# Equality of different definitions of conformal dimension for quasiself-
similar and CLP spaces

Sylvester Eriksson-Bique  Department of Mathematics and Statistics, University
of Jyväskylä, Seminaarinkatu 15, PO Box 35, FI-40014 University of Jyväskylä,
Finland<EMAIL_ADDRESS>
###### Abstract.

We prove that for a quasiself-similar and arcwise connected compact metric
space all three known versions of the conformal dimension coincide: the
conformal Hausdorff dimension, conformal Assouad dimension and Ahlfors regular
conformal dimension. This answers a question posed by Mathav Murugan.
Quasisimilar spaces include all approximately self-similar spaces. As an
example, the standard Sierpiński carpet is quasiself-similar and thus the
three notions of conformal dimension coincide for it.

We also give the equality of the three dimensions for combinatorially
$p$-Loewner (CLP) spaces. Both proofs involve using a new notion of
combinatorial modulus, which lies between two notions of modulus that have
appeared in the literature. The first of these is the modulus studied by Pansu
and Tyson, which uses a Carathéodory construction. The second is the one used
by Keith and Laakso (and later modified and used by Bourdon, Kleiner,
Carrasco-Piaggio, Murugan and Shanmugalingam). By combining these approaches,
we gain the flexibility of giving upper bounds for the new modulus from the
Pansu-Tyson approach, and the ability of getting lower bounds using the Keith-
Laakso approach. Additionally the new modulus can be iterated in self-similar
spaces, which is a crucial, and novel, step in our argument.

The author was partially supported by Finnish Academy Grants n. 345005 and n.
356861. We thank Mathav Murugan for posing the question to us, and for
discussing the problem at the Okinawa Institute of Science and Technology in
June 2023. The work was started at the workshop “Random walks and analysis on
metric spaces”. We thank the institute for its hospitality and care -
especially given the weak typhoon that overlapped the event.

_Key words and phrases_ : Conformal dimension, Assouad Dimension, Ahlfors
Regular, self-similar, combinatorial Loewner property, Modulus

Mathematics Subject Classification (2020): Primary: 30L10, 51F99

###### Contents

  1. 1 Introduction
    1. 1.1 Outline
  2. 2 Notation and Basic properties
    1. 2.1 Basic terminology
    2. 2.2 Relative distance and quasisymmetries
    3. 2.3 Quasiself-similarity
    4. 2.4 $\kappa$-approximations
  3. 3 Discrete moduli
    1. 3.1 Discrete modulus of a collection
    2. 3.2 Modulus of annulus
    3. 3.3 Relationship to Conformal dimension
    4. 3.4 New discrete modulus
  4. 4 Combinatorially Loewner Spaces
    1. 4.1 Definition and basic property
    2. 4.2 Estimates for Modulus
    3. 4.3 Proof of Theorem 1.8
  5. 5 Quasiself-similar Spaces
    1. 5.1 Uniform bound for Annuli
    2. 5.2 Algorithm for pushing down a cover
    3. 5.3 Estimate for Bourdon-Kleiner modulus
    4. 5.4 Proof of main theorem

##  1\. Introduction

In this paper we study the equivalence of three different versions of the
definition of the conformal dimension. First, recall the definition of a
quasisymmetry.

######  Definition 1.1.

We say that a homeomoprhic map $f:X\to Y$ is a quasisymmetry, if there exists
a homeomorphism $\eta:[0,\infty)\to[0,\infty)$ so that for all $x,y,z\in X$
with $x\neq z$, we have

(1.2)
$\frac{d(f(x),f(y))}{d(f(x),f(z))}\leq\eta\left(\frac{d(x,y)}{d(x,z)}\right).$

We write $X\sim_{q.s.}Y$, if there exists a quasisymmetry $f:X\to Y$. We say
that $f$ is an $\eta$-quasisymmetry, if it satisfies (1.2) with this specific
function $\eta$.

We consider the effect of quasisymmetries on the a) Hausdorff dimension, b)
Assouad dimension and c) Ahlfors-regularity of the space. We briefly recall
the definitions of these. If $s\in[0,\infty)$, we define the Hausdorff
$s$-content (at scale $\delta\in(0,\infty]$) of $A\subset X$ as

(1.3)
$\mathcal{H}^{s}_{\delta}(A)=\inf\left\\{\sum_{i\in{\mathbb{N}}}\operatorname{diam}(A_{i})^{s}:A\subset\bigcup_{i\in{\mathbb{N}}}A_{i},\operatorname{diam}(A_{i})\leq\delta\right\\}.$

For future reference, also define Hausdorff measure by

$\mathcal{H}^{s}(A):=\lim_{\delta\to 0}\mathcal{H}^{s}_{\delta}(A).$

The Hausdorff dimension of $X$ can be defined as

${\rm dim}_{H}(X)=\inf\\{s>0:\mathcal{H}^{s}_{\infty}(X)=0\\}.$

Hausdorff dimension is not stable when sequences of spaces “converge” (e.g. in
the Gromov-Hausdorff sense), and this is one reason to introduce Assouad
dimension. If $A\subset X$ is a subset, let

(1.4) $N(A,r)=\inf\\{N:\exists x_{1},\dots,x_{N},A\subset\bigcup
B(x_{i},r)\\}.$

Define the Assouad dimension in terms of the scale-invariant asymptotic
behaviour of this quantity.

${\rm dim}_{A}(X)=\inf\\{s>0:\exists C>0,\forall R>r>0,\forall z\in
X,N(B(z,R),r)\leq CR^{s}r^{-s}.\\}$

Finally, a special setting, where ${\rm dim}_{A}(X)={\rm dim}_{H}(X)$ is when
$X$ is Ahlfors regular for some $Q>0$. We say that $X$ is $Q$-Ahlfors regular,
if there exists a Radon measure $\mu$ on $X$ and some constant $C\geq 1$, with

$C^{-1}r^{Q}\leq\mu(B(z,r))\leq Cr^{Q}.$

for every $z\in X$ and $r\in(0,\operatorname{diam}(X))$. In this case, $Q={\rm
dim}_{A}(X)={\rm dim}_{H}(X)$. Even if the space is not Ahlfors regular, we
always have the inequality ${\rm dim}_{H}(X)\leq{\rm dim}_{A}(X)$, where the
inequality may be strict. Further, while not every space is $Q$-Ahlfors
regular, spaces often can be deformed into such. A more detailed discussion on
this and the three notions of dimension, as well as conformal dimension in
general, is given in [17, Chapter 2].

An example of a quasisymmetric map is the identity map
$(X,d)\to(X,d^{\theta})$, for $\theta\in(0,1)$. Such a map increases all of
the three notions of dimension. It is considerably harder to decrease
dimension, and one is led to defining three quasisymmetric invariants.

Conformal Hausdorff dimension $\displaystyle{\rm dim}_{CH}(X)=\inf\\{{\rm
dim}_{H}(Y):X\sim_{q.s.}Y\\}$ Conformal Assouad dimension $\displaystyle{\rm
dim}_{CA}(X)=\inf\\{{\rm dim}_{A}(Y):X\sim_{q.s.}Y\\}$ Ahlfors regular
conformal dimension $\displaystyle{\rm dim}_{CAR}(X)=\inf\\{Q={\rm
dim}_{H}(Y):X\sim_{q.s.}Y,Y\text{ is $Q$-Ahlfors regular}\\}.$

The first of these was defined in [19], while the final one was used in [5].
The Assouad variant appeared already in [14].

In general, ${\rm dim}_{CH}(X)\leq{\rm dim}_{CA}(X)\leq{\rm dim}_{CAR}(X)$.
For uniformly perfect spaces ${\rm dim}_{CAR}(X)={\rm dim}_{CA}(X)$, see [17,
Proposition 2.2.6.] and [11, Chapters 14 and 15]. The relationship between
${\rm dim}_{CH}(X)$ and ${\rm dim}_{CA}(X)$ has so far not been studied in
detail, beyond giving simple examples such as the following, when they are not
equal.

######  Example 1.5.

Let $X={\mathbb{Z}}\times{\mathbb{R}}$. The conformal Assouad dimension can
only drop under blowing the space down, and thus ${\rm dim}_{CA}(X)\geq{\rm
dim}_{CA}({\mathbb{R}}^{2})=2$. The latter follows since the topological
dimension of the plane is $2$, and the Hausdorff dimension is always greater
than the topological dimension. However, ${\rm dim}_{CH}(X)={\rm
dim}_{H}(X)=1$.

If we set
$X={\mathbb{Z}}\times{\mathbb{R}}\cup{\mathbb{R}}\times{\mathbb{Z}}$, we can
even make $X$ connected without altering the previous argument. It is possible
to make the space compact and connected as well: Let
$X=(\\{\frac{1}{n}:n\in{\mathbb{N}}\\}\cup\\{0\\})\times[0,1]\cup[0,1]\times(\\{\frac{1}{n}:n\in{\mathbb{N}}\\}\cup\\{0\\})$.
In this case, a blow-up of the space is ${\mathbb{R}}^{2}$.

Assouad dimension involves a scale-invariant quantitative condition, while
Hausdorff dimension is merely a qualitative statement on the dimension of the
space. Further, as the previous example indicates ${\rm dim}_{CA}(X)$ has
stability properties under limits, while ${\rm dim}_{CH}(X)$ does not. This
means, that one may only hope for their equality in the case where one assumes
some form of self-similarity. Consequently Mathav Murugan asked if the
different definitions of conformal dimension agree for self-similar spaces
[18]. Our main theorem answers this intuition in the affirmative. The notion
of quasiself-similarity is given in Definition 2.4, and (to our knowledge) was
introduced in [6].

######  Theorem 1.6.

Let $X$ be a compact quasiself-similar metric space, which is connected and
locally connected. Then,

${\rm dim}_{CH}(X)={\rm dim}_{CA}(X)={\rm dim}_{CAR}(X).$

As stated, the equality ${\rm dim}_{CA}(X)={\rm dim}_{CAR}(X)$ for uniformly
perfect spaces was already known, and follows directly from [17, Proposition
2.2.6]. Our contribution is to prove ${\rm dim}_{CH}(X)={\rm dim}_{CA}(X)$.
Indeed, this equality has many further consequences. One may define a zoo of
other conformal dimensions, such as: conformal upper and lower Minkowski
dimension, conformal packing dimension… Since these dimensions lie between the
Hausdorff dimension and the Assouad dimension, one gets equality for the
corresponding notions of conformal dimension as well.

The only other result, which states equality of ${\rm dim}_{CH}(X)$ with ${\rm
dim}_{CAR}(X)={\rm dim}_{CA}(X)$ is that of [22, Theorem 3.4] and [19,
Proposition 2.9.], which apply when $X$ is $Q$-Ahlfors regular and possesses a
curve family with positive continuous $Q$-modulus. We will discuss this
further below. We are not aware of any other instances, where equality of all
notions has been shown.

A concrete corollary of Theorem 1.6 is the following new result. The
$n$-dimensional Sierpiǹski sponge $M_{n}$ is obtained by iteratively
subdividing the side of an $n$-dimensional cube by three, and removing the
central cube.

######  Corollary 1.7.

Let $n\geq 2$. If $M_{n}$ is an $n$-dimensional Sierpsiǹski sponge, then

${\rm dim}_{CH}(M_{n})={\rm dim}_{CA}(M_{n})={\rm dim}_{CAR}(M_{n}).$

We will also give a result for non-self-similar spaces, where self-similarity
is replaced with the combinatorial Loewner property (CLP) from [4, 8]; see
Section 4 for a definition. It is worth noting, that this assumption usually
is verified in the self-similar setting, and thus is not so much more general
than Theorem 1.6. We present this here, since the argument for it is a bit
simpler than for the general self-similar case. Further, it is worth to record
a proof for this result here, since the developed tools may be useful in
tackling the question of Bruce Kleiner, which asks if self-similar
combinatorially Loewner spaces are quasisymmetric to Loewner spaces; see [15]
for further background and the question.

######  Theorem 1.8.

Let $p\in(1,\infty)$. Let $X$ be a compact, doubling and LLC space, which is
$p$-combinatorially Loewner metric space. We have

${\rm dim}_{CH}(X)={\rm dim}_{CA}(X)={\rm dim}_{CAR}(X)=p.$

We note that Corollary 1.7 would also follow from this result, since
Sierpiński sponges are $p$-combinatorially Lowener spaces; see the proofs in
[4]. In the course of the proof of Theorem 1.8 we will in fact present some
stronger results for CLP spaces in Section 4. In fact, while the statement
${\rm dim}_{CH}(X)=p$ is qualitative, we will give a quantitative statement,
Proposition 4.9, which gives a lower bound for the Hausdorff measure of the
images of balls under quasisymmetries. This inequality may be useful in other
settings as well, and is a generalization of an inequality which appeared in
the work of Heinonen and Koskela [12, Theorem 3.6].

Our results here clarify a central point of ambiguity in much of the
literature on conformal dimension, where the equality of the conformal
Hausdorff and conformal Assouad dimensions is not addressed, but rather
avoided and bypassed. We next describe the main idea of the proof.

The key tool in a majority of the research on conformal dimension is a notion
of modulus - in particular discretized versions of moduli of path families.
These generalize the notion of continuous modulus (later, often, modulus), see
e.g. [10, 11, 13] for background. Our proof is also based on defining a new
type of discrete modulus - or rather, discrete admissibility - and relating it
to conformal Hausdorff dimension. At this point, there are several variants of
discrete modulus, each with its own setting and application see e.g. [19, 22,
14, 7, 21, 18, 16, 1]. (There are also other notions, such as trans-boundary
modulus, see e.g. [20, 3], but these are not relevant for our discussion
here.) We will not discuss all these moduli here, but will focus on those
which motivate our approach.

The motivation for our argument and notion of modulus comes from a result of
Pansu [19, Proposition 2.9.], whose dual formulation111As a side note, we
remark that Pansu considers measures on families of curves, while Tyson uses
the notion of curve modulus in [22]. These two notions are roughly dual to
each other, see e.g. [2, 9] for more precise statements. was given by Tyson in
[22, Theorem 3.4]. Tyson shows that if $X$ is a $Q$-Ahlfors regular metric
measure space, and if it possesses a family of curves $\Gamma$ with positive
continuous modulus, then ${\rm dim}_{CH}(X)=Q$. The proof of Tyson uses the
discrete $Q$-modulus as introduced by Pansu in [19]. To be very brief, this
modulus is defined using a Carathèodory construction and involves discrete
sums. One shows, both in [19] and [22], that the discrete $Q$-modulus is
bounded from below by the continuous $Q$-modulus. Further, the discrete
modulus, up to a variation of parameters, is invariant under quasisymmetries.
The final nail in the coffin of the proof is that if ${\rm dim}_{H}(Y)<Q$,
then the discrete $Q$-modulus vanishes on $Y$. Consequently, a family of
curves with positive continuous $Q$-modulus obstructs lowering the dimension
of $X$ by a quasisymmetry below $Q$.

The previous proof relies heavily on the fact that we can use the notion of
continuous modulus to give a lower bound for discrete modulus. In many
settings, such as the Sierpiński sponges mentioned above, the continuous
moduli of all curves vanishes. Thus, we lack this lower bound, and we need
find a way around this by giving a lower bound using a different quantity. In
the quasiself-similar setting, and in the combinatorially Loewner setting, we
can obtain this lower bound by slightly different mechanisms - and by
employing a different modulus.

