*}(R)}\mu(P)\sum_{Q\in\mathrm{Tree}(R),\,2B_{P}\subset
C_{0}B_{Q}}\frac{\ell(P)}{\ell(Q)}\lesssim_{C_{0}}\sum_{P\in\mathsf{Reg}_{*}(R)}\mu(P)\leq\mu(6B_{R})\sim\mu(R),$

where in the last two estimates we used the fact that the cubes in
$\mathsf{Reg}_{*}(R)$ are pairwise disjoint and contained in $6B_{R}$, and
also the doubling property of $\mu$. This finishes the proof of (3.28). ∎

Lemma 3.13 together with (3.26) give the desired estimate

$\sum_{Q\in\mathrm{Tree}(R)}\beta_{\mu,2}(2B_{Q})^{2}\,\mu(Q)\lesssim\theta_{\mu}(R)\,\mu(R).$

This concludes the proof of Lemma 3.3.

##  4\. Traveling Saleseman Theorem

In this section we prove the Analyst’s Traveling Salesman Theorem for sets
with PBP, Theorem 1.10.

###  4.1. Discrete approximation

We recall a simplified version of the corona construction from [AV21].
Throughout this section, $\mathcal{D}$ will denote the Christ-David cubes from
Lemma 2.1 on a fixed lower content regular set $E$.

######  Lemma 4.1 ([AV21], Main Lemma).

Let $k_{0}>0$ be an integer, and $E\subset\mathbb{R}^{n}$ be a compact lower
content $(d,c_{1})$-regular set. Let $Q_{0}\in\mathcal{D}_{0}$ and
$\mathcal{D}(Q_{0},k_{0})=\mathcal{D}(k_{0})=\bigcup_{k=0}^{k_{0}}\\{Q\in\mathcal{D}_{k}\,\mathrel{\mathop{\ordinarycolon}}\,Q\subseteq
Q_{0}\\}$. Then, there exists a family
$\mathsf{Top}(k_{0})\subseteq\mathcal{D}(k_{0})$ such that for any
$R\in\mathsf{Top}(k_{0})$ we have a tree of cubes denoted by
$\mathsf{Tree}(R)$ with root $R$, the trees partition $\mathcal{D}(k_{0})$

$\mathcal{D}(k_{0})=\bigcup_{R\in\mathsf{Top}(k_{0})}\mathsf{Tree}(R),$

and this partition has the following properties:

  1. (1)

There exists a constant $\eta=\eta(d,c_{1})$ so that

$\sum_{R\in\mathsf{Top}(k_{0})}\ell(R)^{d}\leq\eta^{-1}\mathcal{H}^{d}(Q_{0}),$
(4.1)

and $\eta\to 0$ as $c_{1}\to 0$111This is not explicitly stated in [AV21], but
it can be deduced from the proof, specifically see (3.4), (3.5) and (3.10)
there..

  2. (2)

Given $R\in\mathsf{Top}(k_{0})$ let $\mathsf{Stop}=\mathsf{Stop}(R)$ denote
the minimal cubes of $\mathsf{Tree}(R)$ and set

$\displaystyle
d_{R}(x)\mathrel{\mathop{\ordinarycolon}}=\inf_{Q\in\mathsf{Stop}}\left(\ell(Q)+\mathop{\mathrm{missing}}{dist}(x,Q)\right).$
(4.2)

For any $C_{0}>4$ and $\tau>0$, there is a collection
${\mathscr{C}}_{R}\subset\Delta$ of disjoint dyadic cubes covering
$C_{0}B_{R}\cap E$ so that if

$E_{R}\mathrel{\mathop{\ordinarycolon}}=\bigcup_{I\in{\mathscr{C}}_{R}}{\partial}_{d}I,$

where ${\partial}_{d}I$ denotes the $d$-dimensional skeleton of $I$, then the
following hold:

    1. (a)

$E_{R}$ is Ahlfors $d$-regular with constants depending on $C_{0},\tau,d,$ and
$c_{1}$.

    2. (b)

We have

$C_{0}B_{R}\cap E\subseteq\bigcup_{I\in{\mathscr{C}}_{R}}I\subseteq
2C_{0}B_{R}.$ (4.3)

    3. (c)

$E$ is close to $E_{R}$ in $C_{0}B_{R}$ in the sense that

$\mathop{\mathrm{missing}}{dist}(x,E_{R})\lesssim\tau d_{R}(x)\;\;\mbox{ for
all }x\in E\cap C_{0}B_{R}.$ (4.4)

    4. (d)

The dyadic cubes in ${\mathscr{C}}_{R}$ satisfy

$\ell(I)\sim\tau\inf_{x\in I}d_{R}(x)\mbox{ for all }I\in{\mathscr{C}}_{R}.$
(4.5)

The approximating sets $E_{R}$ are analogous to the surfaces $\Gamma$
constructed in §3.4. As before, $E_{R}$ inherits the PBP property from $E$.

######  Lemma 4.2.

Let $E\subset\mathbb{R}^{n}$ be a set with $d$-PBP. Let $E_{R}$ be one of the
sets from Lemma 4.1. Then $E_{R}$ also has $d$-PBP (with possibly a slightly
different constant $\delta^{\prime}$ which equals to $\delta$ up to a
multiplicative constant depending only on $d$).

The proof is analogous to that of Lemma 3.11.

###  4.2. $\beta$-numbers estimates

######  Remark 4.3.

We will keep careful track of the dependence of the various constants on
$\delta$ (the parameteres from the PBP condition). On the other hand, we will
usually not keep track of the dependence on $n,d$.

In this subsection we will prove the following.

######  Proposition 4.4.

If $E\subset\mathbb{R}^{n}$ has $d$-PBP with parameters $\delta>0$, then for
any $Q_{0}\in\mathcal{D}$, we have

$\displaystyle\ell(Q_{0})^{d}+\sum_{Q\in\mathcal{D}(Q_{0})}\beta_{E}^{d,2}(3B_{Q})^{2}\ell(Q)^{d}\lesssim_{n,d}C(\delta)\,\mathcal{H}^{d}(Q_{0}),$
(4.6)

where $C(\delta)\to\infty$ as $\delta\to 0$.

Theorem 1.10 follows immediately from this proposition.

###### Proof of Theorem 1.10.

First, by [AV21, (A.3)], we have that

$\ell(Q_{0})^{d}+\sum_{Q\subset
Q_{0}}\beta_{E}^{d,p}(C_{0}B_{Q})^{2}\,\ell(Q)^{d}\sim_{p,C_{0}}\ell(Q_{0})^{d}+\sum_{Q\subset
Q_{0}}\beta_{E}^{d,2}(3B_{Q})^{2}\ell(Q)^{d}$ (4.7)

whenever $C_{0}>1$ and $1\leq p\leq p(d)$. Together with Proposition 4.4 this
establishes one of the estimates from (1.7). The converse inequality in (1.7)
follows from [AS18, Theorem II], see also [AV21, Theorem A.1(1)] for a more
transparent statement. ∎

We now focus on proving Proposition 4.4. First, recall that for $k_{0}\geq 0$
we defined the truncated dyadic lattice as
$\mathcal{D}(Q_{0},k_{0})=\bigcup_{k=0}^{k_{0}}\\{Q\in\mathcal{D}_{k}\,\mathrel{\mathop{\ordinarycolon}}\,Q\subseteq
Q_{0}\\}$. Observe that to prove (4.6) it suffices to show

$\sum_{Q\in\mathcal{D}(Q_{0},k_{0})}\beta_{E}^{d,2}(3B_{Q})^{2}\ell(Q)^{d}\lesssim_{n,d}C(\delta)\,\mathcal{H}^{d}(Q_{0}),$

with bounds independent of $k_{0}\geq 0$. The estimate
$\ell(Q_{0})^{d}\lesssim\delta^{-1}\mathcal{H}^{d}(Q_{0})$ follows immediately
from lower content regularity of $E$.

Consider the coronization from Lemma 4.1 applied with $C_{0}=6$ and
sufficiently small $\tau$, to be fixed later. Let $R\in\mathsf{Top}(k_{0})$.
We start off by applying Lemma 2.13 with $E_{1}=E$ and $E_{2}=E_{R}$. For
$Q\in\mathcal{D}$, recall that $x_{Q}$ denotes the center of $Q$. By the
definition of $\mathsf{Tree}(R)$, we see that if $Q\in\mathsf{Tree}(R)$, then
there exists a dyadic cube $I\in{\mathscr{C}}_{R}$ with $x_{Q}\subset I$ (by
Lemma 4.1(2.b)). By (4.5), $\ell(I)\lesssim\tau\ell(Q)$. Hence, we may find a
point

$\displaystyle y_{Q}\in E_{R}\quad\mbox{ such that
}\quad|x_{Q}-y_{Q}|\leq\ell(Q),$ (4.8)

and we obtain that

$\displaystyle 3B_{Q}=B(x_{Q},3\ell(Q))\subset
6B(y_{Q},6\ell(Q))=\mathrel{\mathop{\ordinarycolon}}6B^{\prime}_{Q}.$ (4.9)

This implies that for each cube $Q\in\mathsf{Tree}(R)$ the hypotheses of Lemma
2.13 are satisfied (with $E_{1}=E,\ E_{2}=E_{R},\ 3B_{Q},$ and
$6B_{Q}^{\prime}$). Thus,

$\sum_{Q\in\mathsf{Tree}(R)}\beta_{E}^{d,2}(3B_{Q})^{2}\ell(Q)^{d}\lesssim\sum_{Q\in\mathsf{Tree}(R)}\beta_{E_{R}}^{d,2}(6B_{Q}^{\prime})^{2}\,\ell(Q)^{d}\\\
+\sum_{Q\in\mathsf{Tree}(R)}\left(\frac{1}{\ell(Q)^{d}}\int_{6B_{Q}\cap
E}\left(\frac{\mathop{\mathrm{missing}}{dist}(y,E_{R})}{\ell(Q)}\right)^{2}\,d\mathcal{H}^{d}_{\infty}(y)\right)\,\ell(Q)^{d}\eqqcolon
I_{1}+I_{2}.$ (4.10)

We estimate $I_{1}$. We denote by $\mathcal{D}^{E_{R}}$ a family of Christ-
David cubes for $E_{R}$ obtained by applying Lemma 2.1 to $E_{R}$. Using (4.8)
it is immediate to see that for each $Q\in\mathsf{Tree}(R)$ there exists
$P\in\mathcal{D}^{E_{R}}$ with $\ell(P)\sim\ell(R)$ and
$6B_{Q}^{\prime}\subset 3B_{P}$, and that any $P\in\mathcal{D}^{E_{R}}$
corresponds to at most a bounded number of $Q\in\mathsf{Tree}(R)$. Thus, we
get

$\displaystyle
I_{1}\lesssim\sum_{P\in\mathcal{D}^{E_{R}}}\beta_{E_{R}}^{d,2}(3B_{P})^{2}\ell(P)^{d}.$

Recall that $E_{R}$ is Ahlfors $d$-regular (Lemma 4.1) with constant $C$
depending on $C_{0},\tau,d$ and $c_{1}$ (and thus, by Lemma 2.2, on $\delta$),
and that it has $d$-PBP with parameter $\sim\delta$ by Lemma 4.2. It follows
from Corollary 2.9 that for $\mu_{R}=\mathcal{H}^{d}|_{E_{R}}$ we have

$\sum_{P\in\mathcal{D}^{E_{R}}}\beta_{\mu_{R},2}(3B_{P})^{2}\ell(P)^{d}\lesssim_{\tau}C(\delta)\ell(R)^{d}.$

Since $E_{R}$ is Ahlfors regular, we have
$\beta_{\mu_{R},2}(x,r)\sim\beta_{E_{R}}^{d,2}(x,r)$ (see §1.2 in [AS18]), and
so the two estimates above give

$I_{1}\lesssim_{\tau}C(\delta)\ell(R)^{d},$ (4.11)

where $C(\delta)\to\infty$ as $\delta\to 0$.

We move on to estimating $I_{2}$. Recall from Lemma 4.1, (4.3), that

$6B_{R}\cap E=C_{0}B_{R}\cap E\subset\bigcup_{I\in{\mathscr{C}}_{R}}I.$ (4.12)

For each $I\in{\mathscr{C}}_{R}$, set

${\mathscr{F}}(I)\mathrel{\mathop{\ordinarycolon}}=\\{S\in\mathcal{D}\
\mathrel{\mathop{\ordinarycolon}}\ S\cap I\neq\varnothing\mbox{ and
}\rho\ell(I)<\ell(S)\leq\ell(I)\\}.$ (4.13)

Then define ${\mathscr{F}}(R)=\cup_{I\in{\mathscr{C}}_{R}}{\mathscr{F}}(I)$
and set $\widetilde{\mathsf{Stop}}(R)$ to be the subfamily of maximal cubes in
${\mathscr{F}}(R)$ (by maximality, this family is disjoint and covers
${\mathscr{F}}(R)$). Note that

$6B_{R}\cap E\subset\bigcup_{S\in\widetilde{\mathsf{Stop}}(R)}S.$ (4.14)

Indeed, if $x\in 6B_{R}\cap E$, then by (4.12) there is an
$I\in{\mathscr{C}}_{R}$ so that $x\in I$. But $x$ is also contained in some
Christ-David cube $S$ such that $\rho\ell(I)\leq\ell(S)\leq\ell(I)$, and thus
(4.14) holds. Note that (4.14) trivially implies that for any
$Q\in\mathsf{Tree}(R)$ we have $6B_{Q}\cap
E\subset\bigcup_{S\in\widetilde{\mathsf{Stop}}(R)}S$, since $6B_{Q}\cap
E\subset 6B_{R}\cap E$.

For any $Q\in\mathsf{Tree}(R)$ and $x\in 6B_{Q}\cap E$, we claim that there
exists a unique cube $S\in\widetilde{\mathsf{Stop}}(R)$ containing $x$, and it
satisfies

$\mathop{\mathrm{missing}}{dist}(x,E_{R})\lesssim\ell(S).$ (4.15)

Indeed, by (4.14), $x\in S$ for some $S\in\widetilde{\mathsf{Stop}}(R)$, and
uniqueness follows from the maximality of $\widetilde{\mathsf{Stop}}(R)$. By
definition, there is an $I\in{\mathscr{C}}_{R}$ so that $S\cap
I\neq\varnothing$ and $\ell(I)\sim\ell(S)$. But then

$\mathop{\mathrm{missing}}{dist}(x,E_{R})\leq\operatorname{diam}(S)+\operatorname{diam}(I)\lesssim\ell(S).$

We can now estimate

$I_{2}=\sum_{Q\in\mathsf{Tree}(R)}\int_{6B_{Q}\cap
E}\left(\frac{\mathop{\mathrm{missing}}{dist}(y,E_{R})}{\ell(Q)}\right)^{2}\,d\mathcal{H}^{d}_{\infty}(y)\\\
\stackrel{{\scriptstyle\eqref{e:form1001},\eqref{eq:blaaaa}}}{{\lesssim}}\sum_{Q\in\mathsf{Tree}(R)}\sum_{\begin{subarray}{c}S\in\widetilde{\mathsf{Stop}}(R)\\\
S\cap
6B_{Q}\neq\varnothing\end{subarray}}\int_{S}\frac{\ell(S)^{2}}{\ell(Q)^{2}}\,d\mathcal{H}^{d}_{\infty}\lesssim\sum_{S\in\widetilde{\mathsf{Stop}}(R)}\ell(S)^{d+2}\sum_{\begin{subarray}{c}Q\in\mathsf{Tree}(R)\\\
6B_{Q}\cap S\neq\varnothing\end{subarray}}\frac{1}{\ell(Q)^{2}},$ (4.16)

where we may switch the order of summation since both sums are finite.

Fix $S\in\widetilde{\mathsf{Stop}}(R)$. We claim that for any $k\geq 0$ the
number of cubes $Q\in\mathsf{Tree}(R)\cap\mathcal{D}_{k}$ with $6B_{Q}\cap
S\neq\varnothing$ is bounded by some dimensional constant. Indeed, this is
clear as soon as $\ell(S)\leq\ell(Q)=\rho^{k}$. On the other hand, if
$\ell(S)>\rho^{k}$, then there is no $Q\in\mathsf{Tree}(R)\cap\mathcal{D}_{k}$
with $6B_{Q}\cap S\neq\varnothing$. To see this, observe that $6B_{Q}\cap
S\neq\varnothing$ implies
$\mathop{\mathrm{missing}}{dist}(S,Q)\lesssim\ell(Q)$. Also, if
$I\in{\mathscr{C}}_{R}$ is such that $S\in{\mathscr{F}}(I)$, then
$\mathop{\mathrm{missing}}{dist}(I,Q)\lesssim\ell(I)+\mathop{\mathrm{missing}}{dist}(S,Q)\lesssim\ell(S)+\ell(Q)$.
Hence

$\ell(S)\stackrel{{\scriptstyle\eqref{e:form1002}}}{{\sim}}\ell(I)\stackrel{{\scriptstyle\eqref{e:whitney-
like}}}{{\sim}}\tau\inf_{x\in
I}d_{R}(x)\stackrel{{\scriptstyle\eqref{e:d_F}}}{{\lesssim}}\tau(\ell(Q)+\mathop{\mathrm{missing}}{dist}(I,Q))\lesssim\tau\ell(Q)+\tau\ell(S).$

Choosing $\tau>0$ sufficiently small we get that $\ell(S)\leq\ell(Q)$.