In the work [14, 7, 18], the inability to lower the dimension can be converted
to a lower bound on some moduli - see Theorem 3.5 for a precise statement.
Thus, one can use their result to obtain a lower bound for a different
discrete modulus, which we call the Keith-Laakso modulus and which is defined
in subsection 3.1. In the case of combinatorially Loewner spaces, the setting
is a bit simpler and the lower bound is obtained directly by the assumption
that the space is combinatorially Loewner [4, 8]: see Definition 4.1.

At this juncture, we have two moduli: the Keith-Laakso modulus and that of
Pansu and Tyson. For the first we can obtain lower bounds. For the second, one
can show upper bounds. Indeed, for Pansu and Tyson, the notion of discrete
modulus is such that it is very easy to prove that if ${\rm dim}_{H}(Y)<Q$,
the discrete modulus vanishes. In the absence of $Q$-Ahlfors regularity, it is
harder to give lower bounds for the modulus of Pansu and Tyson. On the other
hand, for the Keith-Laakso modulus, one lacks the ability to give good upper
bounds and thus to directly say that the discrete modulus vanishes if one has
Hausdorff dimension lower than $Q$.

The reason for this inability is the following technical, but crucial point.
The definition of Keith-Laakso modulus can be summarized as assigning a value
${\rm Mod}_{p}^{KL}(\Gamma,\mathcal{U})$ for a specific curve family $\Gamma$
and a cover $\mathcal{U}$ of $X$, which [4] calls a $\kappa$-approximation _at
some level $r$_. The key feature of their $\kappa$-approximations is that all
sets in $\mathcal{U}$ have roughly the same size. (See Subsection 2.4 for
details.) This is also a key difference with the work in [19, 22], since there
the Carathéodory construction involves _arbitrary covers_.

Similarly, Assouad dimension involves covering the space by balls of the same
size, whereas Hausdorff dimension involves coverings by sets of various sizes.
To give estimates for Hausdorff dimension, we need to allow arbitrary covers
in the definition of discrete modulus. We bridge this gap, by introducing a
new notion of modulus $\overline{{\rm Mod}}_{p}(\Gamma,\mathcal{U})$ which
lies between those of Pansu and Tyson in [19, 22] and Keith, Laakso and others
in [14, 7, 18]. First, we get more flexibility by allowing arbitrary covers
$\mathcal{U}$ that consist of balls (or, in general, sufficiently round sets).
This forces us to introduce a new admissibility condition, to address several
key technical issues. Similar to Pansu’s discrete modulus, we can show that if
${\rm dim}_{H}(Y)<Q$, then this modulus is very small for a given cover.
Further, in the self-similar and CLP space settings, we can relate the Keith-
Laakso modulus and the new modulus to each other.

For combinatorially Loewner spaces, the story is easier to finish. One can
bound $\overline{{\rm Mod}}_{p}(\Gamma,\mathcal{U})$ from below using the
Keith-Laakso modulus, which in term has a lower bound from the combinatorial
Loewner assumption. This estimate is given in Proposition 4.4. This gives a
contradiction to the previous paragraph’s conclusion of $\overline{{\rm
Mod}}_{p}(\Gamma,\mathcal{U})$ being small. In fact, this argument is somewhat
easier to discover, and it served as a starting point for this paper and
project. For this reason we also include the argument in this paper. Quickly,
however, the author realized that a more technical version of the argument
could be applied for general quasiself-similar spaces.

For _quasiself-similar spaces_ the argument is a bit different. Instead of
directly using a lower bound, we use the fact that the ability to lower
dimension gives an upper bound. Indeed, if there is a quasisymmetric map
$f:X\to Y$ and if $Y$ has small Hausdorff measure, then we obtain a
quantitative statement on moduli of annuli, see Lemma 5.2 and Proposition
5.23. Our quantitative statement can be converted algorithmically by using
iteration to a statement on the smallness of the Keith-Laakso modulus. This
allows us to prove the equality ${\rm dim}_{CA}(X)={\rm dim}_{CH}(X)$ for
quasiself-similar spaces by using the result of Carrasco-Paggio, which we
state below in Theorem 3.5. The iteration is algorithmic, but quite technical.
The basic step of the iteration involves ideas from the proof of the result
for CLP spaces. We will describe it in more detail in Subsection 5.2.

###  1.1. Outline

We will present some general terminology in Section 2. Then, in Section 3 we
introduce the different notions of discrete modulus needed in this paper, and
present some known results on their relationships with the conformal
dimension. For technical reasons, we will use mostly a variant of this
modulus, the Bourdon-Kleiner modulus defined in [4], instead of the Keith-
Laakso modulus. However, we will relate the two moduli to each other. In
Subsection 3.4, we give the new modulus that is key to the approach of this
paper. In Section 4 we focus on CLP spaces. There, we prove Theorem 1.8, which
is the equality of the definitions of conformal dimension for CLP spaces. In
the process, we give some useful stronger results on discrete moduli, and
precise quantitative estimates, which hold for CLP spaces. In Section 5 we
focus on quasiself-similar spaces. There, we study moduli of annuli, and give
an push-down algorithm to adjust the scale of covers. This is then used to
give a relationship between the two moduli used. Finally, in subsection 5.4 we
collect the pieces and complete the proof of Theorem 1.6.

##  2\. Notation and Basic properties

###  2.1. Basic terminology

A compact metric space will be denoted $X$, its metric $d$, and open balls
within it $B(z,r):=\\{w\in X:d(z,w)<r\\}$ for $z\in X,r>0$. An inflation of a
ball $B=B(z,r)$ is denoted $CB:=B(z,Cr)$ for $C>0$. Note that we consider each
ball as having an associated center and radius – and it may happen that a
different center and radius defines the same set. The radius of a ball is
denoted $\mathrm{rad}(B)$. Diameters of sets $A\subset X$ will be denoted
$\operatorname{diam}(A)=\sup_{a,b\in A}d(a,b)$. A curve is a continuous map
$\gamma:I\to X$, where $X$ is a non-empty compact interval in ${\mathbb{R}}$.
We often conflate $\gamma$ and it’s image set ${\rm Image}(\gamma)$.

Recall the definition of $N(A,r)$ from (1.4). We say that a metric space $X$
is metrically doubling, if there exists a constant $D\geq 1$, so that
$N(B(z,r),r/2)\leq D$ for every $z\in X$ and $r>0$.

We will need some connectivity properties. A space $X$ is called locally
connected, if it has a neighborhood basis consising of connected open sets. A
metric space is LLC, if for every $x,y\in X$, there exists a curve $\gamma$
with $x,y\in\gamma$ and $\operatorname{diam}(\gamma)\leq Cd(x,y)$.

We will consider collections of balls, which are often denoted by a script
letter ${\mathcal{B}}$. For these, we define unions by setting
$\bigcup{\mathcal{B}}:=\bigcup_{B\in{\mathcal{B}}}B$, inflations by setting
$C{\mathcal{B}}:=\\{CB:B\in{\mathcal{B}}\\}$ and radii
$\mathrm{rad}({\mathcal{B}})=\sup_{B\in{\mathcal{B}}}\mathrm{rad}(B)$. If $A$
is any finite set, we denote by $|A|$ its cardinality.

###  2.2. Relative distance and quasisymmetries

We need some standard results on quasisymmetries.

######  Lemma 2.1.

If $f:X\to Y$ is an $\eta$-quasisymmetry, then $f^{-1}$ is a
$\tilde{\eta}$-quasisymmetry with
$\tilde{\eta}(t)=\left(\eta^{-1}(t^{-1})\right)^{-1}$.

We note the convention that the value of $\tilde{\eta}$ at zero is given by
$\tilde{\eta}(0)=0$.

###### Proof of Lemma 3.11.

Let $x,y,z\in Y$ and let $x^{\prime},y^{\prime},z^{\prime}\in X$ be such that
$f(x^{\prime})=x,f(y^{\prime})=y,f(z^{\prime})=z$. Since $f$ is an
$\eta$-quasisymmetry, we have

$\frac{d(f(x^{\prime}),f(z^{\prime}))}{d(f(x^{\prime}),f(y^{\prime}))}\leq\eta\left(\frac{d(x^{\prime},z^{\prime})}{d(x^{\prime},y^{\prime})}\right).$

Taking resiprocals and an inverse function, we get

$\frac{d(x,y)}{d(x,z)}\leq\left(\eta^{-1}\left(\left(\frac{d(f(x^{\prime}),f(y^{\prime}))}{d(f(x^{\prime}),f(z^{\prime}))}\right)^{-1}\right)\right)^{-1}.$

Replacing $f(x^{\prime}),f(y^{\prime}),f(z^{\prime})$ with $x,y,z$ and
$x^{\prime},y^{\prime},z^{\prime}$ with $f^{-1}(x),f^{-1}(y),f^{-1}(z)$ yields
that $f^{-1}$ is an $\tilde{\eta}$-quasisymmetry. ∎

Let $X$ be a complete metric space. A continuum $E\subset X$ is a compact
connected set. A continuum is non-degenerate, if it is non-empty. We define
the relative distance between two non-degenerate continua $E,F$ as

$\Delta(E,F):=\frac{d(E,F)}{\min\\{\operatorname{diam}(E),\operatorname{diam}(F)\\}}.$

######  Lemma 2.2.

Let $f:X\to Y$ be an $\eta$-quasisymmetry and let $E,F$ be two non-degenerate
disjoint continua in $X$. Then,

$\frac{1}{2\eta(\Delta(E,F)^{-1})}\leq\Delta(f(E),f(F))\leq\eta(2\Delta(E,F)).$

###### Proof.

Assume by symmetry that $\operatorname{diam}(E)\leq\operatorname{diam}(F)$.
Let $x\in E$ and $y\in F$ be such that $d(E,F)=d(x,y)$. Choose $u\in E,v\in F$
so that $d(x,u),d(y,v)\geq\operatorname{diam}(E)/2$. This is possible by
connectivity. Then, we have

$\frac{d(x,y)}{d(x,u)}\leq 2\Delta(E,F)\quad\text{ and
}\frac{d(y,x)}{d(y,v)}\leq 2\Delta(E,F)$

Let $x^{\prime}:=f(x),y^{\prime}=f(y),u^{\prime}=f(u),v^{\prime}=f(v)$ be the
image points in $Y$. We have, since $\eta$ is increasing and since $f$ is an
$\eta$-quasisymmetry:

$\displaystyle d(f(E),f(F))$ $\displaystyle\leq d(x^{\prime},y^{\prime})$
$\displaystyle\leq\eta\left(\frac{d(x,y)}{d(x,u)}\right)d(x^{\prime},u^{\prime})$
$\displaystyle\leq\eta\left(2\Delta(E,F)\right)\operatorname{diam}(f(E)).$

Similarly,

$\displaystyle d(f(E),f(F))$ $\displaystyle\leq d(y^{\prime},x^{\prime})$
$\displaystyle\leq\eta\left(\frac{d(y,x)}{d(y,v)}\right)d(y^{\prime},v^{\prime})$
$\displaystyle\leq\eta\left(2\Delta(E,F)\right)\operatorname{diam}(f(F)).$

The previous two inequalities combine to gives the inequality:

$\Delta(f(E),f(F))=\frac{d(f(E),f(F))}{\min\\{\operatorname{diam}(f(E)),\operatorname{diam}(f(F))\\}}\leq\eta\left(2\Delta(E,F)\right).$

Applying this to the inverse $f^{-1}$, which by Lemma 2.1 is an
$\tilde{\eta}$-quasisymmetric map, yields the other inequality of the claim. ∎

The following lemma will also prove useful on a few occasions. Note that the
additional assumption on the existence on $y\in B(x,r)$ is automatically
satisfied if $X$ is connected and $r<\operatorname{diam}(X)$.

######  Lemma 2.3.

Let $f:X\to Y$ be a quasisymmetric map and let $B(x,r)$ be a ball in $X$ for
which there exists a $y\in B(x,r)$ with $d(x,y)\geq r/2$. Then, for every
$L\geq 1$, we have

$f(B(x,Lr))\subset B(f(x),\eta(2L)d(f(x),f(y))).$

###### Proof.

Let $z\in B(x,Lr)$, and apply the $\eta$-quasisymmetry to the triple of points
$x,y,z$. This gives

$\frac{d(f(x),f(z))}{d(f(x),f(y))}\leq\eta\left(\frac{d(x,z)}{d(x,y)}\right)\leq\eta(2L).$

Consequently, we get the claim from

$d(f(x),f(z))\leq\eta(2L)d(f(x),f(y)).$

∎

###  2.3. Quasiself-similarity

We define a notion of quasiself-similarity. This is motivated by the notion of
approximate self-similarity discussed in [4].

######  Definition 2.4.

We say that a compact space $X$ is quasiself-similar, if there exists a
homeomorphism $\eta:[0,\infty)\to[0,\infty)$ and a constant $\delta>0$ so that
for $B(x,r)\subset X$ there is a quasisymmetry $f:B(x,r)\to U_{x,r}$ where
$U_{x,r}\subset X$ is an open set with
$\operatorname{diam}(U_{x,r})\geq\delta\operatorname{diam}(X)$. We also say
that $X$ is $\eta$-quasiself-similar, if this property holds for a given
function $\eta$.

The principal advantage of defining quasiself-similar spaces is that they are
more general than approximately self-similar spaces. Further, quasiself-
similarity is an invariant under quasisymmetries: if $X$ is quasiself-similar
and $Y\sim_{q.s.}X$, then $Y$ is also quasiself-similar. The same fails for
approximate self-similarity.

We recall the following result [6, Chapter 2].

######  Lemma 2.5.

If $X$ is a compact quasiself-similar space, which is connected and locally
connected, then $X$ is LLC.

###  2.4. $\kappa$-approximations

We introduce some terminology on approximations. Throughout this paper,
$\mathcal{U}$ and ${\mathcal{V}}$ will denote finite collections of open sets.

######  Definition 2.6.

Let $\kappa\geq 1$. A finite collection of open sets $\mathcal{U}$ of a metric
space $X$ is called a $\kappa$-round collection, if for every
$U\in\mathcal{U}$ there exists a $z_{U}$ so that

$B(z_{U},\kappa^{-1}r_{U})\subset U\subset B(z_{U},r_{U}),$

where $r_{U}=\sup\\{d(z_{U},x):x\in U\\}$. If further there is some $r>0$, so
that $r_{U}=r$ for every $U\in\mathcal{U}$, we call $\mathcal{U}$ a
$\kappa$-round collection at level $r$.

From here on out, if $U$ is any open set and $z_{U}\in U$ has been fixed, we
define $r_{U}:=\sup\\{d(z_{U},x):x\in U\\}$.

######  Definition 2.7.

Let $\kappa\geq 1$. A $\kappa$-round collection of open sets $\mathcal{U}$ of
a metric space $X$ is called a $\kappa$-locally bounded collection, if there
exist $z_{U}\in U$ for every $U\in\mathcal{U}$ for which Definition 2.6 holds
and for which moreover the following two properties hold.

  1. (1)

The balls $\\{B(z_{U},\kappa^{-1}r_{U}),U\in\mathcal{U}\\}$ are pairwise
disjoint.