The observation above allows us to estimate the interior sum in (4.16) as a
geometric sum:

$I_{2}\lesssim\sum_{S\in\widetilde{\mathsf{Stop}}(R)}\ell(S)^{d+2}\sum_{\begin{subarray}{c}Q\in\mathsf{Tree}(R)\\\
6B_{Q}\cap
S\neq\varnothing\end{subarray}}\frac{1}{\ell(Q)^{2}}\lesssim\sum_{S\in\widetilde{\mathsf{Stop}}(R)}\ell(S)^{d}.$

Now, given $S\in\widetilde{\mathsf{Stop}}(R)$, we claim that

$\\#\\{I\in{\mathscr{C}}_{R}\ \mathrel{\mathop{\ordinarycolon}}\
S\in{\mathscr{F}}(I)\\}\sim 1.$ (4.17)

This follows immediately from the fact that for dyadic cubes $I,J$ as above we
have $\ell(I)=\ell(J)$ and
$\mathop{\mathrm{missing}}{dist}(I,J)\lesssim\ell(I)$. Hence, for each
$S\in\widetilde{\mathsf{Stop}}(R)$ there is a bounded number of cubes in
${\mathscr{C}}_{R}$ of sidelength $\sim\ell(S)$ which intersect it. Finally,
we may conclude that

$\displaystyle I_{2}$
$\displaystyle\lesssim\sum_{S\in\widetilde{\mathsf{Stop}}(R)}\ell(S)^{d}\sim\sum_{S\in\widetilde{\mathsf{Stop}}(R)}\sum_{\begin{subarray}{c}I\in{\mathscr{C}}_{R}\\\
S\in{\mathscr{F}}(I)\end{subarray}}\ell(I)^{d}$
$\displaystyle=\sum_{I\in{\mathscr{C}}_{R}}\ell(I)^{d}\\#\\{S\in\widetilde{\mathsf{Stop}}(R)\
\mathrel{\mathop{\ordinarycolon}}\
S\in{\mathscr{F}}(I)\\}\lesssim\sum_{I\in{\mathscr{C}}_{R}}\ell(I)^{d},$

and hence, using the Ahlfors $d$-regularity of $E_{R}$,

$I_{2}\lesssim\sum_{I\in{\mathscr{C}}_{R}}\ell(I)^{d}\sim\sum_{I\in{\mathscr{C}}_{R}}\mathcal{H}^{d}(\partial_{d}I)\sim\mathcal{H}^{d}(E_{R})\lesssim_{\tau,\delta}\ell(R)^{d}.$
(4.18)

Putting the estimates for $I_{1}$ and $I_{2}$ together with (4.10) gives

$\displaystyle\sum_{Q\in\mathsf{Tree}(R)}\beta_{E}^{d,2}(3B_{Q})^{2}\ell(Q)^{d}\lesssim_{\tau}C(\delta)\ell(R)^{d},$
(4.19)

where $C(\delta)\to\infty$ as $\delta\to 0$. We conclude that

$\sum_{Q\in\mathcal{D}(Q_{0},k_{0})}\beta_{E}^{d,2}(3B_{Q})^{2}\ell(Q)^{d}\stackrel{{\scriptstyle{\rm
Lemma}\,\ref{l:corona}}}{{\lesssim}}\sum_{R\in\mathsf{Top}(k_{0})}\sum_{Q\in\mathsf{Tree}(R)}\beta_{E}^{d,2}(3B_{Q})^{2}\ell(Q)^{d}\\\
\stackrel{{\scriptstyle\eqref{e:TreeEst}}}{{\lesssim_{\tau}}}C(\delta)\sum_{R\in\mathsf{Top}(k_{0})}\ell(R)^{d}\stackrel{{\scriptstyle\eqref{e:ADR-
packing}}}{{\lesssim}}C(\delta)\eta^{-1}\mathcal{H}^{d}(Q_{0})\sim
C(\delta)\mathcal{H}^{d}(Q_{0}).$ (4.20)

This finishes the proof of Proposition 4.4.

##  5\. Dimension of wiggly sets

In this section we establish our estimate for the dimension of wiggly sets
with PBP, Theorem 1.12.

Through this section we will work with David-Christ cubes (Lemma 2.1)
associated to different sets. To avoid confusion we will write
$\mathcal{D}^{E}$ to denote the system of cubes associated to a given set
$E\subset\mathbb{R}^{n}$.

Recall that $p(d)=\tfrac{2d}{d-2}$ for $d>2$ and $p(d)=\infty$ if $d\leq 2$.
Theorem 1.12 will follow as an easy corollary of the following proposition.

######  Proposition 5.1.

Let $E\subset\mathbb{R}^{n}$ be a closed set with $d$-PBP with constant
$\delta>0$ and which is uniformly wiggly of dimension $d$, constant
$\beta_{0}$, and with respect to the Azzam-Schul $\beta_{E}^{d,p}$-numbers,
where $1\leq p<p(d)$. That is, $E$ satisfies

$\beta_{E}^{d,p}(x,r)\geq\beta_{0}\mbox{ for all }x\in E,\
0<r<\operatorname{diam}(E).$

Then

$\mathrm{dim}_{H}(E)\geq d+c\beta_{0}^{2}$ (5.1)

with $c$ depending on $\delta,p,n,d$.

###### Proof of Theorem 1.12 using Proposition 5.1.

Let $E$ be as in the statement of the theorem. In particular,
$\beta_{E,\infty}^{d}(B_{Q})>\beta_{0}$. By [AS18, Lemma 2.12, Lemma 2.13], we
have that

$\beta_{E,\infty}^{d}(B_{Q})\lesssim_{d}\beta_{E}^{d,p}(B_{Q})^{\frac{1}{d+1}}.$

Hence $E$ is uniformly wiggly of dimension $d$ with respect to
$\beta_{E}^{d,p}$ with constant $\beta_{0}^{\prime}\sim\beta_{0}^{d+1}$, and
(1.10) follows immediately from (5.1) with $\beta_{0}$ replaced by
$\beta_{0}^{\prime}$. ∎

We begin the proof of Proposition 5.1. Let $E\subset\mathbb{R}^{n}$ be a
closed set with $d$-PBP, with constant $\delta>0$. Without loss of generality
we assume $E$ is compact, $\operatorname{diam}(E)=1/2$, so that
$\mathcal{D}_{0}=\\{E\\}$, and $E\subset[0,1)^{n}$.

Recall that $\Delta$ is the standard dyadic lattice on $\mathbb{R}^{n}$, and
$\Delta_{k}=\\{I\in\Delta\ \mathrel{\mathop{\ordinarycolon}}\
\ell(I)=2^{-k}\\}$. Given a dyadic cube $I\in\Delta$ and $A\geq 1$, we will
denote by $AI$ the cube with the came center as $I$ and sidelength $A\ell(I)$.

Recall that $\rho=1/1000$ and that for $Q\in\mathcal{D}^{E}_{m}$ we have
$\ell(Q)=5\rho^{m}$, see Lemma 2.1. Given $k\geq 0$ let $j(k)$ be the unique
integer such that $2^{-j(k)}\leq 5\rho^{k}<2^{-j(k)+1}$. We set

$\displaystyle\Delta_{k}(E)\mathrel{\mathop{\ordinarycolon}}=\left\\{I\in\Delta_{j(k)}\,\mathrel{\mathop{\ordinarycolon}}\,I\cap
E\neq\varnothing\right\\}.$ (5.2)

We defined $j(k)$ in such a way that if $I\in\Delta_{k}(E)$ and
$Q\in\mathcal{D}^{E}_{k}$ then $\ell(I)\leq\ell(Q)\leq 2\ell(I)$. In
particular, $\ell(I)\sim\rho^{k}$.

Set

$E_{k}\mathrel{\mathop{\ordinarycolon}}=\bigcup_{I\in\Delta_{k}(E)}\partial_{d}I,$
(5.3)

where $\partial_{d}I$ is the $d$-dimensional dyadic skeleton of $I$. Just like
in Lemma 3.11 and Lemma 4.2, one may show that since $E$ has $d$-PBP with
constant $\delta$, then $E_{k}$ also has $d$-PBP (with constant
$\delta^{\prime}\sim\delta$).

######  Lemma 5.2.

Let $k\geq j\geq 1$ be integers, and let $I\in\Delta_{j}(E)$. We have

$\sum_{m=j}^{k}\sum_{\begin{subarray}{c}Q\in\mathcal{D}^{E}_{m}\\\ Q\cap
3I\neq\varnothing\end{subarray}}\beta_{E}^{d,p}(B_{Q})^{2}\ell(Q)^{d}\leq
C\mathcal{H}^{d}(E_{k}\cap AI),$ (5.4)

where $A=24$, and $C$ depends on $\delta,n,d$ and $p$.

###### Proof.

Without loss of generality we may assume that $p=2$ (see [AV21, (A.3)]).
Consider a cube $Q\in\mathcal{D}^{E}_{m}$ with $j\leq m\leq k$ and $Q\cap
3I\neq\varnothing$. Let $P\in\mathcal{D}^{E_{k}}$ be a cube with
$\ell(P)=\ell(Q)$ and such that $B_{Q}\cap B_{P}\neq\varnothing$ (such a cube
exists because $\mathop{\mathrm{missing}}{dist}(x_{Q},E_{k})\leq
2^{-j(k)}\leq\ell(Q)$). In particular, we have $2B_{Q}\subset 4B_{P}$ and we
may apply Lemma 2.13 with $E_{1}=E$ and $E_{2}=E_{k}$ to obtain

$\displaystyle\beta_{E}^{d,2}(B_{Q})\lesssim\beta_{E}^{d,2}(2B_{Q})\lesssim\beta^{d,2}_{E_{k}}(4B_{P})+\left(\frac{1}{\ell(Q)^{d}}\int_{4B_{Q}\cap
E}\left(\frac{\mathop{\mathrm{missing}}{dist}(y,E_{k})}{\ell(Q)}\right)^{2}\,d\mathcal{H}_{\infty}^{d}(y)\right)^{\frac{1}{2}}.$

Note that we have $B_{P}\cap 11I\neq\varnothing$ because $\ell(Q)\leq
2\ell(I),\ Q\cap 3I\neq\varnothing$ and $B_{P}\cap B_{Q}\neq\varnothing$.

Multiplying by $\ell(Q)^{d}$ and summing over $Q\in\mathcal{D}^{E}_{m}$ with
$j\leq m\leq k$ and $Q\cap 3I\neq\varnothing$ yields

$\sum_{m=j}^{k}\sum_{\begin{subarray}{c}Q\in\mathcal{D}^{E}_{m}\\\ Q\cap
3I\neq\varnothing\end{subarray}}\beta_{E}^{d,2}(B_{Q})^{2}\ell(P)^{d}\lesssim\sum_{m=j}^{k}\sum_{\begin{subarray}{c}P\in\mathcal{D}_{m}^{E_{k}}\\\
B_{P}\cap
12I\neq\varnothing\end{subarray}}\beta^{d,2}_{E_{k}}(4B_{P})^{2}\ell(P)^{d}\\\
+\sum_{m=j}^{k}\sum_{\begin{subarray}{c}Q\in\mathcal{D}^{E}_{m}\\\ Q\cap
3I\neq\varnothing\end{subarray}}\left(\frac{1}{\ell(Q)^{d}}\int_{4B_{Q}\cap
E}\left(\frac{\mathop{\mathrm{missing}}{dist}(y,E_{k})}{\ell(Q)}\right)^{2}\,\mathcal{H}^{d}_{\infty}(y)\right)\ell(P)^{d}\eqqcolon
S_{1}+S_{2}.$ (5.5)

We begin by estimating $S_{1}$. Let

$\mathcal{R}=\\{R\in\mathcal{D}^{E_{k}}_{j}\
\mathrel{\mathop{\ordinarycolon}}\ B_{R}\cap 11I\neq\varnothing\\}.$

Note that for each cube $P$ appearing in $S_{1}$ there exists a unique
$R\in\mathcal{R}$ such that $P\subset R$. Observe also that all the cubes
$R\in\mathcal{R}$ are contained in $19I$. Applying these observations together
with Theorem 1.10 to the set $E_{k}$ and cubes $R\in\mathcal{R}$ yields

$S_{1}\lesssim\sum_{R\in\mathcal{R}}\sum_{P\in\mathcal{D}(R)}\beta^{d,2}_{E_{k}}(4B_{P})^{2}\ell(P)^{d}\lesssim_{\delta}\sum_{R\in\mathcal{R}}\mathcal{H}^{d}(R)\leq\mathcal{H}^{d}(E_{k}\cap
19I).$ (5.6)

We move on to estimating $S_{2}$. This estimate is similar to that of
"$I_{2}$" in (4.10). First, we use the definition of $E_{k}$ and the
subadditivity of Hausdorff content to get

$\int_{4B_{Q}\cap
E}\left(\frac{\mathop{\mathrm{missing}}{dist}(y,E_{k})}{\ell(P)}\right)^{2}\,d\mathcal{H}^{d}_{\infty}(y)\lesssim\sum_{\begin{subarray}{c}J\in\Delta_{k}(E)\\\
J\cap
4B_{Q}\neq\varnothing\end{subarray}}\frac{\ell(J)^{2}}{\ell(Q)^{2}}\mathcal{H}^{d}_{\infty}(J\cap
4B_{Q}\cap E)\lesssim\sum_{\begin{subarray}{c}J\in\Delta_{k}(E)\\\ J\cap
4B_{Q}\neq\varnothing\end{subarray}}\frac{\ell(J)^{d+2}}{\ell(Q)^{2}}.$

It is easy to check that if $Q\in\mathcal{D}^{E}_{m}$ with $j\leq m\leq k$ and
$Q\cap 3I\neq\varnothing$, then for $J$ as above we have $J\subset 24I$. It
follows that

$S_{2}\leq\sum_{m=j}^{k}\sum_{\begin{subarray}{c}Q\in\mathcal{D}^{E}_{m}\\\
Q\cap
3I\neq\varnothing\end{subarray}}\sum_{\begin{subarray}{c}J\in\Delta_{k}(E)\\\
J\cap
4B_{Q}\neq\varnothing\end{subarray}}\frac{\ell(J)^{d+2}}{\ell(Q)^{2}}\lesssim\sum_{\begin{subarray}{c}J\in\Delta_{k}(E)\\\
J\subset
24I\end{subarray}}\ell(J)^{d+2}\sum_{\begin{subarray}{c}Q\in\mathcal{D}^{E}\\\
\ell(Q)\geq\rho^{k}\mathrel{\mathop{\ordinarycolon}}\,4B_{Q}\cap
J\neq\varnothing\end{subarray}}\frac{1}{\ell(Q)^{2}}\\\
\lesssim\sum_{\begin{subarray}{c}J\in\Delta_{k}(E)\\\ J\subset
24I\end{subarray}}\ell(J)^{d}.$

We also have

$\sum_{\begin{subarray}{c}J\in\Delta_{k}(E)\\\ J\subset
24I\end{subarray}}\ell(J)^{d}\sim\sum_{\begin{subarray}{c}J\in\Delta_{k}(E)\\\
J\subset
24I\end{subarray}}\mathcal{H}^{d}(\partial_{d}J)\sim\mathcal{H}^{d}(E_{k}\cap
24I).$

Hence, $S_{2}\lesssim\mathcal{H}^{d}(E_{k}\cap 24I)$. Putting this together
with (5.5) and (5.6) gives (5.4).

∎

######  Lemma 5.3.

Let $k\geq j>0$ be integers and $I\in\Delta_{j}(E)$. We have

$\sum_{m=j}^{k}\sum_{Q\in\mathcal{D}^{E}_{m}\atop Q\cap
3I\neq\varnothing}\beta_{E}^{d,p}(B_{Q})^{2}\ell(Q)^{d}\geq(k-j)C\delta\beta_{0}^{2}\rho^{jd},$
(5.7)

where $C$ is a constant depending only on $n,d$, but not on $\delta$.

###### Proof.

Recall first that $E$ is lower $(d,c_{1})$-regular (with $c_{1}\sim\delta$,
see Lemma 2.2), so that in particular $\mathcal{H}^{d}_{\infty}(E\cap
3I)\gtrsim\delta\rho^{dj}$. Let $m\geq j$. Since the balls $\\{B_{Q}\
\mathrel{\mathop{\ordinarycolon}}\ Q\in\mathcal{D}^{E}_{m},\,Q\cap
3I\neq\varnothing\\}$ are a cover of $E\cap 3I$, we get

$\sum_{Q\in\mathcal{D}^{E}_{m}\atop Q\cap
3I\neq\varnothing}\ell(Q)^{d}\gtrsim\delta\rho^{dj}.$

Recalling that $E$ is uniformly wiggly we arrive at

$\sum_{Q\in\mathcal{D}^{E}_{m}\atop Q\cap
3I\neq\varnothing}\beta_{E}^{d,p}(B_{Q})^{2}\ell(Q)^{d}\geq\beta_{0}^{2}\sum_{Q\in\mathcal{D}^{E}_{m}\atop
Q\cap 3I\neq\varnothing}\ell(Q)^{d}\gtrsim\delta\beta_{0}^{2}\rho^{dj}.$

Summing over $j\leq m\leq k$ gives (5.7). ∎

Recall that our aim was to prove that

$\displaystyle{\rm dim}_{H}(E)\geq d+c\beta_{0}^{2}.$

This estimate follows immediately from Theorem 8.8 in [Mat95] and the next
lemma.

######  Lemma 5.4.

There exists a non-zero measure $\mu$ with $\operatorname{supp}\mu\subset E$
and such that

$\mu(B(x,r))\lesssim r^{d+c\beta_{0}^{2}}\quad\text{for $x\in E$ and $r>0$},$

where $c$ depends on $\delta,p,n,d$.

###### Proof.

The construction of $\mu$ will is similar to the proof of Frostman’s lemma, as
in [Mat95, Theorem 8.8]. First, we need to define a tree-like collection of
dyadic cubes intersecting $E$ which we will use in our construction.