  2. (2)

For every $L\geq 1$, there exists a constant $\kappa_{L}$ so that if
$B(z_{U},Lr_{U})\cap B(z_{V},Lr_{V})\neq\emptyset$, then
$r_{U}\leq\kappa_{L}r_{V}$.

If $\mathcal{U}$ also covers $X$, then we call it a $\kappa$-approximation. If
further there is some $r>0$, so that $r_{U}=r$ for every $U\in\mathcal{U}$, we
call $\mathcal{U}$ a $\kappa$-approximation at level $r$.

Let $\mathrm{rad}(\mathcal{U})=\sup\\{r_{U}:U\in\mathcal{U}\\}$. A standard
way to obtain a $\kappa$-approximation is the following. A set $N\subset X$ is
called $r$-separated if for all $x,y\in X$ we have $d(x,y)\geq r$. A maximal
$r$-separated set is called an $r$-net. Given any $r$-net $N$ in a connected
space $X$, with $r\in(0,\operatorname{diam}(X)/2)$, it is straightforward to
show that the collection $\mathcal{U}=\\{B(x,2r):x\in N\\}$ is a
$\kappa$-approximation at level $r$ with $r_{U}=2r$ and $z_{U}=x$ for every
$U=B(x,2r)\in\mathcal{U}$, and $\kappa=1,\kappa_{L}=4$ for all $L\geq 1$.

We note that we have made some adjustments in the notation and terminology to
bridge small differences in the literature, and in order to connect more
directly to our work. The following remark explains some of these choices and
how the other definitions/concepts can be expressed in our framework.

######  Remark 2.8.

We briefly explain the relationships between different definitions used in
[14, 7, 18, 21] and [4]. In the first four of these, one takes $\alpha\geq 2$
and considers a sequence $N_{k}$ of $\alpha^{-k}$ nets and a parameter
$\lambda>1$, and defines graphs $G_{n}$ whose vertex set is $N_{k}$, and with
edges $v,w$ if $B(v,\lambda 2^{-k})\cap B(w,\lambda 2^{-k})\neq\emptyset$. In
our case, this would correspond to the $\kappa$-approximation given by
$\mathcal{U}=\\{B(v,\lambda 2^{-k})\\}$, and setting $\kappa=2\lambda$. Doing
so, the incidence graph associated to $\mathcal{U}$ is isomorphic to that of
$G_{n}$. This isomorphism is relevant in Section 3, since we will define
discrete moduli using incidences of sets in $\mathcal{U}$, while in [14, 7,
18, 21] the moduli are defined in the graphs $G_{n}$. Since these two graphs
are isomorphic, the relevant notions of moduli coincide.

On the other hand, compared to [4] we use a slightly more general framework of
arbitrary $\kappa$-approximations in order to ensure the quasisymmetry
invariance of our definitions. In their work, one only uses
$\kappa$-approximations at a given level $r$.

Let ${\mathcal{V}}$ be a $\kappa$-round collection in $X$, and let $f:X\to Y$
be a quasisymmetry. Then define the image collection
$f({\mathcal{V}}):=\\{f(V):V\in{\mathcal{V}}\\}$. We then then have the
following.

######  Lemma 2.9.

Let ${\mathcal{V}}$ be a $\kappa$-round collection in a space $X$ and if
$f:X\to Y$ is an $\eta$-quasisymmetric map, then $f({\mathcal{V}})$ is a
$\kappa^{\prime}$-round collection with $\kappa^{\prime}=2\eta(\kappa)$.

Moreover, if ${\mathcal{V}}$ is a $\kappa$-approximation, then
$f({\mathcal{V}})$ is a $\kappa^{\prime}$-approximation with
$\kappa^{\prime}=2\eta(\kappa)$.

###### Proof.

For every $V\in{\mathcal{V}}$ let $z_{V}\in V,r_{V}>0$ be the center and
radius specified in Definition 2.6. Define $z_{f(V)}=f(z_{V})$ and
$r_{f(V)}=\sup\\{d(y,f(z_{V})):y\in f(V)\\}$.

Suppose first that ${\mathcal{V}}$ is $\kappa$-round and let
$\kappa^{\prime}=2\eta(\kappa)$. We will show that $V$ is
$\kappa^{\prime}$-round, that is, we prove

(2.10) $B(z_{f(V)},\kappa^{\prime-1}r_{f(V)})\subset f(V)\subset
B(z_{f(V)},r_{f(V)}).$

The second of these inclusions follows from the definition of $r_{f(V)}$. Now,
let $y\in B(z_{f(V)},\kappa^{\prime-1}r_{f(V)})$, and let $b\in X$ be such
that $f(b)=y$. Choose a point $w\in f(V)$ so that $d(w,z_{f(V)})\geq
2^{-1}r_{f(V)}$ and let $c\in V$ be such that $f(c)=w$. Since $f$ is a
quasisymmetry, we get

$\kappa^{\prime}/2=\frac{2^{-1}r_{f(V)}}{\kappa^{\prime-1}r_{f(V)}}\leq\frac{d(z_{f(V)},w)}{d(z_{f(V)},y)}\leq\eta\left(\frac{d(z_{V},c)}{d(z_{V},b)}\right).$

Thus,

$d(z_{V},b)\leq d(z_{V},c)\eta^{-1}(\kappa^{\prime}/2)^{-1}\leq
r_{V}\kappa^{-1}.$

Therefore $b\in B(z_{V},r_{V}\kappa^{-1})\subset V$ and $y\in f(V)$. This
yields the first of the inclusions in (2.10). Thus, $f({\mathcal{V}})$ is
$\kappa$-round.

Let us know assume further that ${\mathcal{V}}$ is a $\kappa$-approximation.
Indeed, it is $\kappa$-locally bounded and covers $X$. Clearly
$f({\mathcal{V}})$ covers $Y$. Thus, it suffices to prove that
$f({\mathcal{V}})$ is $\kappa^{\prime}$-locally bounded.

The proof above showed in fact that

(2.11) $B(z_{f(V)},\kappa^{\prime-1}r_{f(V)})\subset
f(B(z_{V},\kappa^{-1}r_{f(V)})).$

Thus, the balls
$\\{B(z_{f(V)},\kappa^{\prime-1}r_{f(V)}):V\in{\mathcal{V}}\\}$ are pairwise
disjoint. Therefore, we are left to show that for every $L\geq 1$ there exists
a $\kappa_{L}^{\prime}$ so that if

$B(z_{f(V)},Lr_{f(V)})\cap B(z_{f(U)},Lr_{f(U)})\neq\emptyset$

for some $U,V\in{\mathcal{V}}$, then
$r_{f(U)}\leq\kappa_{L}^{\prime}r_{f(V)}$. This is obtained by first finding
an $L^{\prime}\geq 1$ so that $B(z_{V},L^{\prime}r_{V})\cap
B(z_{U},Lr_{U})\neq\emptyset$, which yields an estimate for $d(z_{U},z_{V})$
in terms of $r_{V}$, and then using the quasisymmetry to translate this into a
bound for $r_{f(U)}$ in terms of $r_{f(V)}$.

Let $w\in B(z_{f(V)},Lr_{f(V)})\cap B(z_{f(U)},Lr_{f(U)})$ and let $u\in
B(z_{f(U)},r_{f(U)}),v\in B(z_{f(V)},r_{f(V)})$ be points with
$d(u,z_{f(U)})\geq r_{f(U)}/2$ and $d(v,z_{f(V)})\geq r_{f(V)}/2$. Let $a\in
X,b_{U}\in U,b_{V}\in V$ points so that $f(a)=w,f(b_{U})=u,f(b_{V})=v$.

By Lemma 3.11, the map $f^{-1}$ is a $\tilde{\eta}$-quasisymmetry, with
$\tilde{\eta}(t)=\left(\eta^{-1}(t^{-1})\right)^{-1}$. By the quasisymmetry
condition applied to the three points $b_{U},a,z_{U}$, we have

$d(w,z_{U})\leq\tilde{\eta}\left(\frac{d(w,z_{f(U)})}{d(u,z_{f(U)})}\right)d(z_{U},b_{U})\leq\tilde{\eta}(2L)r_{U}.$

Thus, $a\in B(z_{U},\tilde{\eta}(2L)r_{U})$. Similarly, we get $a\in
B(z_{V},\tilde{\eta}(2L)r_{V}).$ Consequently

(2.12) $a\in B(z_{U},\tilde{\eta}(2L)r_{U})\cap
B(z_{V},\tilde{\eta}(2L)r_{V}).$

Therefore, since $\mathcal{U}$ is locally bounded, there exists a constant
$\kappa_{\tilde{\eta}(2L)}$ for which

(2.13) $\tilde{\eta}(2L)^{-1}r_{V}\leq
r_{U}\leq\kappa_{\tilde{\eta}(2L)}r_{V}.$

From (2.12) and (2.13) we get

$d(z_{U},z_{V})\leq
d(z_{U},a)+d(z_{V},a)\leq\tilde{\eta}(2L)(1+\kappa_{\tilde{\eta}(2L)})r_{V}.$

We have $z_{U}\in
B(z_{V},\tilde{\eta}(2L)(1+\kappa_{\tilde{\eta}(2L)})r_{V})$. Again, by Lemma
2.3, we get that

$z_{f(U)}=f(z_{U})\in
B(z_{f(V)},\eta(2\tilde{\eta}(2L)(1+\kappa_{\tilde{\eta}(2L)}))r_{f(V)}).$

In particular,

(2.14)
$d(z_{f(U)},z_{f(V)})\leq\eta(2\tilde{\eta}(2L)(1+\kappa_{\tilde{\eta}(2L)}))r_{f(V)}.$

We also have $z_{V}\not\in B(z_{U},\kappa^{-1}r_{U})$, and thus

(2.15) $d(z_{U},z_{V})\geq\kappa^{-1}r_{U}.$

Finally, apply the $\eta$-quasisymmetry to the points $z_{U},z_{V}$ and
$b_{U}$ and use (2.15) to give

(2.16)
$\frac{r_{f(U)}}{2d(z_{f(V)},z_{f(U)})}\leq\frac{d(z_{f(U)},u)}{d(z_{f(V)},z_{f(U)})}\leq\eta\left(\frac{d(z_{U},b_{U})}{d(z_{U},z_{V})}\right)\leq\eta\left(\frac{\kappa
r_{U}}{r_{U}}\right)\leq\eta(\kappa).$

Thus, by applying (2.14) we get

$r_{f(U)}\leq
2\eta(\kappa)\eta(2\tilde{\eta}(2L)(1+\kappa_{\tilde{\eta}(2L)}))r_{f(V)}.$

This is the desired estimate with
$\kappa_{L}^{\prime}=2\eta(\kappa)\eta(2\tilde{\eta}(2L)(1+\kappa_{\tilde{\eta}(2L)}))$
and yields the local boundedness.

∎

##  3\. Discrete moduli

###  3.1. Discrete modulus of a collection

We will define all the relevant discrete moduli in this section. First, we
define a discrete modulus of a collection of discrete subsets. Let
$\mathcal{U}$ be a $\kappa$ round collection and let $\mathcal{P}$ be a
collection of subsets of $\mathcal{U}$. (Indeed, in general $\mathcal{U}$
could be any finite collection of objects, but in our application, we will
restrict to such collections.) We say that $\rho:\mathcal{U}\to[0,\infty)$ is
discretely admissible for $\mathcal{P}$, and write
$\rho\wedge_{\mathcal{U}}\mathcal{P}$, if

$\sum_{U\in\mathcal{P}}\rho(U)\geq 1,\text{ for all }P\in\mathcal{P}.$

Define the discrete modulus by

${\rm
Mod}_{p}^{D}(\mathcal{P},\mathcal{U}):=\inf_{\rho\wedge_{\mathcal{U}}\mathcal{P}}\sum_{U\in\mathcal{U}}\rho(U)^{p}.$

The sum on the right will often also be called the _$p$ -energy_ of $\rho$.

We recall some basic properties of modulus, whose proofs are standard. For
similar arguments, see e.g. [10, Section 1]. The existence of minimizers
follows directly from the fact that $\mathcal{U}$ must be finite, and the
optimization is done in a finite dimensional space.

######  Lemma 3.1.

Let $\mathcal{U}$ be a $\kappa$-round collection of $X$ and let $p\geq 1$.

  1. (1)

Monotonicity: If $\mathcal{P}\subset\mathcal{P}^{\prime}$ are two collections
of sets, then

${\rm Mod}_{p}^{D}(\mathcal{P},\mathcal{U})\leq{\rm
Mod}_{p}(\mathcal{P}^{\prime},\mathcal{U}).$

  2. (2)

Sub-additivity: If $\mathcal{P},\mathcal{P}^{\prime}$ are two collections of
curves, then

${\rm Mod}_{p}^{D}(\mathcal{P}\cup\mathcal{P}^{\prime},\mathcal{U})\leq{\rm
Mod}^{D}_{p}(\mathcal{P}^{\prime},\mathcal{U})+{\rm
Mod}^{D}_{p}(\mathcal{P},\mathcal{U}).$

  3. (3)

Majorization: If $\mathcal{P},\mathcal{P}^{\prime}$ are two collections of
curves so that every set $P\in\mathcal{P}$ contains a subset in
$\mathcal{P}^{\prime}$, then

${\rm Mod}_{p}^{D}(\mathcal{P},\mathcal{U})\leq{\rm
Mod}^{D}_{p}(\mathcal{P}^{\prime},\mathcal{U}).$

  4. (4)

Existence of minimizers: If $X$ is compact, then there exists a
$\rho\wedge_{\mathcal{U}}\mathcal{P}$ with

${\rm
Mod}^{D}_{p}(\mathcal{P},\mathcal{U})=\sum_{U\in\mathcal{U}}\rho(U)^{p}.$

In what follows, since these properties are so standard, we will often simply
apply these facts without explicit reference to this Lemma.

###  3.2. Modulus of annulus

Let $B$ be a ball in $X$ and $L>1$. Consider a $\kappa$-round collection
$\mathcal{U}$. We say that $P=\\{U_{1},\dots,U_{n}\\}\subset\mathcal{U}$ is a
$(\mathcal{U},B,L)$-path, if $U_{1}\cap B\neq\emptyset$, $U_{n}\cap X\setminus
LB\neq\emptyset$ and if $U_{i}\cap U_{i+1}\neq\emptyset$ for all
$i=1,\dots,n-1$. Let $\mathcal{P}_{\mathcal{U},B,L}$ be the collection of all
$(\mathcal{U},B,L)$-paths.

Then, we define the Keith-Laakso modulus as

${\rm Mod}_{p,L,\mathcal{U}}^{KL}:={\rm
Mod}_{p}^{D}(\mathcal{P}_{\mathcal{U},B,L},\mathcal{U}).$

This notion of modulus coincides with that of [14, 7, 18] if we use the
collection $\mathcal{U}$ indicated in Remark 2.8.

In [4], a slightly different form of modulus is obtained by using a more
restrictive admissibility constraint. Let $\Gamma$ be a family of curves, and
let $\mathcal{P}_{\Gamma}=\\{P_{\gamma}:\gamma\in\Gamma\\}$, where
$P_{\gamma}=\\{U\in\mathcal{U}:U\cap\gamma\neq\emptyset\\}$. We define the
(Bourdon-Kleiner) modulus of the curve family as

${\rm Mod}_{p,\mathcal{U}}(\Gamma)={\rm
Mod}_{p}^{D}(\mathcal{P}_{\Gamma},\mathcal{U}).$

We relate the two moduli via the following lemma. This justifies using the
Bourdon-Kleiner modulus, instead of the Keith-Laakso modulus in the context of
conformal dimension, see Theorem 3.5. Let $\Gamma_{L,B}$ be the collection of
curves $\gamma$ connecting $B$ to $X\setminus LB$.