Let $k>j\geq 1$ be integers, and let $I\in\Delta_{j}(E)$. Using (5.4) and
(5.7) we get that

$\displaystyle\mathcal{H}^{d}(E_{k}\cap
AI)\gtrsim_{\delta}(k-j)\,\beta_{0}^{2}\,\rho^{dj}.$ (5.8)

Let

$D_{k}(I)\coloneqq\Delta_{k}(E\cap AI)=\\{J\in\Delta_{k}\
\mathrel{\mathop{\ordinarycolon}}\ J\cap AI\cap E\neq\varnothing\\}.$

It follows easily from the definition of $E_{k}$ (5.3) that

$E_{k}\cap AI\subset\bigcup_{J\in D_{k}(I)}{\partial}_{d}J.$

Hence,

$\displaystyle\mathcal{H}^{d}(E_{k}\cap AI)\lesssim\rho^{dk}\,\\#D_{k}(I).$

Together with (5.8) this gives

$\displaystyle\\#D_{k}(I)\gtrsim_{\delta}(k-j)\,\beta_{0}^{2}\,\rho^{d(j-k)}.$
(5.9)

Observe that $\Delta(E)=\bigcup_{k\geq 0}\Delta_{k}(E)$ can be endowed with a
natural tree structure, and this structure is used in the usual proof of the
Frostman’s lemma. We would like to define a similar structure for the
collections $D_{k}(I)$ for $I\in\Delta_{j}(E),k\geq j\geq 0$, but we need to
take extra care due to possible overlaps between collections
$D_{k}(I),D_{k}(J)$ when $I,J\in\Delta_{j}(E)$.

To overcome this technicality, for any integers $j,k$ with $k\geq j\geq 1$ and
$I\in\Delta_{j}(E)$ let $D^{\prime}_{k}(I)\subset D_{k}(I)$ be a maximal
subfamily of $D_{k}(I)$ such that for any $J_{1},J_{2}\in D^{\prime}_{k}(I)$
we have $A^{\prime}J_{1}\cap A^{\prime}J_{2}=\varnothing,$ where
$A^{\prime}=A+1$. Note that since for each $J\in D^{\prime}_{k}(I)$ we have
$J\subset AI$, we get

$\bigcup_{J\in D^{\prime}_{k}(I)}A^{\prime}J\subset A^{\prime}I.$

Moreover, since all the cubes in $D_{k}(I)$ have equal sidelength, the cubes
$\\{A^{\prime}J\\}_{J\in D_{k}(I)}$ have bounded overlap, and so it follows
that

$\\#D^{\prime}_{k}(I)\gtrsim\\#D_{k}(I).$

Recalling (5.9), we get that

$\\#D^{\prime}_{k}(I)\geq C_{1}(k-j)\,\beta_{0}^{2}\,\rho^{d(j-k)}.$ (5.10)

for some $0<C_{1}<1$ depending on $\delta,p,n,d$.

Let

$\displaystyle\kappa\coloneqq\lceil 10^{6}\,C_{1}^{-1}\beta_{0}^{-2}\rceil,$

and observe that for $c\coloneqq C_{1}/10^{6}$ we have

$\kappa\,C_{1}\beta_{0}^{2}\geq
10^{6}=\rho^{-2}\geq\rho^{-\kappa\,c\beta_{0}^{2}},$

where in the last inequality we use also the fact that
$C_{1},\beta_{0}\in(0,1)$. Hence, if $k$ is such that $k=j+\kappa$, the
inequality (5.10) gives

$\displaystyle\\#D^{\prime}_{j+\kappa}(I)\geq\rho^{-\kappa(d+c\beta_{0}^{2})}.$
(5.11)

We are ready to define the tree structure we will use to construct $\mu$. Set

$\displaystyle{\mathcal{S}}_{0}=\Delta_{0}(E)=\\{[0,1)^{n}\\},$

where we used our assumption $E\subset[0,1)^{n}$. Let $j\geq 0$ and assume
that ${\mathcal{S}}_{j}$ has already been defined, that
${\mathcal{S}}_{j}\subset\Delta_{j\kappa}(E)$, and that for every
$I,J\in{\mathcal{S}}_{j},\ I\neq J,$ we have $A^{\prime}I\cap
A^{\prime}J=\varnothing$. For each $I\in{\mathcal{S}}_{j}$ we set

${\mathcal{S}}(I)=D^{\prime}_{(j+1)\kappa}(I),$

and

${\mathcal{S}}_{j+1}=\bigcup_{I\in{\mathcal{S}}_{j}}{\mathcal{S}}(I).$

Note that for every $J\in{\mathcal{S}}_{j+1}$ there is a unique “parent”
$I\in{\mathcal{S}}_{j}$ such that $J\subset AI$ (and $A^{\prime}J\subset
A^{\prime}I$). Moreover, for every $I,J\in{\mathcal{S}}_{j+1},\ I\neq J,$ we
have $A^{\prime}I\cap A^{\prime}J=\varnothing$ (either $I$ and $J$ have
distinct parents, or they have the same parent $I^{\prime}$ in which case we
use the definition of $D^{\prime}_{(j+1)\kappa}(I^{\prime})$). Finally,
observe that if $s\coloneqq d+c\beta_{0}^{2}$, then by (5.11)

$\\#{\mathcal{S}}_{j}\geq\rho^{-j\kappa
s}\quad\text{and}\quad\\#{\mathcal{S}}(I)\geq\rho^{-\kappa s}\text{ for each
$I\in{\mathcal{S}}_{j}$}.$ (5.12)

We are ready to construct the measure $\mu$. It is obtained as a weak limit of
measures $\mu_{j},\,j\geq 0$. The definition of $\mu_{j}$ follows the usual
“bottom-to-top” construction of the Frostman measure, as in Theorem 8.8 of
[Mat95].

First, we set

$\mu_{j}^{j}=\rho^{j\kappa
s}\sum_{I\in{\mathcal{S}}_{j}}\frac{\mathcal{L}^{n}|_{A^{\prime}I}}{\mathcal{L}^{n}(A^{\prime}I)}.$

Assume that the measure $\mu_{j}^{i}$ with $1\leq i\leq j$ has already been
defined, and that for each $I\in{\mathcal{S}}_{i}$ we have
$\mu_{j}^{i}(A^{\prime}I)=\rho^{i\kappa s}$. We define $\mu_{j}^{i-1}$ by
modifying $\mu_{j}^{i}$ at the level of ${\mathcal{S}}_{i-1}$: for each
$I\in{\mathcal{S}}_{i-1}$ we set

$\mu_{j}^{i-1}|_{A^{\prime}I}\coloneqq\rho^{(i-1)\kappa
s}\mu_{j}^{i}(A^{\prime}I)^{-1}\mu_{j}^{i}|_{A^{\prime}I}.$

Finally, we define $\mu_{j}=\mu_{j}^{0}$. Clearly, $\mu_{j}([0,1)^{n})=1$.

Observe that the quantity used in the definition of
$\mu_{j}^{i-1}|_{A^{\prime}I}$ satisfies

$\rho^{(i-1)\kappa s}\mu_{j}^{i}(A^{\prime}I)^{-1}=\rho^{\kappa
s}(\\#{\mathcal{S}}(I))^{-1}\overset{\eqref{eq:cardSI}}{\leq}1.$

It follows that for every $i\leq k\leq j$ and $I\in{\mathcal{S}}_{k}$ we have
$\mu_{j}^{i}(A^{\prime}I)\leq\ell(I)^{s}.$ In particular,

$\mu_{j}(A^{\prime}I)\leq\ell(I)^{s}\quad\text{for all
$I\in{\mathcal{S}}_{k},\,0\leq k\leq j$.}$ (5.13)

Let $\mu_{j_{k}}$ be a subsequence of $\mu_{j}$ converging weakly, and define
$\mu$ to be the weak limit of $\mu_{j_{k}}$. Since $\mu_{j}(\mathbb{R}^{n})=1$
for all $j$, we also have $\mu(\mathbb{R}^{n})=1$. Using (5.13), a standard
argument gives

$\mu(B(x,r))\lesssim r^{s}\quad\text{for $x\in\mathbb{R}^{n},\,r>0$},$

see p.114 in [Mat95] for details. Finally, $\operatorname{supp}\mu\subset E$
because

$\operatorname{supp}\mu_{j}\subset\bigcup_{I\in\Delta_{j\kappa}(E)}I.$

∎

This completes the proof of Proposition 5.1.

##  6\. Denjoy’s conjecture

In this section we prove Theorem 1.7. Let $\Sigma\subset\mathbb{R}^{d+1}$ be a
compact set with $d$-PBP with parameter $\delta>0$, and let $E\subset\Sigma$
be compact. Let $\mathcal{D}$ denote the Christ-David cubes on $\Sigma$, as in
Lemma 2.1. We are going to use Theorem 2.10 to get a lower bound on
$\Gamma_{n,d}(E)$.

By Frostman’s lemma (see [Tol14, Lemma 1.23]), there exists a measure $\mu$
supported on $E$ such that $\mu(B(x,r))\leq r^{d}$ for all
$x\in\mathbb{R}^{n},\,r>0$, and $\mu(E)\sim\mathcal{H}_{\infty}^{d}(E)$. In
fact, it follows immediately from the proof of [Tol14, Lemma 1.23] that one
even has $\mu(B(x,r))\leq\mathcal{H}^{d}_{\infty}(E\cap B(x,r))$.

Recall that $\Sigma$ is lower content regular with constant $\sim\delta$ (see
Lemma 2.2). Using the fact that for any ball $x\in E$ and $0<r<\infty$ we have

$\mu(B(x,r))\leq\mathcal{H}^{d}_{\infty}(E\cap
B(x,r))\leq\mathcal{H}^{d}_{\infty}(\Sigma\cap B(x,r))$

it follows from the definitions of $\beta_{\mu,2}$ and $\beta_{\Sigma}^{d,2}$
that

$\displaystyle\beta_{\mu,2}(x,r)\lesssim\beta_{\Sigma}^{d,2}(x,r).$ (6.1)

Indeed, given a plane $L\in\mathcal{A}(n,d)$ infimising $\beta_{E}^{d,2}(x,r)$
set

$E_{t}=E_{t}(x,r)\mathrel{\mathop{\ordinarycolon}}=\\{y\in B(x,r)\cap
E\,\mathrel{\mathop{\ordinarycolon}}\,\mathop{\mathrm{missing}}{dist}(y,L)^{2}>tr^{2}\\}.$

Let ${\mathcal{C}}$ be a covering of $E_{t}$ such that
$\sum_{B\in{\mathcal{C}}}\operatorname{diam}(B)^{d}\leq
2\mathcal{H}^{d}_{\infty}(E_{t})$. We may assume that the covering consists of
balls (see [Mat95], Chapter 5). Since ${\mathcal{C}}$ covers $E_{t}$, and
$\mu$ is a Frostman measure, we have

$\mu(E_{t})\leq\sum_{B\in{\mathcal{C}}}\mu(B)\leq\sum_{B\in{\mathcal{C}}}r(B)^{d}\lesssim\mathcal{H}^{d}_{\infty}(E_{t}).$

Hence,

$\beta_{\mu,2}(x,r)\leq\int_{0}^{\infty}\mu(E_{t})\,t\,dt\lesssim\int_{0}^{\infty}\mathcal{H}^{d}_{\infty}(E_{t})\,t\,dt\sim\beta_{E}^{d,2}(x,r),$

and this gives (6.1) since $E\subset\Sigma$.

Now recall that $\theta_{\mu}(x,r)=\mu(B(x,r))\,r^{-d}$ is the $d$-density of
$\mu$ in the ball $B(x,r)$. Since $\mu(B(x,r))\leq r^{d}$, we have
$\theta_{\mu}(x,r)\leq 1$. Arguing as we did below (3.3) and using (6.1) we
get

$\beta^{2}(\mu)\mathrel{\mathop{\ordinarycolon}}=\iint_{0}^{\infty}\beta_{\mu,2}(x,r)^{2}\theta_{\mu}(x,r)\,\frac{dr}{r}d\mu(x)\\\
\leq\iint_{0}^{1}\beta_{\mu,2}(x,r)^{2}\,\frac{dr}{r}d\mu(x)+\iint_{1}^{\infty}\beta_{\mu,2}(x,r)^{2}\,\frac{dr}{r}d\mu(x)\\\
\lesssim\sum_{Q\in\mathcal{D}}\beta_{\mu,2}(3B_{Q})^{2}\mu(Q)+\iint_{1}^{\infty}\theta_{\mu}(x,r)^{2}\,\frac{dr}{r}d\mu(x)\\\
\lesssim\sum_{Q\in\mathcal{D}}\beta_{\Sigma}^{d,2}(3B_{Q})^{2}\mu(Q)+\mu(E)\lesssim\sum_{Q\in\mathcal{D}_{\Sigma}}\beta_{\Sigma}^{d,2}(3B_{Q})^{2}\ell(Q)^{d}+\mu(E).$

Applying Theorem 1.10 to estimate the last sum above yields

$\beta^{2}(\mu)\lesssim\sum_{Q\in\mathcal{D}}\beta_{\Sigma}^{d,2}(3B_{Q})^{2}\ell(Q)^{d}+\mu(E)\lesssim
C(\delta)\mathcal{H}^{d}(\Sigma)+\mathcal{H}^{d}_{\infty}(E)\lesssim
C(\delta)\mathcal{H}^{d}(\Sigma),$

so that $\beta^{2}(\mu)\leq C_{1}(\delta)\mathcal{H}^{d}(\Sigma)$ for some
$C_{1}(\delta)>1$.

Set

$C_{2}\mathrel{\mathop{\ordinarycolon}}=\left(\frac{\mu(E)}{C_{1}(\delta)\mathcal{H}^{d}(\Sigma)}\right)^{\frac{1}{2}},$

and define the measure $\sigma\mathrel{\mathop{\ordinarycolon}}=C_{2}\mu$.
Since $C_{2}\in(0,1)$, we have $\sigma(B(x,r))\leq\mu(B(x,r))\leq r^{d}$.
Moreover,

$\beta^{2}(\sigma)=C_{2}^{3}\beta^{2}(\mu)\leq\left(\frac{\mu(E)}{C_{1}(\delta)\mathcal{H}^{d}(\Sigma)}\right)^{\frac{3}{2}}C_{1}(\delta)\mathcal{H}^{d}(\Sigma)=C_{2}\,\mu(E)=\sigma(E).$

Then, by Theorem 2.10,

$\displaystyle\Gamma_{n,d}(E)\geq\sigma(E)=C_{2}\mu(E)=\left(\frac{\mu(E)}{C_{1}(\delta)\mathcal{H}^{d}(\Sigma)}\right)^{\frac{1}{2}}\mu(E)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad=C_{1}(\delta)^{\frac{1}{2}}\frac{\mu(E)^{\frac{3}{2}}}{\mathcal{H}^{d}(\Sigma)^{\frac{1}{2}}}\sim
C_{1}(\delta)^{\frac{1}{2}}\frac{\mathcal{H}^{d}_{\infty}(E)^{\frac{3}{2}}}{\mathcal{H}^{d}(\Sigma)^{\frac{1}{2}}}.$

This finishes the proof of Theorem 1.7.

##  Appendix A Frostman’s lemma for lower content regular sets

We prove the following version of the classical Frostman’s lemma.

######  Theorem A.1.

Let $E\subset[0,1]^{n}$ be a compact, lower content $(d,c_{1})$-regular set.
Then, there exists a measure $\mu$ satisfying the following properties:

  1. (1)

$\operatorname{supp}\mu=E$,

  2. (2)

$\mu(E)=\mathcal{H}^{d}_{\infty}(E)$,

  3. (3)

$\mu$ has polynomial growth, that is, there exists a constant $C_{1}\geq 1$
such that for all $x\in E$ and $0<r<\operatorname{diam}(E)$ we have

$\mu(B(x,r))\leq C_{1}r^{d},$

  4. (4)

$\mu$ is doubling, that is, there exists a constant $C_{db}\geq 1$ such that
for all $x\in E$ and $0<r<\operatorname{diam}(E)$ we have

$\mu(B(x,2r))\leq C_{db}\,\mu(B(x,r)).$ (A.1)

  5. (5)

the $d$-dimensional density of $\mu$ is almost monotone, that is, there exists
a constant $A\geq 1$ such that if $P,Q\in\mathcal{D}$, and $P\subset Q$, then

$\theta_{\mu}(P)\leq A\,\theta_{\mu}(Q).$

In the above, $C_{1}$ depends only on $d,n$, while $C_{db}$ and $A$ also
depend on the lower regularity constant $c_{1}$.

The usual proof of Frostman’s lemma is “bottom-to-top”: it starts from small
scales and goes up (see [Mat95, Theorem 8.8]). An alternative, simpler proof
is due to Tolsa [Tol14, Theorem 1.23], who came up with a “top-to-bottom”
construction. To prove Theorem A.1 we will modify Tolsa’s construction so that
the resulting Frostman measure is doubling. Roughly speaking, at each step we
are going to modify the measure by redistributing the mass between cubes where
the doubling condition fails.

###  A.1. Construction of a sequence of measures

In this subsection we construct a sequence of measures $\mu_{k}$. The measure
$\mu$ from Theorem A.1 is going to be the weak limit of $\mu_{k}$.