######  Lemma 3.2.

Suppose that $X$ is compact, metrically doubling and LLC. Then, for any
$\kappa$-approximation $\mathcal{U}$, any ball $B\subset X$ and any $L>1$, we
have

${\rm Mod}_{p,\mathcal{U}}(\Gamma_{B,L})\sim{\rm
Mod}_{p,B,L,\mathcal{U}}^{KL}.$

###### Proof.

We have $\mathcal{P}_{\Gamma}\subset\mathcal{P}_{\mathcal{U},B,L}$. From the
definition of the modulus, and Lemma 3.1 it is thus direct that

${\rm Mod}_{p,\mathcal{U}}(\Gamma_{B,L})={\rm
Mod}^{D}_{p}(\mathcal{P}_{\Gamma},\mathcal{U})\leq{\rm
Mod}^{D}_{p}(\mathcal{P}_{\mathcal{U},B,L},\mathcal{U}))={\rm
Mod}_{p,B,L,\mathcal{U}}^{KL}.$

For the other direction of the proof we need the assumptions of $X$ being LLC
and metrically doubling. Let $X$ be $C$-LLC and $D$-metrically doubling. Next,
let $\rho\wedge_{\mathcal{U}}\mathcal{P}_{\Gamma}$ be arbitrary. We will
define another $\tilde{\rho}$ so that
$\tilde{\rho}\wedge_{\mathcal{U}}\mathcal{P}_{\mathcal{U},B,L}$ and so that

(3.3) $\sum_{U\in\mathcal{U}}\tilde{\rho}(U)^{p}\leq
M\sum_{U\in\mathcal{U}}\rho(U)^{p}$

for a constant $M$ depending on $C$, the local boundedness constants and $D$.
From these the claim of the Lemma follows by taking an infimum over all
$\rho\wedge_{\mathcal{U}}\mathcal{P}_{\Gamma}$. The rest of the proof consists
of defining $\tilde{\rho}$, showing (3.3) and proving
$\tilde{\rho}\wedge_{\mathcal{U}}\mathcal{P}_{\mathcal{U},B,L}$.

We do a small preliminary estimate. Fix $V\in\mathcal{U}$ and any constant
$S\geq 1$ and let

$\mathcal{U}_{V,S}=\\{U\in\mathcal{U}:U\cap
B(z_{V},(1+2S)r_{V})\neq\emptyset\\}.$

If $U_{1},U_{2}\in\mathcal{U}_{V,S}$ are distinct, we have by local
boundedness

(3.4) $\kappa_{1+2S}^{-1}r_{V}\leq r_{U_{1}},r_{U_{2}}\leq\kappa_{1+2S}r_{V}.$

Thus, the balls $B(z_{U_{i}},r_{V}\kappa_{1+2S}^{-1}\kappa^{-1})$ are disjoint
for $i=1,2$ and are contained in $B(z_{V},(1+2S+2\kappa_{1+2S})r_{V})$. Thus,
by metric doubling, there are at most $D^{m}$ sets contained in
$\mathcal{U}_{V,S}$ for any $V\in\mathcal{U}$ as long as $2^{m}\geq
4\kappa\kappa_{1+2S}(1+2S+2\kappa_{1+2S}).$

We will next consider $\mathcal{U}_{V,C}$. Let $L=1+\kappa_{1+2C}(1+2C)$.
Choose $k,l\in{\mathbb{N}}$ with $l\leq k$ and so that
$4\kappa\kappa_{1+2C}(1+2C+2\kappa_{1+2C})\leq 2^{l}$ and so that
$4\kappa\kappa_{1+2L}(1+2L+2\kappa_{1+2L})\leq 2^{k}$, and let $M=D^{2kp}$. By
the argument after (3.4) with $S=C$, we have that $|\mathcal{U}_{V,C}|\leq
2^{l}\leq 2^{k}$. Let

$\tilde{\rho}(V)=D^{k}\max\\{\rho(U):U\in\mathcal{U}_{V,C}\\}.$

For each $V\in\mathcal{U}$ choose a $U_{V}\in\mathcal{U}_{V,C}$ so that
$\tilde{\rho}(V)=D^{k}\rho(U_{V})^{p}$. Let
$\tilde{\mathcal{U}}_{U,C}=\\{V\in\mathcal{U}:U_{V}=U\\}$. For every
$V\in\tilde{\mathcal{U}}_{U,C}$, we have $U_{V}=U$ and thus
$U\in\mathcal{U}_{V,C}$. Thus $B(z_{U},r_{U})\cap
B(z_{V},(1+2C)r_{V})\neq\emptyset$, and $V\subset B(z_{V},r_{V})\subset
B(z_{U},r_{U}+(1+2C)r_{V})$. Consequently, from using (3.4) we get $V\cap
B(z_{U},(1+\kappa_{1+2C}(1+2C))r_{U})\neq\emptyset$. In particular, we have
$\tilde{\mathcal{U}}_{U,C}\subset\mathcal{U}_{U,L}$. Thus, we have by the
argument after (3.4) with $S$ replaced with $L$ that
$|\tilde{\mathcal{U}}_{U,C}|\leq D^{k}$.

Let $P=\\{U_{1},\dots,U_{n}\\}\in\mathcal{P}_{\mathcal{U},B,L}$. Define a
sequence of points $(x_{i})_{i=1}^{n+1}$ as follows. Let $x_{1}\in U_{1}\cap
B,x_{n+1}\in U_{n}\cap X\setminus LB$, and let $x_{i}\in U_{i}\cap U_{i-1}$
for $i=2,\dots n$. Since $X$ is $C$-LLC, we can find curves $\gamma_{i}$
connecting $x_{i}$ to $x_{i+1}$ with $\operatorname{diam}(\gamma_{i})\leq
Cd(x_{i},x_{i+1})\leq\operatorname{diam}(U_{i})$ for $i=1,\dots,n$. Let
$\gamma$ be the concatenation of $\gamma_{i}$. We have that
$\gamma\in\Gamma_{B,L}$.

Now, let $P_{\gamma}=\\{U\in\mathcal{U}:U\cap\gamma\neq\emptyset\\}$. We have

$\sum_{U\in P_{\gamma}}\rho(U)\geq 1,$

since $\rho\wedge_{\mathcal{U}}\mathcal{P}_{\Gamma}$. Now, for each $U\in
P_{\gamma}$ we have some $i=1,\dots,n$ so that $U\cap\gamma_{i}\neq\emptyset$.
Therefore, we have $d(U,U_{i})\leq C\operatorname{diam}(U_{i})\leq
2Cr_{U_{i}}$. Thus, $U\in\mathcal{U}_{U_{i},C}$ for some $i$. Let
$P_{\gamma,i}=\mathcal{U}_{U_{i},C}\cap P_{\gamma}$. Since
$|\mathcal{U}_{U_{i},C}|\leq D^{k}$, we get

$\tilde{\rho}(U_{i})\geq\sum_{U\in P_{\gamma,i}}\rho(U).$

Summing these, we get

$\displaystyle 1\leq\sum_{U\in P_{\gamma}}\rho(U)$
$\displaystyle\leq\sum_{i=1}^{n}\sum_{U\in P_{\gamma,i}}\rho(U)$
$\displaystyle\leq\sum_{i=1}^{n}\tilde{\rho}(U_{i}).$

Thus, $\tilde{\rho}\wedge\mathcal{P}_{\mathcal{U},B,L}$, since $P$ was
arbitrary.

Finally, we show (3.3) for $M=D^{k(1+p)}$. We have by the size bound for
$\tilde{U}_{U,C}$ that

$\displaystyle\sum_{V\in\mathcal{U}}\tilde{\rho}(V)^{p}$
$\displaystyle\leq\sum_{U\in\mathcal{U}}\sum_{V\in\tilde{\mathcal{U}}_{U,C}}\tilde{\rho}(V)^{p}$
$\displaystyle\leq\sum_{U\in\mathcal{U}}D^{k}D^{kp}\rho(U)^{p}\leq
D^{k(1+p)}\sum_{U\in\mathcal{U}}\rho(U)^{p}.$

∎

###  3.3. Relationship to Conformal dimension

The proof of the main Theorem 1.6 is based on the following Theorem of
Carrasco-Piaggio. An interested reader may see also [21] and [18] for slightly
different versions and proofs of this statement. We have used Lemma 3.2 and
Remark 2.8 to reformulate the theorem using our notion of
$\kappa$-approximations and moduli.

######  Theorem 3.5 (Theorem 1.3 in [7]).

Suppose that $X$ is a compact, metrically doubling LLC space, and let
$\mathcal{U}_{k}$ be $\kappa$-approximations at level $2^{-k}$. Then

${\rm dim}_{CAR}(X)=\inf\left\\{Q>0:\liminf_{m\to\infty}\sup_{z\in X,k\geq
0}{\rm Mod}_{Q,\mathcal{U}_{m+k}}(\Gamma_{B(z,2^{-k}),2})=0\right\\}.$

###  3.4. New discrete modulus

In this subsection, we introduce a new notion of modulus, which allows for an
arbitrary $\kappa$-round collection $\mathcal{U}$, which may or may not be a
$\kappa$-approximation. Indeed, formally we shall permit that the collection
even fails to be a cover. This brings the definition closer to that considered
by Pansu and Tyson in [19, 22]. We note that it may be interesting to study
more carefully the relationships between their modulus and the one presented
here. However, since it would be a side track in the present paper, we do not
pursue this here.

######  Definition 3.6.

Fix $\tau\geq 4$. Let $\mathcal{U}$ be a $\kappa$-round, and let $\Gamma$ a
family of sets in $X$. We say that $\rho:\mathcal{U}\to[0,\infty)$ is strongly
discretely $\tau$-admissible for $\Gamma$, and write
$\rho\overline{\wedge}_{\tau,\mathcal{U}}\Gamma$, if for every
$\gamma\in\Gamma$ there exists a collection
$\mathcal{U}_{\gamma}\subset\mathcal{U}$ with the following properties:

  1. i)

$\\{B(z_{U},\tau r_{U}):U\in\mathcal{U}_{\gamma}\\}$ pairwise disjoint;

  2. ii)

$U\cap\gamma\neq\emptyset$ for all $U\in\mathcal{U}_{\gamma}$; and

  3. iii)

we have

$\sum_{U\in\mathcal{U}_{\gamma}}\rho(U)\geq 1.$

Define

$\overline{{\rm
Mod}}_{p,\kappa,\tau}(\Gamma,\mathcal{U})=\inf_{\rho\overline{\wedge}_{\mathcal{U},\tau}\Gamma}\sum_{U\in\mathcal{U}}\rho(U)^{p}.$

The first result we prove, is that the new modulus bounds from above the
notion of discrete modulus defined before, when the collections are roughly at
the same level.

######  Proposition 3.7.

Let $k\in{\mathbb{N}}$. Assume that $X$ is metrically doubling, and that
$\kappa\geq 1$, $\tau\geq 4$. There exists a constant $C>0$ so that the
following holds for $r>0$.

Suppose that $\mathcal{U}$ is a $\kappa$-approximation at level $r$ and
${\mathcal{V}}$ is a $\kappa$-round collection with $\kappa^{-1}r\leq
r_{V}\leq r$ for every $V\in{\mathcal{V}}$. If $\Gamma$ is a collection of
curves in $X$, then

${\rm Mod}_{p,\mathcal{U}}(\Gamma)\leq C\overline{{\rm
Mod}}_{p,\kappa,\tau}(\Gamma,{\mathcal{V}}).$

###### Proof.

Since $X$ is metrically doubling, and by an argument similar to that in Lemma
3.2, there is a constant $D$ so that each $U\in\mathcal{U}$ intersects at most
$D$ pairwise disjoint sets in ${\mathcal{V}}$. Similarly, for each
$V\in{\mathcal{V}}$ there are at most $D$ many $U\in\mathcal{U}$ with $U\cap
V\neq\emptyset$.

Without loss of generality, assume that $\overline{{\rm
Mod}}_{p,\kappa,\tau}(\Gamma,{\mathcal{V}})<\infty$. Let
$\overline{\rho}\wedge_{\tau,{\mathcal{V}}}\Gamma$ be any admissible function
so that

$\sum_{V\in{\mathcal{V}}}\overline{\rho}(V)^{p}<\infty.$

For each $U\in\mathcal{U}$, define

$\rho(U)=D\max\\{\overline{\rho}(V):U\cap
V\neq\emptyset,V\in{\mathcal{V}}\\},$

if there exists some $V\in{\mathcal{V}}$ with $U\cap V\neq\emptyset$. If there
does not exist any $V\in{\mathcal{V}}$ with $V\cap U\neq\emptyset$, set
$\rho(U)=0.$

For each $U\in\mathcal{U}$, for which it is possible, choose one
$V_{U}\in{\mathcal{V}}$ so that $U\cap V\neq\emptyset$ and
$\rho(U)=D\overline{\rho}(V)$. Let
$\mathcal{U}_{V}=\\{U\in\mathcal{U}:V_{U}=U\\}$ for $V\in{\mathcal{V}}$. We
have, by the first paragraph of the proof, that $|\mathcal{U}_{V}|\leq D$.

We claim that $\rho\wedge_{\mathcal{U}}\mathcal{P}_{\Gamma}$. Let
$\gamma\in\Gamma$ be arbitrary. Let
${\mathcal{V}}_{\gamma}\subset{\mathcal{V}}$ be the collection, where each set
intersects $\gamma$ and such that the collection $\\{B(z_{V},\tau
r_{V}):V\in{\mathcal{V}}_{\gamma}\\}$ is disjoint with

$\sum_{V\in{\mathcal{V}}_{\gamma}}\overline{\rho}(V)\geq 1.$

Let $\mathcal{U}_{\gamma}:=\\{U\in\mathcal{U}:U\cap\gamma\neq\emptyset\\}$.
Since $\mathcal{U}$ is a cover of $X$, for each $V\in{\mathcal{V}}_{\gamma}$,
we may choose a $U^{V}\in\mathcal{U}$ so that
$U\cap(V\cap\gamma)\neq\emptyset$. For each $U\in\mathcal{U}$, let
${\mathcal{V}}_{\gamma,U}=\\{V\in{\mathcal{V}}_{\gamma}:U^{V}=U\\}$. This
means that

(3.8)
${\mathcal{V}}_{\gamma}\subset\bigcup_{U\in\mathcal{U}_{\gamma}}{\mathcal{V}}_{\gamma,U}$

We also have by the first paragraph of the proof that
$|{\mathcal{V}}_{\gamma,U}|\leq D$ for every $U\in\mathcal{U}_{\gamma}$. Thus,
for every $U\in\mathcal{U}_{\gamma}$, we have

(3.9) $\rho(U)\geq\sum_{V\in{\mathcal{V}}_{\gamma,U}}\overline{\rho}(V).$

By applying (3.8) and (3.9) we get:

$\displaystyle\sum_{U\cap\gamma\neq\emptyset}\rho(U)$
$\displaystyle\geq\sum_{U\cap\gamma\neq\emptyset}\sum_{V\in{\mathcal{V}}_{\gamma,U}}\overline{\rho}(V)$
$\displaystyle\geq\sum_{V\in{\mathcal{V}}_{\gamma}}\overline{\rho}(V)\geq 1.$

Note that $\mathcal{U}=\bigcup_{V\in{\mathcal{V}}}\mathcal{U}_{V}$. By using
this, we estimate the $p$-energy of $\rho$ using the bound
$|\mathcal{U}_{V}|\leq D$ for every $V\in\mathcal{U}$.