Let $\mathcal{D}$ be the David-Christ lattice on $E$, as in Lemma 2.1. For a
cube $R\in\mathcal{D}_{k}$, we denote by $\mathsf{Ch}(R)$ the cubes $Q$ in
$\mathcal{D}_{k+1}$ with $Q\subset R$. By $\mathsf{Nbd}(Q)$ we denote the
cubes $P\in\mathcal{D}_{k}$ so that
$\mathop{\mathrm{missing}}{dist}(P,Q)\leq\ell(Q)$. Note that there exists a
dimensional constant $c_{n}>1$ such that

$\\#\mathsf{Ch}(Q)+\\#\mathsf{Nbd}(Q)\leq c_{n}.$ (A.2)

Recall that $Q^{1}$ denotes the parent cube of $Q$, and also that the balls
$B(Q)=B(x_{Q},c_{0}\ell(Q))$ and $B_{Q}=B(x_{Q},\ell(Q))$ satisfy $B(Q)\cap
E\subset Q\subset B_{Q}$, where $\ell(Q)=\rho^{k}$ for $Q\in\mathcal{D}_{k}$.

######  Proposition A.2.

Let $E\subset[0,1]^{n}$ be a compact, lower content $(d,c_{1})$-regular set.
There exists a constant $C_{0}=C_{0}(n,d,c_{1})>1$ and sequence of Radon
measures $\mu_{k}$, $k\geq 0$, absolutely continuous with respect to
$\mathcal{L}^{n}$, such that, for each $k$, we have

$\displaystyle\mu_{k}(\mathbb{R}^{n})=\mathcal{H}^{d}_{\infty}(E),$ (A.3)
$\displaystyle\operatorname{supp}\mu_{k}=\bigcup_{Q\in\mathcal{D}_{k}}B(Q),$
(A.4)

and moreover for all $Q\in\mathcal{D}_{k}$

$\displaystyle\mu_{k}(B(Q))$ $\displaystyle\leq\mathcal{H}^{d}_{\infty}(Q)\leq
2\ell(Q)^{d},$ (A.5) $\displaystyle\mu_{k}(B(Q))$ $\displaystyle\leq
C_{0}\,\mu_{k}(B(P))\quad\text{for all $P\in\mathsf{Nbd}(Q)$},$ (A.6)
$\displaystyle C_{0}^{-1}\mu_{k-1}(B(Q^{1}))$
$\displaystyle\leq\mu_{k}(B(Q))\lesssim\mu_{k-1}(B(Q^{1}))\quad\text{if $k\geq
1$}.$ (A.7)

Note that since for all $P\in\mathsf{Nbd}(Q)$ we have $Q\in\mathsf{Nbd}(P)$,
if the property (A.6) holds for all $Q\in\mathcal{D}_{k}$, then

$C_{0}^{-1}\mu_{k}(B(P))\leq\mu_{k}(B(Q))\leq
C_{0}\,\mu_{k}(B(P))\quad\text{for all $P\in\mathsf{Nbd}(Q)$}.$ (A.8)

We construct $\mu_{k}$ inductively. Let $Q_{0}=E\in\mathcal{D}_{0}$; that is,
$Q_{0}$ is the top cube. We define

$\mu_{0}\coloneqq\mathcal{H}^{d}_{\infty}(Q_{0})\cdot\frac{\mathcal{L}^{n}|_{B(Q_{0})}}{\mathcal{L}^{n}(B(Q_{0}))}.$

Clearly, $\mu_{0}$ satisfies all the properties from Proposition A.2. Now,
assume that $\mu_{k-1}$ has already been defined and satisfies the properties
(A.3)–(A.7).

####  A.1.1. Auxiliary measure $\eta_{k}$

For $R\in\mathcal{D}_{k-1}$ and for each $Q\in\mathsf{Ch}(R)$, set

$\eta_{k}|_{B(Q)}\coloneqq\left(\frac{\mathcal{H}^{d}_{\infty}(Q)}{\sum_{P\in\mathsf{Ch}(R)}\mathcal{H}^{d}_{\infty}(P)}\cdot\mu_{k-1}(B(R))\right)\frac{\mathcal{L}^{n}|_{B(Q)}}{\mathcal{L}^{n}(B(Q))}.$
(A.9)

Observe that

$\sum_{Q\in\mathsf{Ch}(R)}\eta_{k}(B(Q))=\mu_{k-1}(B(R)).$ (A.10)

######  Lemma A.3.

We have
$\eta_{k}(\mathbb{R}^{n})=\mu_{k-1}(\mathbb{R}^{n})=\mathcal{H}^{d}_{\infty}(E)$,
so that condition (A.3) holds for $\eta_{k}$.

###### Proof.

We have

$\eta_{k}(\mathbb{R}^{n})=\sum_{Q\in\mathcal{D}_{k}}\eta_{k}(B(Q))=\sum_{R\in\mathcal{D}_{k-1}}\sum_{Q\in\mathsf{Ch}(R)}\eta_{k}(B(Q))\\\
\overset{\eqref{eq:etachildrencons}}{=}\sum_{R\in\mathcal{D}_{k-1}}\mu_{k-1}(B(R))=\mu_{k-1}(\mathbb{R}^{n})=\mathcal{H}^{d}_{\infty}(E).$

∎

######  Lemma A.4.

The polynomial growth condition (A.5) holds for $\eta_{k}$.

###### Proof.

Let $R\in\mathcal{D}_{k-1}$ and $Q\in\mathsf{Ch}(Q)$. Since (A.5) holds for
$\mu_{k-1}$, we have
$\mu_{k-1}(B(R))\leq\mathcal{H}^{d}_{\infty}(R)\leq\sum_{P\in\mathsf{Ch}(R)}\mathcal{H}^{d}_{\infty}(P)$,
by subadditivity of $\mathcal{H}^{d}_{\infty}$. Then,

$\eta_{k}(B(Q))=\mathcal{H}^{d}_{\infty}(Q)\cdot\frac{\mu_{k-1}(B(R))}{\sum_{P\in\mathsf{Ch}(R)}\mathcal{H}^{d}_{\infty}(P)}\leq\mathcal{H}^{d}_{\infty}(Q)\leq
2\ell(Q)^{d}.$ (A.11)

∎

######  Lemma A.5.

The doubling condition (A.6) holds for $\eta_{k}$ whenever both $Q$ and $P$ in
$\mathcal{D}_{k}$ have a common parent $R\in\mathcal{D}_{k-1}$. Moreover, for
all $Q\in\mathcal{D}_{k}$

$\eta_{k}(B(Q))\geq c_{2}\,\mu_{k-1}(B(Q^{1}))$ (A.12)

for $0<c_{2}<1$ depending on $c_{1},n,d$. In particular (A.7) holds for
$\eta_{k}$ assuming $C_{0}$ is big enough.

###### Proof.

Let $R\in\mathcal{D}_{k-1}$ and $Q,P\in\mathsf{Ch}(R)$. Using lower content
regularity of $E$ we have

$\mathcal{H}^{d}_{\infty}(Q)\geq\mathcal{H}^{d}_{\infty}(B(Q)\cap E)\geq
c_{1}(c_{0}\ell(Q))^{d}=\frac{c_{1}(c_{0})^{d}}{2\rho^{d}}\cdot
2(\ell(R))^{d}\geq\frac{c_{1}(c_{0})^{d}}{2\rho^{d}}\mathcal{H}^{d}_{\infty}(R).$
(A.13)

Recall that $\\#\mathsf{Ch}(R)\leq c_{n}$. We then compute

$\eta_{k}(B(Q))=\frac{\mathcal{H}^{d}_{\infty}(Q)}{\sum_{P^{\prime}\in\mathsf{Ch}(R)}\mathcal{H}^{d}_{\infty}(P^{\prime})}\cdot\mu_{k-1}(B(R))\\\
\geq\frac{c_{1}(c_{0})^{d}}{2\rho^{d}}\cdot\frac{\mathcal{H}^{d}_{\infty}(R)}{c_{n}\mathcal{H}^{d}_{\infty}(R)}\cdot\mu_{k-1}(B(R))\
=\frac{c_{1}(c_{0})^{d}}{2c_{n}\rho^{d}}\mu_{k-1}(B(R)).$

This shows (A.12). To see (A.6) note that

$\eta_{k}(B(P))\leq\mu_{k-1}(B(R))\leq c_{2}\eta_{k}(B(Q))\leq
C_{0}\eta_{k}(B(Q)),$

so (A.6) holds true whenever $P$ is a sibling of $Q$. ∎

The lemmas above say that $\eta_{k}$ satisfies almost all the required
properties, except that instead of the full doubling condition it only
satisfies a “dyadic doubling” condition. We need to make some adjustments.