$\displaystyle\sum_{U\in\mathcal{U}}\rho(U)^{p}$
$\displaystyle\leq\sum_{V\in{\mathcal{V}}}\sum_{U\in\mathcal{U}_{V}}\rho(U)^{p}$
$\displaystyle\leq\sum_{V\in{\mathcal{V}}}D^{p}|\mathcal{U}_{V}|\overline{\rho}(V)^{p}$
$\displaystyle\leq\sum_{B\in{\mathcal{B}}}D^{p+1}\overline{\rho}(V)^{p}.$

Thus, the claim holds for $C=D^{p+1}$ after we take an infimum over
$\overline{\rho}\wedge_{\tau,{\mathcal{V}}}\Gamma$. ∎

One of the benefits of this notion of modulus, is that we can give simple
bounds for it in terms of the Hausdorff measure of the space. The following
will be an example of such a bound, which we will use. Recall the definition
of Hausdorff content $\mathcal{H}_{\delta}^{p}$ from (1.3).

######  Proposition 3.10.

Let $\kappa\geq 1,\tau\geq 4$. Let $X$ be any connected compact metric space,
and suppose that $\Gamma$ is a family of curves, where each curve in $\Gamma$
is contained in a ball $B(x,R)\subset X$ and has diameter at least $r$. Then,
for every $\epsilon\in(0,1),\delta\in(0,\operatorname{diam}(X)/2)$ there
exists a $\kappa$-round collection ${\mathcal{V}}$ for which

$\overline{{\rm
Mod}}_{p,\tau}(\Gamma,{\mathcal{V}})\leq\frac{(20\tau)^{p}\mathcal{H}_{\delta}^{p}(B(x,R))+\epsilon}{r^{p}},$

and $\sup_{V\in{\mathcal{V}}}r_{V}\leq\delta$ and each ball in ${\mathcal{V}}$
intersects $B(x,R)$ as well as some curve in $\Gamma$.

###### Proof.

Fix $\epsilon>0$. From the definition of Hausdorff content in (1.3), and by
replacing each set in the cover by an enclosing ball, we can find a covering
${\mathcal{V}}$ of $B(x,R)$ by balls $V=B=B(z_{V},r_{V})$ with
$r_{V}\leq\delta$ so that

$\sum_{V\in{\mathcal{V}}}\operatorname{diam}(V)^{p}\leq
2^{p}\mathcal{H}_{\delta}^{p}(B(x,R))+(10\tau)^{-p}\epsilon.$

Moreover, by possibly making the collection smaller, we assume that each ball
$V\in{\mathcal{V}}$ intersects $B(x,R)$ and some curve in $\Gamma$. This
modified collection still covers every curve $\gamma\in\Gamma$. Now,
${\mathcal{V}}$ is $\kappa$-round with $\kappa=1$. Let
$\rho(V)=10\tau\operatorname{diam}(V)/r$. We have by the choice of
${\mathcal{V}}$ and $\rho$ that

$\sum_{V\in{\mathcal{V}}}\rho(V)^{p}\leq\sum_{V\in{\mathcal{V}}}(10\tau)^{p}\operatorname{diam}(V)^{p}r^{-p}\leq\frac{(20\tau)^{p}\mathcal{H}_{\delta}^{p}(B(x,R))+\epsilon}{r^{p}}.$

Therefore, the claim will follow once we show that
$\rho\overline{\wedge}_{\tau,{\mathcal{V}}}\Gamma$. Let $\gamma\in\Gamma$. We
need to find a collection ${\mathcal{V}}_{\gamma}$ so that the properties i,ii
and iii from Definition 3.6 hold. Since ${\mathcal{V}}$ is a cover of
$\gamma$, we have that $\\{B(z_{V},\tau r_{V})\\}$ is a cover of $\gamma$.
Applying the $5r$-covering lemma, we get a finite subcollection
${\mathcal{V}}_{\gamma}\subset{\mathcal{V}}$ so that i) $\\{B(z_{V},\tau
r_{V}):V\in{\mathcal{V}}_{\gamma}\\}$ is pairwise disjoint, ii) so that
$\gamma\cap V\neq\emptyset$ for all $V\in{\mathcal{V}}_{\gamma}$ and so that
we have that ${\mathcal{V}}^{\prime}=\\{B(z_{V},5\tau
r_{V}):V\in{\mathcal{V}}_{\gamma}\\}$ is a covering of $\gamma$. Note that

$\operatorname{diam}(B(z_{V},5\tau r_{V}))\leq 10\tau r_{V}\leq
10\tau\operatorname{diam}(V)=\rho(V)r.$

Since the balls $\\{B(z_{V},5\tau r_{V}):V\in{\mathcal{V}}_{\gamma}\\}$ cover
$\gamma$, we get

$\sum_{V\in{\mathcal{V}},V\cap\gamma\neq\emptyset}\rho(V)\geq\sum_{V\in{\mathcal{V}}^{\prime}}\operatorname{diam}(B(z_{V},5\tau
r_{V}))/r\geq\operatorname{diam}(\gamma)/r\geq 1.$

Thus, $\rho\overline{\wedge}_{\tau,{\mathcal{V}}}\Gamma$ and the claim
follows. ∎

The new notion of modulus is also invariant under quasisymmetries, except for
adjusting the $\tau$ parameter. In the following, if $\Gamma$ is a collection
of curves in $X$ and $f:X\to Y$ is a homeomorphism, we write
$f(\Gamma)=\\{f\circ\gamma:\gamma\in\Gamma\\}$. The opposite inequality can be
obtained by adjusting $\tau$, and applying this lemma to the inverse mapping
$f^{-1}$.

######  Lemma 3.11.

Let $\tau\geq 4$ and let $f:X\to Y$ be an $\eta$-quasisymmetry. If
${\mathcal{V}}$ is a $\kappa$-round collection, and if $\Gamma$ is any
collection of curves in $X$, then

$\overline{{\rm Mod}}_{p,\tau}(\Gamma,{\mathcal{V}})\leq\overline{{\rm
Mod}}_{p,\max\\{4,\eta(\tau)\\}}(f(\Gamma),f({\mathcal{V}})).$

###### Proof.

Let $\tau^{\prime}=\max\\{4,\eta(2\tau)\\}$. By Lemma 2.9, we have that
$f({\mathcal{V}})$ is a $\kappa^{\prime}$-round collection for some
$\kappa^{\prime}$. Let
$\rho\overline{\wedge}_{\tau^{\prime},f({\mathcal{V}})}f(\Gamma)$. Define
$\overline{\rho}(V)=\rho(f(V))$ for $V\in{\mathcal{V}}$. We clearly have

$\sum_{V\in{\mathcal{V}}}\overline{\rho}(V)^{p}=\sum_{V\in
f({\mathcal{V}})}\rho(V)^{p}.$

Thus, the claim will follow, if we can show that
$\overline{\rho}\overline{\wedge}_{\tau,{\mathcal{V}}}\Gamma$.

In the following, elements of $f({\mathcal{V}})$ will be written as $f(V)$,
where $V\in{\mathcal{V}}$. Let $\gamma\in\Gamma$. Then, $f\circ\gamma\in
f(\Gamma)$ and, since
$\rho\overline{\wedge}_{\tau^{\prime},f({\mathcal{V}})}f(\Gamma)$, there
exists a collection $\mathcal{U}_{f(\gamma)}\subset f({\mathcal{V}})$ so that

  1. i)

$\\{B(z_{f(V)},\tau^{\prime}r_{f(V)}):f(V)\in\mathcal{U}_{f(\gamma)}\\}$ is
pairwise disjoint;

  2. ii)

$f(V)$ intersects $\gamma$ for every $f(V)\in\mathcal{U}_{f(\gamma)}$;

  3. iii)

we have

$\sum_{U\in\mathcal{U}_{f(\gamma)}}\rho(U)\geq 1.$

Let
$\mathcal{U}_{\gamma}=\\{V\in{\mathcal{V}}:f(V)\in\mathcal{U}_{f(\gamma)}\\}$.
We need to check the three properties from Definition 3.6 of
$\overline{\rho}\overline{\wedge}_{\tau,{\mathcal{V}}}\Gamma$:

  1. a)

$\\{B(z_{V},\tau r_{V}):V\in\mathcal{U}_{\gamma}\\}$ is pairwise disjoint;

  2. b)

$V$ intersects $\gamma$ for every $V\in\mathcal{U}_{\gamma}$;

  3. c)

we have

$\sum_{U\in\mathcal{U}_{\gamma}}\overline{\rho}(U)\geq 1.$

From these, b) and c) follow immediately from the properties ii) and iii)
above. By Lemma 2.3, we have

$f(B(z_{V},\tau r_{V}))\subset f(B(z_{f(V)},\eta(2\tau)r_{f(V)}),$

for every $V\in\mathcal{U}_{\gamma}$. Since $\tau^{\prime}\geq\eta(2\tau)$,
the disjointness in a) follows from that in i). ∎

##  4\. Combinatorially Loewner Spaces

###  4.1. Definition and basic property

For two closed sets $E,F$, let $\Gamma(E,F)$ be the collection of curves which
join them. We adapt the definition of Bourdon and Kleiner of the combinatorial
Loewner property slightly, as modified by Clais in [8, Definition 2.6]. Let
$\mathcal{U}_{k}$ be a sequence of $\kappa$-approximations at level $2^{-k}$.

######  Definition 4.1.

Fix $p>1$. We say that a compact LLC space $X$ satisfies the combinatorial
$p$-Loewner property, if there exist some increasing continuous functions
$\phi,\psi:(0,\infty)\to(0,\infty)$ with $\lim_{t\to 0}\psi(t)=0$, with the
following two properties.

  1. (1)

For every pair of disjoint continua $E,F\subset X$ and all $k\geq 0$ with
$2^{-k}\leq\min\\{\operatorname{diam}(E),\operatorname{diam}(F)\\}$, we have

$\phi(\Delta(E,F)^{-1})\leq{\rm Mod}_{p,\mathcal{U}_{k}}(\Gamma(E,F)).$

  2. (2)

For every $z\in X$ and $0<r<R$ and all $k\geq 0$ with $2^{-k}\leq r$, we have

${\rm Mod}_{p,\mathcal{U}_{k}}(\Gamma(\overline{B(z,r)},X\setminus
B(z,R)))\leq\psi\left(\frac{r}{R-r}\right).$

Spaces with the combinatorial $p$-Loewner property are also called CLP -spaces
or $p$-CLP spaces, if we wish to explicate the exponent $p>1$.

We first note that a combinatorially $p$-Loewner space has conformal Hausdorff
dimension equal to $p$. This theorem is quite well known and is a rather
direct consequence from the known Theorem 3.5. However, we present a proof for
the sake of clarity, and since its proof does not appear to have been
published elsewhere. Later, we will prove Theorem 1.8, which is one of our
main contributions, and which improves the following statement by showing that
also ${\rm dim}_{CH}(X)=p$.

######  Lemma 4.2.

For a compact LLC space $X$, which is combinatorially $p$-Loewner, it holds
that

${\rm dim}_{CA}(X)={\rm dim}_{CAR}(X)=p.$

###### Proof.

Let $\psi$ and $\psi$ be the functions appearing in Definition 4.1.

Since $X$ is a compact LLC space, it is uniformly perfect, and by [17,
Proposition 2.2.6] and [11, Chapters 14 and 15] we have ${\rm
dim}_{CA}(X)={\rm dim}_{CAR}(X)$. Next, let $\mathcal{U}_{k}$ be a sequence of
$\kappa$-approximations at levels $2^{-k}$ for $k\in{\mathbb{N}}$. Let $z\in
X$ and $0<r\leq\operatorname{diam}(X)/4$. Then, by the LLC property, there
exists a continuum $E\subset\overline{B(z,r)}$ with
$\operatorname{diam}(E)\geq r$ and another continuum
$F\subset\overline{B(z,3r)}\setminus B(z,2r)$ with $\operatorname{diam}(F)\geq
r$. Since every curve connecting $E$ to $F$ contains a sub-curve within
$\Gamma(\overline{B(z,r)},X\setminus B(z,2r))$, we have

${\rm Mod}_{p}(\Gamma(E,F),\mathcal{U}_{k})\leq{\rm
Mod}_{p}(\Gamma(\overline{B(z,r)},X\setminus B(z,2r)),\mathcal{U}_{k}).$

However, by the CLP property and since $\Delta(E,F)\leq 6$, we get for all
$k\geq 0$ such that $2^{-k}\leq r$ that

$\phi(6^{-1})\leq{\rm Mod}_{p}(\Gamma(E,F),\mathcal{U}_{k})\leq{\rm
Mod}_{p}(\Gamma(\overline{B(z,r)},X\setminus B(z,2r)),\mathcal{U}_{k}).$

We thus get:

$\liminf_{m\to\infty}\sup_{z\in X,k\geq 0}{\rm
Mod}_{p,\mathcal{U}_{m+k}}(\Gamma_{B(z,2^{-k}),2})\geq\phi(6^{-1}).$

Thus, $p\leq{\rm dim}_{CAR}(X)$ by Theorem 3.5.

The inequality $p\geq{\rm dim}_{CAR}(X)$ follows by showing that for all
$\epsilon>0$, we have

$\lim_{m\to\infty}\sup_{z\in X,k\geq 0}{\rm
Mod}_{p+\epsilon,\mathcal{U}_{m+k}}(\Gamma_{B(z,2^{-k}),2})=0.$

The idea in showing this is to compare the discrete moduli with exponents
$p+\epsilon$ and $p$. Indeed, we will show that for all $m\geq 3$ we have

(4.3) ${\rm
Mod}_{p+\epsilon,\mathcal{U}_{m+k}}(\Gamma_{B(z,2^{-k}),2})\leq\psi(2^{2-m})^{\epsilon}{\rm
Mod}_{p,\mathcal{U}_{m+k}}(\Gamma_{B(z,2^{-k}),2})\leq\psi(2^{2-m})^{\epsilon}\psi(1).$

Then, since $\lim_{t\to 0}\psi(t)=0$, the claim follows.

Let $\rho$ be the optimal function for ${\rm
Mod}_{p,\mathcal{U}_{m+k}}(\Gamma_{B(z,2^{-k}),2})$, which exists by Lemma
3.2. We will show that $\rho(U)\leq\psi(2^{1-m})$ for every
$U\in\mathcal{U}_{m+k}$. This uses a bound for modulus coming from [4, Lemma
2.3], which in turn relies on estimating the modulus of the curves which pass
through the set $U$. Let $U\in\mathcal{U}_{m+k}$. Let $\Gamma_{U}$ be the
collection of curves in $\Gamma_{B(z,2^{-k}),2}$ which intersect $U$. Then any
curve in $\Gamma_{B(z,2^{-k}),2}$ which intersects $U$ will contain a sub-
curve connecting $\overline{B(z_{U},r_{U})}$ to $X\setminus
B(z_{U},2^{m-1}r_{U})$. Thus,

${\rm Mod}_{p,\mathcal{U}_{k}}(\Gamma_{U})\leq{\rm
Mod}_{p,\mathcal{U}_{k}}(\Gamma(\overline{B(z_{U},r_{U})},X\setminus
B(z_{U},2^{m-1}r_{U}))\leq\psi(2^{2-m}).$

By [4, Lemma 2.3], we get for all $U\in\mathcal{U}_{m+k}$

$\rho(U)\leq{\rm Mod}_{p,\mathcal{U}_{k}}(\Gamma_{U})\leq\psi(2^{2-m}).$

This, together with the optimality of $\rho$ yields

$\sum_{U\in\mathcal{U}}\rho(U)^{p+\epsilon}\leq\max_{U\in\mathcal{U}}\rho(U)^{\epsilon}\sum_{U\in\mathcal{U}}\rho(U)^{p}\leq\psi(2^{2-m})^{\epsilon}{\rm
Mod}_{p,\mathcal{U}_{m+k}}(\Gamma_{B(z,2^{-k}),2}),$

which is the desired estimate (4.3).

∎

###  4.2. Estimates for Modulus

If the space is combinatorially Loewner, then we can give a lower bound of our
modulus, which we introduced in Subsection 3.4, in terms of the Bourdon-
Kleiner modulus. This is a strengthening of the Proposition 3.7. In a sense,
the following Proposition is the starting point of our paper, since its
argument was the first to be discovered.

######  Proposition 4.4.

Let $k\in{\mathbb{N}}$, $p>1$. Assume that $X$ is metrically doubling, LLC and
combinatorially $p$-Loewner, and that $\kappa\geq 1$, $\tau\geq 4$. There
exists a constant $C>0$ so that the following holds for $r>0$.

Suppose that $\mathcal{U}$ is a $\kappa$-approximation at level $r$ and
${\mathcal{V}}$ is a $\kappa$-round collection with
$\inf\\{r_{V}:V\in{\mathcal{V}}\\}\geq 2r$. If $\Gamma$ is a collection of
curves in $X$ with
$2\tau\sup_{V\in{\mathcal{V}}}r_{V}\leq\operatorname{diam}(\gamma)$ for all
$\gamma\in\Gamma$, then

${\rm Mod}_{p,\mathcal{U}}(\Gamma)\leq C\overline{{\rm
Mod}}_{p,\tau}(\Gamma,{\mathcal{V}}).$

###### Proof.

Again, assume that $\overline{{\rm
Mod}}_{p,\tau}(\Gamma,{\mathcal{V}})<\infty$, and that
$\overline{\rho}\wedge_{\tau,{\mathcal{V}}}\Gamma$ with

$\sum_{V\in{\mathcal{V}}}\overline{\rho}^{p}(V)<\infty.$

For each $V\in{\mathcal{V}}$ consider the collection of curves
$\Gamma_{V}=\Gamma(\overline{B(z_{V},r_{V})},X\setminus
B(z_{V},(\tau-1)r_{V}))$. By the $p$-combinatorial Loewner assumption and
since $r\leq r_{V}/2$, we have

(4.5) ${\rm Mod}_{p,\mathcal{U}}(\Gamma_{V})\leq C,$

for $C=\psi(\frac{1}{\tau-1})>0$, where $\psi$ is from Definition 4.1. Let
$\rho_{V}:\mathcal{U}\to[0,\infty)$ be such that
$\rho_{V}\wedge_{\mathcal{U}}\Gamma_{V}$ and so that

(4.6) $\sum_{U\in\mathcal{U}}\rho_{V}(U)^{p}\leq 2C.$

Let

$\rho(U)=\max\\{\rho_{V}(U)\overline{\rho}(V):V\in{\mathcal{V}}\\}.$

We claim that $\rho\wedge_{\mathcal{U}}\Gamma$. Let $\gamma\in\Gamma$. Since
$\overline{\rho}\wedge_{\tau,{\mathcal{V}}}\Gamma$, there exists a collection
${\mathcal{V}}_{\gamma}$ of $V\in{\mathcal{V}}$ with

  1. (1)

$V\cap\gamma\neq\emptyset$ for all $V\in{\mathcal{V}}_{\gamma}$;

  2. (2)

$\\{B(z_{V},\tau r_{V}):V\in{\mathcal{V}}_{\gamma}\\}$ is a pairwise disjoint
collection of balls; and

  3. (3)

(4.7) $\sum_{V\in{\mathcal{V}}_{\gamma}}\overline{\rho}(V)\geq 1.$

For each $V\in{\mathcal{V}}_{\gamma}$, let $\gamma|_{V}$ be a minimal subcurve
which connected $B(z_{V},r_{V})$ to $B(z_{V},(\tau-1)r_{V})$. Such a subcurve
exists since $\operatorname{diam}(\gamma)\geq 2\tau r_{V}$ and $\gamma\cap
B(z_{V},r_{V})\neq\emptyset$. These subcurves are disjoint and
$d(\gamma|_{V},\gamma|_{V^{\prime}})\geq 2\min\\{r_{V},r_{V^{\prime}}\\}\geq
4r$, for distinct $V,V^{\prime}\in{\mathcal{V}}_{\gamma}$. Therefore, if we
let $\mathcal{U}_{V}=\\{U\in\mathcal{U}:U\cap\gamma|_{V}\neq\emptyset\\}$ for
$V\in{\mathcal{V}}_{\gamma}$, then
$\mathcal{U}_{V}\cap\mathcal{U}_{V^{\prime}}=\emptyset$ for distinct
$V,V^{\prime}\in{\mathcal{V}}_{\gamma}$. We also have, since
$\rho_{V}\wedge\Gamma_{V}$ and $\rho\geq\rho_{V}\overline{\rho}(V)$ that

(4.8)
$\sum_{U\in\mathcal{U}_{V}}\rho(U)\geq\sum_{U\in\mathcal{U},U\cap\gamma|_{V}\neq\emptyset}\rho_{V}(U)\overline{\rho}(V)\geq\overline{\rho}(V).$

Now, let
$\mathcal{U}_{\gamma}=\\{U\in\mathcal{U}:U\cap\gamma\neq\emptyset\\}$. We also
have

$\bigcup_{V\in{\mathcal{V}}_{\gamma}}\mathcal{U}_{V}\subset\mathcal{U}_{\gamma}.$

By the disjointness of the collections $\mathcal{U}_{V}$, for distinct
$V\in{\mathcal{V}}_{\gamma}$, and by applying (4.7), (4.8) and the choice of
$\rho$, we get

$\displaystyle\sum_{U\in\mathcal{U}_{\gamma}}\rho(U)$
$\displaystyle\geq\sum_{V\in{\mathcal{V}}_{\gamma}}\sum_{U\in\mathcal{U}_{V}}\rho(U)$
$\displaystyle\geq\sum_{V\in{\mathcal{V}}_{\gamma}}\overline{\rho}(V)\geq 1.$

Thus, since $\gamma$ is arbitrary, $\rho\wedge_{\mathcal{U}}\Gamma$.

Next, we show a mass-bound for $\rho$. For each $U\in\mathcal{U}$ let
$V_{U}\in{\mathcal{V}}$ be such that
$\rho(U)=\rho_{V_{U}}(U)\overline{\rho}(V_{U})$. This yields a partition of
$\mathcal{U}$ into sets $\mathcal{U}^{V}=\\{U\in\mathcal{U}:V_{U}=V\\}.$ Thus,
we have, since $\mathcal{U}^{V}\subset\mathcal{U}$

$\displaystyle{\rm
Mod}_{p}(\Gamma,\mathcal{U})\leq\sum_{U\in\mathcal{U}}\rho(U)^{p}$
$\displaystyle=\sum_{V\in{\mathcal{V}}}\sum_{U\in\mathcal{U}^{V}}\rho_{V}(U)^{p}\overline{\rho}(V)^{p}$
$\displaystyle\leq\sum_{V\in{\mathcal{V}}}\overline{\rho}(V)^{p}\sum_{U\in\mathcal{U}}\rho_{V}(U)^{p}$
$\displaystyle\leq\sum_{V\in{\mathcal{V}}}2C\overline{\rho}(V)^{p}=2C\sum_{V\in{\mathcal{V}}}\overline{\rho}(V)^{p}.$

By infimizing over $\overline{\rho}$ such that
$\overline{\rho}\wedge_{\tau,{\mathcal{V}}}\Gamma$ the claim follows. ∎

We obtain the following proposition, which give a lower bound for the
Hausdorff measure of the image space. It is worth noting, that $f$ can be the
identity map. In this way, this generalizes to combinatorially Loewner spaces
the classical estimate of Heinonen and Koskela, [12, Theorem 3.6]. That result
is much easier to show using continuous modulus. For discrete modulus one
needs to do some extra work.

In the following proof we will need balls in $X$ and $Y$. The space in which a
ball is taken is indicated by a subscript e.g. $B_{Y}$.

######  Proposition 4.9.

Let $X$ be a $p$-combinatorially Loewner LLC and metrically doubling space,
and let $f:X\to Y$ be a quasisymmetric homeomorphism. Then, there exists a
constant $C\geq 1$ so that for every $r\in(0,\operatorname{diam}(Y))$ and any
$y\in Y$ we have

$\mathcal{H}^{p}(B_{Y}(y,r))\geq Cr^{p}.$

###### Proof.

Let $\eta:[0,\infty)\to[0,\infty)$ be a homeomorphism and let $f$ be an
$\eta$-quasisymmetry. Let
$\tilde{\eta}(t)=\left(\eta^{-1}(t^{-1})\right)^{-1}$ be such that $f^{-1}$ is
an $\tilde{\eta}$-quasisymmetry by Lemma 2.1.

Let $e_{Y}\in Y$. It is sufficient to prove

(4.10) $\mathcal{H}^{p}(B_{Y}(e_{Y},2L^{\prime}r))\geq C_{Y}r^{p}.$

for some uniform constants $L^{\prime}\geq 1,C_{Y}>0$ for all
$r\in(0,\operatorname{diam}(Y)/8)$. Since $Y$ is LLC, we can find a continuum
$E_{Y}\subset B(y,r)$ with $r\geq\operatorname{diam}(E_{Y})\geq r/2$ and with
$y\in E_{Y}$. Further, there exists a continuum
$F_{Y}\subset\overline{B(y,4r)}\setminus B(y,3r)$ with
$8r\geq\operatorname{diam}(F_{Y})\geq r$. We have

$1\leq\Delta(E_{Y},F_{Y})\leq 16.$

Let $\Gamma_{Y}$ be the collection of continuous curves connecting $E_{Y}$ to
$F_{Y}$. Let
$\Gamma_{X}=f^{-1}(\Gamma_{Y})=\\{f^{-1}\circ\gamma:\gamma\in\Gamma_{Y}\\}$,
which consists of curves connecting $E_{X}:=f^{-1}(E_{Y})$ to
$F_{X}:=f^{-1}(F_{Y})$.

First, we give some preliminary estimates for $E_{X}$ and $E_{Y}$. We have, by
Lemma 2.2 that

$\eta^{-1}(1)/2\leq\eta^{-1}(\Delta(E_{Y},F_{Y}))/2\leq\Delta(E_{X},F_{X})\leq\tilde{\eta}(2\Delta(E_{Y},F_{Y}))\leq\tilde{\eta}(32).$

Let $s_{X}=\operatorname{diam}(E_{X})$ and let $e_{X}\in E_{X}$ be such that
$e_{Y}=f(e_{X})$. Let $a\in F_{X}$ be the closest point to $E_{X}$. We have
$d(a,e_{X})\leq\operatorname{diam}(E_{X})+d(E_{X},F_{X})\leq(1+\Delta(E_{X},F_{X}))s_{X}\leq(1+\eta(32))s_{X}$.
If $b\in F_{X}$ is such that $d(a,b)\geq\operatorname{diam}(F_{X})/2$. We have

$8^{-1}=\frac{r}{8r}\leq\frac{d(f(a),f(e_{X}))}{d(f(a),f(b))}\leq\eta\left(\frac{d(a,e_{X})}{d(a,b)}\right).$

Thus,
$d(a,b)\leq(\eta^{-1}(8^{-1}))^{-1}d(a,e_{X})\leq(1+\eta(32))\tilde{\eta}(8)s_{X}$.
In particular, $\operatorname{diam}(F_{X})\leq
2(1+\eta(32))\tilde{\eta}(8)s_{X}$. Let $S=(1+\eta(32))(1+2\tilde{\eta}(8))$.
Then, we get $F_{X}\subset B(e_{X},Ss_{X})$.

Next, choose $a\in F_{X},b\in E_{X}$ so that $d(a,b)=d(F_{X},E_{X})$. Let
$c\in E_{X}$ be such that $d(b,c)\geq s_{X}/2$. We have

$2^{-1}\leq\frac{r}{2r}\leq\frac{d(f(a),f(b))}{d(f(b),f(c))}\leq\eta\left(\frac{d(a,b)}{d(b,c)}\right).$

Thus,

(4.11)
$d(E_{X},F_{X})=d(a,b)\geq\eta^{-1}(2^{-1})d(b,c)\geq\eta^{-1}(2^{-1})2^{-1}s_{X}.$

Next, our strategy in proving (4.10) is to show three estimates. We will show
that.

  1. A)

There is a collection $\Gamma_{B}$ so that for any $\kappa$-approximation
$\mathcal{U}$ at a small enough level the quantity ${\rm
Mod}_{p,\mathcal{U}}(\Gamma_{X}\setminus\Gamma_{B})$ can be bounded from below
by using the CLP property, and each curve in the image of
$\Gamma_{X}\setminus\Gamma_{B}$ is contained in a ball. It is this
requirement, that determines $\Gamma_{B}$.

  2. B)

Proposition 3.10 gives a lower bound for the Hausdorff measure in terms of the
discrete modulus $\overline{{\rm Mod}}_{p,\tau^{\prime}}(\Gamma_{Y}\setminus
f(\Gamma_{B}),{\mathcal{V}})$ of a subcollection of $\Gamma_{Y}$.

  3. C)

We use quasi-invariance Lemma 3.11 to bound $\overline{{\rm
Mod}}_{p,\tau^{\prime}}(\Gamma_{Y}\setminus f(\Gamma_{B}),{\mathcal{V}})$ from
below by $\overline{{\rm
Mod}}_{p,\tau}(\Gamma_{X}\setminus\Gamma_{B},f^{-1}({\mathcal{V}}))$.

  4. D)

Finally, Proposition 4.4 is used to find a collection $\mathcal{U}$ for which
we can bound $\overline{{\rm
Mod}}_{p,\tau}(\Gamma_{X}\setminus\Gamma_{B},f^{-1}({\mathcal{V}}))$ from
below by ${\rm Mod}_{p,\mathcal{U}}(\Gamma_{X}\setminus\Gamma_{B})$ for some
$\kappa$-approximation $\mathcal{U}$ at a small enough level. These estimates
together yield the desired bound.