####  A.1.2. Definition of $\mu_{k}$

We define families of cubes where the doubling condition (A.8) fails. For each
$Q\in\mathcal{D}_{k}$ let

$\mathsf{Rich}(Q)=\\{P\in\mathsf{Nbd}(Q)\ \mathrel{\mathop{\ordinarycolon}}\
\eta_{k}(B(Q))<4C_{0}^{-1}\,\eta_{k}(B(P))\\},$

and similarly

$\mathsf{Poor}(Q)=\big{\\{}P\in\mathsf{Nbd}(Q)\
\mathrel{\mathop{\ordinarycolon}}\
\eta_{k}(B(Q))>\frac{C_{0}}{4}\,\eta_{k}(B(P))\big{\\}}.$

Note that

$\\#\mathsf{Poor}(Q)+\\#\mathsf{Rich}(Q)\leq 2\,\\#\mathsf{Nbd}(Q)\leq
2\,c_{n}.$ (A.14)

Set

$\displaystyle\mathsf{Poor}_{k}$ $\displaystyle=\\{Q\in\mathcal{D}_{k}\
\mathrel{\mathop{\ordinarycolon}}\ \mathsf{Rich}(Q)\neq\varnothing\\},$
$\displaystyle\mathsf{Rich}_{k}$ $\displaystyle=\\{Q\in\mathcal{D}_{k}\
\mathrel{\mathop{\ordinarycolon}}\ \mathsf{Poor}(Q)\neq\varnothing\\}.$

######  Lemma A.6.

We have $\mathsf{Poor}_{k}\cap\mathsf{Rich}_{k}=\varnothing$.

###### Proof.

Suppose that $Q\in\mathsf{Poor}_{k}\cap\mathsf{Rich}_{k}$. Then, there exist
$P_{p},P_{r}\in\mathsf{Nbd}(Q)$ such that

$\frac{C_{0}}{4}\,\eta_{k}(B(P_{p}))<\eta_{k}(B(Q))<4C_{0}^{-1}\,\eta_{k}(B(P_{r})).$

In particular, $\eta_{k}(B(P_{p}))<16C_{0}^{-2}\,\eta_{k}(B(P_{r}))$. Let
$R_{p},R_{r}$ be the parents of $P_{p},P_{r}$, respectively. Note that
$R_{p}\in\mathsf{Nbd}(R_{r})$. Then, by (A.12)

$\mu_{k-1}(B(R_{p}))\leq
c_{2}^{-1}\,\eta_{k}(B(P_{p}))<16c_{2}^{-1}C_{0}^{-2}\eta_{k}(B(P_{r}))\leq
C_{0}^{-1}\mu_{k-1}(B(R_{r})),$

assuming $C_{0}$ big enough. This contradicts the doubling property (A.8) for
$\mu_{k-1}$. Hence, $\mathsf{Poor}_{k}\cap\mathsf{Rich}_{k}=\varnothing$. ∎

We define the measure $\mu_{k}$ with
$\operatorname{supp}\mu_{k}=\operatorname{supp}\eta_{k}=\bigcup_{Q\in\mathcal{D}_{k}}B(Q)$
in the following way. For each $Q\in\mathcal{D}_{k}$ we set

$\mu_{k}|_{B(Q)}=\begin{cases}\big{(}1-C_{0}^{-1}\cdot\\#\mathsf{Poor}(Q)\big{)}\eta_{k}|_{B(Q)}&\text{if
$Q\in\mathsf{Rich}_{k}$,}\\\
\big{(}1+C_{0}^{-1}\sum_{P\in\mathsf{Rich}(Q)}\frac{\eta_{k}(B(P))}{\eta_{k}(B(Q))}\big{)}\eta_{k}|_{B(Q)}&\text{if
$Q\in\mathsf{Poor}_{k}$,}\\\ \eta_{k}|_{B(Q)}&\text{if
$Q\notin\mathsf{Rich}_{k}\cup\mathsf{Poor}_{k}$.}\end{cases}$

######  Lemma A.7.

Property (A.7) holds for $\mu_{k}$.

###### Proof.

Observe that by (A.14), as soon as $C_{0}$ is large enough depending on $n$,
we have

$\mu_{k}\geq\frac{1}{2}\eta_{k}.$ (A.15)

Thus, we immediately get from (A.12) that $\mu_{k}$ satisfies

$\mu_{k}(B(Q))\geq\frac{c_{2}}{2}\mu_{k-1}(B(Q^{1}))\geq
C_{0}^{-1}\mu_{k-1}(B(Q^{1})).$

This shows one of the inequalities in (A.7). Now we prove that

$\mu_{k}(B(Q))\lesssim\mu_{k-1}(B(Q^{1})).$ (A.16)

For $Q\notin\mathsf{Poor}_{k}$ this is trivial, because then by the definition
of $\mu_{k}$ and $\eta_{k}$ we have
$\mu_{k}(B(Q))\leq\eta_{k}(B(Q))\leq\mu_{k-1}(B(Q^{1}))$. On the other hand,
for $Q\in\mathsf{Poor}_{k}$

$\mu_{k}(B(Q))=\eta_{k}(B(Q))+C_{0}^{-1}\sum_{P\in\mathsf{Rich}(Q)}\eta_{k}(B(P))\\\
\leq\mu_{k-1}(B(Q^{1}))+C_{0}^{-1}\sum_{P\in\mathsf{Rich}(Q)}\mu_{k-1}(B(P^{1})).$

For each $P\in\mathsf{Rich}(Q)$ we have $P^{1}\in\mathsf{Nbd}(Q^{1})$, and so
by (A.6) applied to $\mu_{k-1}$ we get

$\mu_{k-1}(B(Q^{1}))+C_{0}^{-1}\sum_{P\in\mathsf{Rich}(Q)}\mu_{k-1}(B(P^{1}))\\\
\leq\mu_{k-1}(B(Q^{1}))+\\#\mathsf{Rich}(Q)\mu_{k-1}(B(Q^{1}))\lesssim\mu_{k-1}(B(Q^{1})).$

This finishes the proof. ∎

######  Lemma A.8.

We have
$\mu_{k}(\mathbb{R}^{n})=\eta_{k}(\mathbb{R}^{n})=\mathcal{H}^{d}_{\infty}(E),$
so that condition (A.3) holds for $\mu_{k}$.

###### Proof.

We have

$\mu_{k}(\mathbb{R}^{n})=\sum_{Q\in\mathcal{D}_{k}}\mu_{k}(B(Q))=\sum_{Q\in\mathsf{Rich}_{k}}\big{(}1-C_{0}^{-1}\cdot\\#\mathsf{Poor}(Q)\big{)}\eta_{k}(B(Q))\\\
+\sum_{Q\in\mathsf{Poor}_{k}}\big{(}\eta_{k}(B(Q))+C_{0}^{-1}\sum_{P\in\mathsf{Rich}(Q)}\eta_{k}(B(P))\big{)}+\sum_{Q\notin\mathsf{Rich}_{k}\cup\mathsf{Poor}_{k}}\eta_{k}(B(Q))\\\
=\sum_{Q\in\mathcal{D}_{k}}\eta_{k}(B(Q))-C_{0}^{-1}\sum_{Q\in\mathsf{Rich}_{k}}\\#\mathsf{Poor}(Q)\eta_{k}(B(Q))+C_{0}^{-1}\sum_{Q\in\mathsf{Poor}_{k}}\sum_{P\in\mathsf{Rich}(Q)}\eta_{k}(B(P))\\\
=\sum_{Q\in\mathcal{D}_{k}}\eta_{k}(B(Q))=\eta_{k}(\mathbb{R}^{n})=\mathcal{H}^{d}_{\infty}(E).$

∎

######  Lemma A.9.

The polynomial growth condition (A.5) holds for $\mu_{k}$.

###### Proof.

Observe that for $Q\in\mathcal{D}_{k}\setminus\mathsf{Poor}_{k}$ we have
$\mu_{k}(B(Q))\leq\eta_{k}(B(Q))$, so in this case (A.5) follows from Lemma
A.4. Assume that $Q\in\mathsf{Poor}_{k}$. Then, there exists
$P\in\mathsf{Nbd}(Q)$ such that

$\eta_{k}(B(Q))\leq 2C_{0}^{-1}\eta_{k}(B(P))\leq 4C_{0}^{-1}\ell(Q)^{d},$

where in the last inequality we used again Lemma A.4. Then,

$\mu_{k}(B(Q))=\eta_{k}(B(Q))+C_{0}^{-1}\sum_{P\in\mathsf{Rich}(Q)}\eta_{k}(B(P))\\\
\leq 4C_{0}^{-1}\ell(Q)^{d}+\\#\mathsf{Rich}(Q)\cdot
2C_{0}^{-1}\ell(Q)^{d}\overset{\eqref{eq:poorcard}}{\lesssim_{n}}C_{0}^{-1}\ell(Q)^{d}.$

Recalling that $\mathcal{H}^{d}_{\infty}(Q)\gtrsim c_{1}\ell(Q)^{d}$ by
(A.13), we get that $\mu_{k}(B(Q))\leq\mathcal{H}^{d}_{\infty}(Q)\leq
2\ell(Q)^{d}$ assuming $C_{0}$ large enough depending on $c_{1},n,d$. ∎

######  Lemma A.10.

The doubling condition (A.6) holds for $\mu_{k}$.

###### Proof.

We want to show that for any $Q\in\mathcal{D}_{k}$ and $P\in\mathsf{Nbd}(Q)$