We focus on A) first and determine $\Gamma_{B}$. Let $\mathcal{U}$ be a
$\kappa$-approximation at level $2^{-k}$ for some $k\in{\mathbb{N}}$ s.t.
$2^{-k}\leq\min\\{\operatorname{diam}(E_{X}),\operatorname{diam}(F_{X})\\}$.
We have

${\rm Mod}_{p,\mathcal{U}}(\Gamma_{X})\geq\phi(\tilde{\eta}(32)^{-1}).$

Let $L\geq 2$ be such that $\psi(2L^{-1})\leq
2^{-1}\phi(\tilde{\eta}(32)^{-1})$. Let $\Gamma_{B}$ be the collection of
curves $\gamma\in\Gamma_{X}$ with a subcurve in
$\Gamma(\overline{B_{X}(e_{X},s_{X})},X\setminus B_{X}(e_{X},Ls_{X}))$. We
have, since $X$ is CLP, that

${\rm Mod}_{p,\mathcal{U}}(\Gamma_{B})\leq{\rm
Mod}_{p,\mathcal{U}}(\Gamma(\overline{B_{X}(e_{X},s_{X})},X\setminus
B_{X}(e_{X},Ls_{X})))\leq\psi(2L^{-1})\leq\frac{\phi(\tilde{\eta}(32)^{-1})}{2}.$

Thus, by subadditivity of modulus, we get for
$\Gamma_{X,G}:=\Gamma_{X}\setminus\Gamma_{B}$ the estimate

(4.12) ${\rm
Mod}_{p,\mathcal{U}}(\Gamma_{X,G})\geq\frac{\phi(\tilde{\eta}(32)^{-1})}{2}.$

Now, let $\Gamma_{Y,G}=f(\Gamma_{X}\setminus\Gamma_{B})\subset\Gamma_{Y}$ be a
collection of good curves in $\Gamma_{Y}$. Each curve $\gamma\in\Gamma_{Y,G}$
connects $E_{Y}$ to $F_{Y}$ and so $\operatorname{diam}(\gamma)\geq r$. Next,
we will find a constant $L^{\prime}\geq 1$ so that

(4.13) $\gamma\subset B(e_{Y},L^{\prime}r)$

for every $\gamma\in\Gamma_{Y,G}$. Indeed, it will suffice to show that

(4.14) $f(B_{X}(e_{X},Ls_{X}))\subset B_{Y}(e_{Y},L^{\prime}r),$

since every curve in $\Gamma_{X,G}$ is contained in $B_{X}(e_{X},Ls_{X})$. To
do this, consider $z\in B_{X}(e_{X},Ls_{X})$. Let $w\in E_{X}$ be such that
$d(f(w),e_{Y})\geq r/2$. If $a\in E_{X}$ is such that $d(a,w)\geq
2^{-1}s_{X}$, we get

$2^{-1}=\frac{r}{2r}\leq\frac{d(f(w),e_{Y})}{d(f(w),f(a))}\leq\eta\left(\frac{d(w,e_{X})}{d(a,w)}\right).$

Thus,

$2^{-1}s_{X}\leq d(a,w)\leq\tilde{\eta}(2)d(w,e_{X}).$

Then, by using the quasisymmetry condition, we have the following.

$\frac{d(e_{Y},f(z))}{d(e_{Y},f(w))}\leq\eta\left(\frac{d(e_{X},z)}{d(e_{X},w)}\right)\leq\eta\left(\frac{2L\tilde{\eta}(2)s_{X}}{s_{X}}\right).$

Thus,

$d(e_{Y},f(z))\leq
d(e_{Y},f(w))\eta\left(2L\tilde{\eta}(2)\right)\leq\eta\left(2L\tilde{\eta}(2)\right)r.$

Thus, by choosing $L^{\prime}=\eta\left(2L\tilde{\eta}(2)\right)$ we obtain
(4.13).

Next, we deduce B) in our strategy. Let $\tau\geq 4$ and let
$\tau^{\prime}=\max\\{4,\eta(2\tau)\\}$. Choose $\delta>0$ so that

$\delta<\frac{r}{4\eta\left(\frac{32L^{\prime}\tau}{\eta^{-1}(2^{-1})}\right)}.$

This choice ensures an estimate that we need later:

(4.15)
$8\frac{L^{\prime}s_{X}}{\eta^{-1}(\frac{r}{4\delta})}\leq\frac{\eta^{-1}(2^{-1})s_{X}}{4\tau}$

Given the lower bound on the diameter of the curves in $\Gamma_{Y,G}$, which
are all contained in $B_{Y}(e_{Y},L^{\prime}r)$, we can apply Proposition 3.10
to find for any $\delta,\epsilon>0$ a $1$-round collection ${\mathcal{V}}$ of
balls which intersect $B_{Y}(e_{Y},L^{\prime}r)$ and some curve in
$\Gamma_{Y,G}$ with $\mathrm{rad}({\mathcal{V}})\leq\delta$ and with

(4.16) $\overline{{\rm
Mod}}_{p,\tau^{\prime}}(\Gamma_{Y,G},{\mathcal{V}})\leq(20\tau^{\prime})^{p}\left(\mathcal{H}^{p}_{\delta}(B_{Y}(e_{Y},L^{\prime}r))+\epsilon\right)r^{-p}\leq(20\tau^{\prime})^{p}\left(\mathcal{H}^{p}(B_{Y}(e_{Y},L^{\prime}r))+\epsilon\right)r^{-p}.$

Then, by Lemma 3.11, we have C):

(4.17) $\overline{{\rm
Mod}}_{p,\tau}(\Gamma_{X,G},f^{-1}({\mathcal{V}}))\leq\overline{{\rm
Mod}}_{p,\tau^{\prime}}(\Gamma_{Y,G},{\mathcal{V}}).$

Finally, we deduce D). Each curve in $\Gamma_{X,G}$ connects $E_{X}$ to
$F_{X}$, and thus by (4.11) we get for all $\gamma\in\Gamma_{X,G}$

(4.18) $\operatorname{diam}(\gamma)\geq\eta^{-1}(2^{-1})2^{-1}s_{X}.$

Moreover, if $V\in{\mathcal{V}}$ and $f^{-1}(V)\in f^{-1}({\mathcal{V}})$,
then we can bound $r_{f^{-1}(V)}$ as follows. Let $a\in f^{-1}(V)\cap
B_{X}(e_{X},Ls_{X})$, which exists, since $V$ intersects a curve in
$\Gamma_{G,Y}$. Choose $b\in f^{-1}(V)$, with $d(a,b)\geq r_{f^{-1}(V)}/4$.
Then, let $c\in E_{X}$ be such that $d(f(c),f(a))\geq r/2$. We then get from
the $\eta$-quasisymmetry of $f$, and since $f(a),f(b)\in V$ that

$\frac{r}{4\delta}\leq\frac{d(f(a),f(c))}{d(f(a),f(b))}\leq\eta\left(\frac{d(a,c)}{d(a,b)}\right)\leq\eta\left(\frac{2L^{\prime}s_{X}}{d(a,b)}\right).$

Thus,

$r_{f^{-1}(V)}\leq 4d(a,b)\leq
8\frac{L^{\prime}s_{X}}{\eta^{-1}(\frac{r}{4\delta})}\stackrel{{\scriptstyle\eqref{eq:deltachoice}}}{{\leq}}\frac{\eta^{-1}(2^{-1})s_{X}}{4\tau}.$

This means that
$2\tau\sup_{V\in{\mathcal{V}}}r_{f^{-1}(V)}\leq\inf_{\gamma\in\Gamma_{X,G}}\operatorname{diam}(\gamma)$.
By Lemma 2.9 $f^{-1}({\mathcal{V}})$ is $\kappa^{\prime}$-round. Thus, by
Proposition 4.4, there exists a constant $C$ so that

(4.19) ${\rm Mod}_{p,\mathcal{U}}(\Gamma_{X,G})\leq C\overline{{\rm
Mod}}_{p,\tau}(\Gamma_{X,G},f^{-1}({\mathcal{V}})),$

if $k$ is so large so that $\inf\\{r_{f^{-1}(V)}:f^{-1}(V)\in
f^{-1}({\mathcal{V}})\\}\geq 2^{-k-1}$.

By combining Estimates A-D), we get the following once $k$ is large enough

$\displaystyle\frac{\phi(\tilde{\eta}(32)^{-1})}{2}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:A)}}}{{\leq}}{\rm
Mod}_{p,\mathcal{U}}(\Gamma_{X,G})$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:D)}}}{{\leq}}C\overline{{\rm
Mod}}_{p,\tau}(\Gamma_{X,G},f^{-1}({\mathcal{V}}))$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:C)}}}{{\leq}}C\overline{{\rm
Mod}}_{p,\tau^{\prime}}(\Gamma_{Y,G},{\mathcal{V}})$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:B)}}}{{\leq}}(20\tau^{\prime})^{p}C(\mathcal{H}^{p}(B(e_{Y},L^{\prime}r))+\epsilon)r^{-p}.$

Consequently, since this holds for all $\epsilon>0$, we get

$\frac{\phi(\tilde{\eta}(32)^{-1})}{2C(20\tau^{\prime})^{p}}r^{p}\leq\mathcal{H}^{p}(B(e_{Y},L^{\prime}r)).$

This yields the desired estimate (4.10).

∎

###  4.3. Proof of Theorem 1.8

Using the previous properties, we are able to prove the equality of different
forms of conformal dimension for CLP spaces.

###### Proof of Theorem 1.8.

Assume that $X$ is combinatorially $p$-Loewner. By Lemma 4.2, we have

${\rm dim}_{CA}(X)={\rm dim}_{CAR}(X)=p.$

We also have ${\rm dim}_{CH}(X)\leq{\rm dim}_{CAR}(X)=p$. Thus, we only need
to show that ${\rm dim}_{CH}(X)\geq p$. Let $f:X\to Y$ be a quasisymmetry.
Then, by Proposition 4.9 there exists a constant $C$ so that we have for every
$y\in Y$ and any $r\in(0,\operatorname{diam}(Y))$ that

$\mathcal{H}^{p}(B(y,r))\geq Cr^{p}>0.$

From the definition of Hausdorff dimension, and since
$\mathcal{H}^{p}(B(y,r))>0$ if and only if
$\mathcal{H}^{p}_{\infty}(B(y,r))>0$, we have ${\rm dim}_{H}(Y)\geq p$.
Consequently, by taking an infimum over all $Y$ which are quasisymmetric to
$X$, we get ${\rm dim}_{CH}(X)\geq p$. ∎

##  5\. Quasiself-similar Spaces

###  5.1. Uniform bound for Annuli

Define an annulus as $A(x,r,R):=B(x,R)\setminus\overline{B(x,r)}$.

######  Definition 5.1.

Let $p\in(1,\infty)$. Let $\tau\geq 4$. We say that a metric space $X$ has
uniformly small $p$-moduli of annuli, if there exists $\epsilon\in(0,1)$ and
constants $0<\delta_{-}<\delta_{+}<\tau^{-1}$, so that the following holds.
For every annulus $A(x,r,(\tau-2)r)$ in $X$, with $x\in
X,r\in(0,2^{-1}\tau^{-1}\operatorname{diam}(X))$, there exists a finite
collection of balls ${\mathcal{V}}_{x,r}$ contained in $B(x,\tau r)$ and which
intersect $B(x,(\tau-2)r)$, with $r_{V}\in[\delta_{-}r,\delta_{+}r]$ for each
$V\in{\mathcal{V}}_{x,r}$, and there exists a function
$\rho_{x,r}:{\mathcal{V}}_{x,r}\to[0,\infty)$ with

$\rho_{x,r}\overline{\wedge}_{\tau,{\mathcal{V}}_{x,r}}\Gamma(\overline{B(x,r)},X\setminus
B(x,(\tau-2)r))$

and with

$\sum_{B\in{\mathcal{V}}_{B}}\rho_{x,r}(B)^{p}\leq\epsilon.$

The following Lemma is a refinement of Proposition 3.10 to the quasiself-
similar setting.

######  Lemma 5.2.

Suppose that ${\rm dim}_{CH}(X)<p$, that $p\in(1,\infty)$, and that $X$ is an
arcwise connected quasiself-similar compact metric space. Then $X$ has
uniformly small $p$-moduli of annuli.

###### Proof.

Assume that $X$ is $\eta$-quasiself-similar and let $\tau\geq 4$. Fix any
$\delta_{+}\in(0,\tau^{-1})$. Since ${\rm dim}_{CH}(X)<p$, there exists a
compact space $Y$ with ${\rm dim}_{H}(Y)<p$ and a quasisymmetry $g:X\to Y$.
Fix $C\geq 1,\sigma\in(0,2^{-1})$ to be determined. By adjusting $\eta$, we
may assume that $g$ is an $\eta$-quasisymmetry. Let $\epsilon>0$, and choose a
covering of $Y$ by a collection of balls ${\mathcal{B}}_{Y}$ with

$\sum_{B\in{\mathcal{B}}_{Y}}\operatorname{diam}(B)^{p}\leq\epsilon
C^{-p}\operatorname{diam}(Y)^{p},$

and for which $\mathrm{rad}(B)\leq\sigma\operatorname{diam}(Y)$ for every
$B\in{\mathcal{B}}_{Y}$. Let $A(x,r,(\tau-2)r)$ be an annulus in $X$ with
$x\in X$ and $r\in(0,2^{-1}\tau^{-1}\operatorname{diam}(X))$. There is a
homeomorphism $f:B(x,2\tau r)\to U$, for some open set $U\subset X$, which is
an $\eta$-quasisymmetry, where
$\operatorname{diam}(U)\geq\delta\operatorname{diam}(X)$.

We first define the collection ${\mathcal{V}}_{x,r}$ used in Definition 5.1.
For each $B=B(y,s)\in{\mathcal{B}}_{Y}$ with

$B\cap g(f(B(x,(\tau-2)r)))\neq\emptyset,$

choose $x_{V_{B}}\in(g\circ f)^{-1}(B)\cap B(x,(\tau-2)r)$, and let
$r_{V_{B}}=\sup\\{d(x,x_{B}):x\in(g\circ f)^{-1}(2B)\cap B(x,\tau r)\\}$.
Define $V_{B}:=B(x_{V_{B}},r_{V_{B}})$. Let

${\mathcal{V}}_{x,r}:=\\{V_{B}:B\in{\mathcal{B}}_{Y},B\cap
g(f(B(x,(\tau-2)r)))\neq\emptyset\\}$

be the collection of balls we seek. Next, we give bounds for $r_{V_{B}}$ by
using the fact that $X$ is connected and that $g\circ f$ is a
$\tilde{\eta}$-quasisymmetry with $\tilde{\eta}=\eta\circ\eta$.

Since $\operatorname{diam}(U)\geq\delta\operatorname{diam}(X)$, we can choose
$a,b\in U$ with $d(a,b)\geq 2^{-1}\delta\operatorname{diam}(X)$. Choose a
point $c\in X$ so that $d(g(c),g(a))\geq\operatorname{diam}(Y)2^{-1}$. Since
$g$ is an $\eta$-quasisymmetry, we have

$\frac{d(g(a),g(c))}{d(g(a),g(b))}\leq\eta\left(\frac{d(c,a)}{d(b,a)}\right)\leq\eta(2\delta^{-1}).$

Thus,

(5.3) $d(g(a),g(b))\geq\eta(2\delta^{-1})^{-1}2^{-1}\operatorname{diam}(Y).$

We will use (5.3) to give an upper bound for $r_{V_{B}}$ for each
$V_{B}\in{\mathcal{V}}_{x,r}$, where $B\in{\mathcal{B}}_{Y}$. Let $u,v\in
B(x,2\tau r)$ be such that $f(u)=a,f(v)=b$. Choose $s,t\in(g\circ
f)^{-1}(2B)\cap B(x,\tau r)$ so that $d(s,t)\geq r_{V_{B}}/2$. Up to possibly
switching $u$ and $v$, and $a,b$, we can assume by (5.3) that

(5.4)
$d(g(f(s)),g(a))\geq\frac{d(g(a),g(b))}{2}\geq\eta(2\delta^{-1})^{-1}\operatorname{diam}(Y)2^{-2}.$

We have

(5.5)
$\frac{d(g(f(s)),g(f(u)))}{d(g(f(s)),g(f(t)))}\leq\tilde{\eta}\left(\frac{d(s,u)}{d(s,t)}\right).$

Since $g(f(s)),g(f(t))\in 2B$, we get $d(g(f(s)),g(f(t)))\leq
4\mathrm{rad}(B)\leq 4\sigma\operatorname{diam}(Y)$. Thus, from (5.4), we get

$\frac{1}{2^{4}\eta(2\delta^{-1})\sigma}\leq\frac{\operatorname{diam}(Y)}{2^{4}\mathrm{rad}(B)\eta(2\delta^{-1})}\leq\frac{d(g(f(s)),g(a))}{d(g(f(s)),g(f(t)))}.$

By combining this with (5.5), we deduce

$\tilde{\eta}^{-1}\left(\frac{1}{2^{4}\sigma\eta(2\delta^{-1})}\right)\leq\frac{d(s,u)}{d(s,t)}\leq\frac{4\tau
r}{r_{V_{B}}}.$

Thus,

$r_{V_{B}}\leq\frac{4\tau}{\tilde{\eta}^{-1}\left(\frac{1}{2^{4}\sigma\eta(2\delta^{-1})}\right)}r.$

Choose now
$\sigma\leq\tilde{\eta}(\frac{4\tau}{\delta_{+}})^{-1}\eta(2\delta^{-1})^{-1}2^{-4}$.
We then have, $r_{V_{B}}\leq\delta_{+}r$. Since $\delta_{+}<1$, we also have
$r_{V_{B}}\leq r$ and since $x_{V_{B}}\in B(x,(\tau-2)r)$ we clearly have
$V_{B}\subset B(x,\tau r)$.

Next, we give a uniform lower bound for the radii $r_{V_{B}}$ for
$V_{B}\in{\mathcal{V}}_{x,r}$ . Since ${\mathcal{B}}_{Y}$ is finite, there
exists a constant $\beta>0$ so that
$\mathrm{rad}(B)\geq\beta\operatorname{diam}(Y)$ for all
$B\in{\mathcal{B}}_{Y}$. Choose $\delta_{-}=\tilde{\eta}^{-1}(\beta)/2$. Let
$c\in B(x_{V_{B}},\delta_{-}r)$ be an arbitrary point. Also, choose $b\in
B(x,2\tau r)$ with $d(b,x_{V_{B}})\geq r$, which is possible by connectivity.
Then, by the quasisymmetry condition, we get

$\frac{d(g(f(c)),g(f(x_{V_{B}})))}{d(g(f(b)),g(f(x_{V_{B}})))}\leq\tilde{\eta}\left(\frac{d(c,x_{V_{B}})}{d(b,x_{V_{B}})}\right)\leq\tilde{\eta}\left(\delta_{-}\right).$

The choice of $\delta_{-}$ guarantees $\tilde{\eta}(\delta_{-})\leq\beta$, and
thus

$d(g(f(c)),g(f(x_{V_{B}})))\leq\tilde{\eta}(\delta_{-})\operatorname{diam}(Y)\leq
r_{B}.$

Therefore, since $g(f(x_{V_{B}}))\in B$, we get $g(f(c))\in 2B$. This holds
for all $c\in B(x_{V_{B}},\delta_{-}r)$, and thus

$g(f(B(x_{V_{B}},\delta_{-}r)))\subset 2B.$

This yields, by connectivity and the definition of $r_{V_{B}}$ that
$r_{V_{B}}\geq\delta_{-}r$.

Finally, we define the admissible function $\rho$. Define
$\rho(V)=\max\\{C\operatorname{diam}_{Y}(B)\operatorname{diam}_{Y}(Y)^{-1}:V_{B}=V,B\in{\mathcal{B}}_{Y}\\}$
for $V\in{\mathcal{V}}_{x,r}$. We have

(5.6) $\sum_{V\in{\mathcal{V}}_{B}}\rho(V)^{p}\leq
C^{p}\sum_{B\in{\mathcal{B}}_{Y}}\operatorname{diam}_{Y}(B)^{p}\operatorname{diam}_{Y}(Y)^{-p}\leq\epsilon,$

since for every $V\in{\mathcal{V}}_{x,r}$ there exists at least one
$B\in{\mathcal{B}}_{Y}$ so that $V_{B}=V$, and for every
$B\in{\mathcal{B}}_{Y}$ there is only one $V\in{\mathcal{V}}_{x,r}$ for which
$V_{B}=V$.

Next, we show that
$\rho\overline{\wedge}_{\tau,{\mathcal{V}}_{x,r}}\Gamma(\overline{B(x,r)},X\setminus
B(x,(\tau-2)r))$. Let $\gamma\in\Gamma(\overline{B(x,r)},X\setminus
B(x,(\tau-2)r))$ be arbitrary. Let $\sigma$ be a sub-curve of $\gamma$ so that
$\sigma\subset\overline{(\tau-2)B}$ and
$\sigma\in\Gamma(\overline{B(x,r)},X\setminus B(x,(\tau-2)r))$. To show
admissibility, we will combine the fact that ${\mathcal{B}}_{Y}$ covers
$g\circ f\circ\sigma$ with a lower bound for the diameter of $g\circ
f\circ\sigma$.

Since $\sigma$ connects $B(x,r)$ to $X\setminus B(x,(\tau-2)r)$ there exist
$j,k\in\sigma$ with $d(j,k)\geq(\tau-3)r$. Let $a,b$ and $u,v$ be as before.
By possibly switching $j$ and $k$, we can assume that $d(j,u)\geq d(j,k)/2\geq
2^{-1}(\tau-3)r$. We get

$\frac{d(g(f(j)),g(f(u)))}{d(g(f(j)),g(f(k))}\leq\tilde{\eta}\left(\frac{d(j,u)}{d(j,k)}\right)\leq\tilde{\eta}\left(\frac{4\tau
r}{(\tau-3)r}\right)\leq\tilde{\eta}(16).$

Thus,

(5.7) $\operatorname{diam}(g\circ f\circ\sigma)\geq
d\left((g(f(j)),g(f(k))\right)\geq
d\left(g(f(j)),g(f(u))\right)\tilde{\eta}(16)^{-1}.$

Next, $d(j,u)\geq 2^{-1}(\tau-3)r\geq d(u,v)/8$. Thus, by a similar reasoning
that uses the quasisymmetry of $g\circ f$ and by employing (5.3), we get

(5.8) $d(g(f(j)),g(f(u)))\geq
d(g(f(u),g(f(v)))\tilde{\eta}(8)^{-1}\geq\tilde{\eta}(48)^{-1}\eta(2\delta^{-1})^{-1}2^{-1}\operatorname{diam}(Y).$

By combining (5.7) and (5.8), we obtain

(5.9) $\operatorname{diam}(g\circ f\circ\sigma)\geq
d(g(f(j)),g(f(k))\geq\tilde{\eta}(16)^{-1}\tilde{\eta}(8)^{-1}\eta(2\delta^{-1})^{-1}2^{-1}\operatorname{diam}(Y).$

Recall that ${\mathcal{V}}_{x,r}$ consists of balls. The open sets
${\mathcal{V}}_{x,r}$ cover the ball $B(x,2(\tau-2))$, and thus the curve
$\sigma$. Therefore, by the Vitali covering theorem, there exists a finite
collection of balls ${\mathcal{V}}_{\gamma}$ with $\sigma\subset\bigcup
5\tau{\mathcal{V}}_{\gamma}$, and for which $\tau{\mathcal{V}}_{\gamma}$ are
disjoint, and so that each ball in ${\mathcal{V}}_{\gamma}$ intersects
$\gamma$. For each $V\in{\mathcal{V}}_{\gamma}$, choose a ball
$B(V)\in{\mathcal{B}}_{Y}$ so that $V=V_{B(V)}$ and
$\rho(V)=C\operatorname{diam}_{Y}(B(V))\operatorname{diam}_{Y}(Y)^{-1}$.

First, we note that the quasisymmetry condition and Lemma 2.3, we have
$g(f(5\tau V))\subset\tilde{\eta}(10\tau)B(V)$. Therefore, we get that the
balls $\tilde{\eta}(10\tau)B(V)$ for $V\in{\mathcal{V}}_{\gamma}$ cover
$g(f(\sigma))$. Thus,

$\displaystyle\sum_{V\in{\mathcal{V}}_{\gamma}}\rho(V)$
$\displaystyle=\sum_{V\in{\mathcal{V}}_{\gamma}}C\operatorname{diam}_{Y}(B(V))\operatorname{diam}_{Y}(Y)^{-1}$
$\displaystyle\geq\sum_{V\in{\mathcal{V}}_{\gamma}}C(2\tilde{\eta}(10\tau))^{-1}\operatorname{diam}_{Y}(Y)^{-1}\operatorname{diam}_{Y}(\tilde{\eta}(10\tau)B(V))$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:diamsigma-3}}}{{\geq}}\operatorname{diam}(g\circ
f\circ\sigma)C(2\tilde{\eta}(10\tau))^{-1}\operatorname{diam}_{Y}(Y)^{-1}$
$\displaystyle\geq
C2^{-2}\tilde{\eta}(16)^{-1}\tilde{\eta}(8)^{-1}\eta(2\delta^{-1})^{-1}\tilde{\eta}(10\tau)^{-1}.$

If $C\geq
4\tilde{\eta}(16)\tilde{\eta}(8)\eta(2\delta^{-1})\tilde{\eta}(10\tau)$, then
$\rho\overline{\wedge}_{\tau,{\mathcal{V}}_{x,r}}\Gamma(\overline{B(x,r)},X\setminus
B(x,(\tau-2)r))$ is admissible and the claim follows.

∎

###  5.2. Algorithm for pushing down a cover

The following lemma describes a “push down” algorithm. It uses admissible
functions for annuli in order to push down a collection of balls
${\mathcal{B}}$ and a strongly discretely $\tau$-admissible function $\rho$.
This is done by replacing a ball $\mathbf{B}\in{\mathcal{B}}$ by a collection
${\mathcal{B}}_{\mathbf{B}}$ and an associated function $\rho_{\mathbf{B}}$. A
new admissible function $\overline{\rho}$ is defined by taking a maximum over
$\mathbf{B}\in{\mathcal{B}}$, and a new collection by taking a union of all
the new balls. This arguments for admissibility and the construction of
$\overline{\rho}$ are similar to Proposition 4.4. To distinguish the “parent”
balls from the “descendant balls”, we will bold the parent balls. This
replacement algorithm is depicted and explained more in Figure 1. As seen in
this figure, we permit all sorts of overlaps, and balls of different sizes.
This is one of the technical reasons for using the new modulus from Subsection
3.4.

Recall that $\Gamma_{L,B}$ denotes the collection of curves $\gamma$
connecting $B$ to $X\setminus LB$.

######  Lemma 5.10.

Let $\epsilon,\eta\in(0,1)$. Assume that ${\mathcal{B}}$ is a finite
collection of balls, $\Gamma$ is a collection of curves,
$2(\tau-2)\mathrm{rad}({\mathcal{B}})\leq\inf_{\gamma\in\Gamma}\operatorname{diam}(\gamma)$
and $\rho\wedge_{{\mathcal{B}}}\Gamma$. Suppose further that
${\mathcal{C}}\subset{\mathcal{B}}$ is any finite collection of balls, and for
every $\mathbf{B}\in{\mathcal{C}}$, there exists a finite collection of balls
${\mathcal{B}}_{\mathbf{B}}$ and a function
$\rho_{\mathbf{B}}:{\mathcal{B}}_{\mathbf{B}}\to[0,\infty)$ with

  1. (1)

$\mathrm{rad}({\mathcal{B}}_{\mathbf{B}})\leq\tau^{-1}\mathrm{rad}(\mathbf{B})$,

  2. (2)

$\rho_{\mathbf{B}}\overline{\wedge}_{\tau,{\mathcal{B}}_{\mathbf{B}}}\Gamma_{\mathbf{B},(\tau-2)}$,

  3. (3)

every ball in ${\mathcal{B}}_{\mathbf{B}}$ intersects $(\tau-2)\textbf{B}$,
and satisfies

$\sum_{B\in{\mathcal{B}}_{\mathbf{B}}}\rho_{\mathbf{B}}(B)^{p}\leq\eta.$

For $\mathbf{B}\not\in{\mathcal{C}}$ assume that
${\mathcal{B}}_{\mathbf{B}}=\\{\mathbf{B}\\}$ and
$\rho_{\mathbf{B}}(\mathbf{B})=1$.

For the collection
$\overline{{\mathcal{B}}}:=\bigcup_{\mathbf{B}\in{\mathcal{B}}}{\mathcal{B}}_{\mathbf{B}}$,
and function

$\overline{\rho}(B):=\max\\{\rho(\mathbf{B})\rho_{\mathbf{B}}(B):\textbf{B}\in{\mathcal{B}}\
\text{s.t.}\ B\in{\mathcal{B}}_{\mathbf{B}}\\},$

we have
$\overline{\rho}\overline{\wedge}_{\tau,\overline{{\mathcal{B}}}}\Gamma$ and

$\sum_{B\in\overline{{\mathcal{B}}}}\overline{\rho}(B)^{p}\leq\sum_{\mathbf{B}\in{\mathcal{C}}}\eta\rho(\mathbf{B})^{p}+\sum_{\mathbf{B}\in{\mathcal{B}}\setminus{\mathcal{C}}}\rho(\mathbf{B})^{p}.$

Figure 1. The push-down algorithm: As input we are given a collection of balls
${\mathcal{B}}$, and a function $\rho$ which is strongly discretely admissible
for some collection of curves $\Gamma$. For each ball $B$ in some sub-
collection ${\mathcal{C}}\subset{\mathcal{B}}$ we have a “replacement
collection” ${\mathcal{B}}_{\textbf{B}}$, and a replacement function
$\rho_{\textbf{B}}$. We replace each ball in ${\mathcal{C}}$ by the associated
collection ${\mathcal{B}}_{B}$, and keep the remaining balls fixed. A new
$\overline{\rho}$ is defined using $\rho_{\textbf{B}}$ for this new
collection, and is shown to be admissible under appropriate assumptions. In
the figure, the non-filled balls are the collection ${\mathcal{B}}$, and
${\mathcal{C}}$ consists of the empty balls with dashed boundaries. The figure
shows how the dashed lined empty balls are replaced by the filled balls, where
each filled ball comes from a collection ${\mathcal{B}}_{\textbf{B}}$ for some
$\textbf{B}\in{\mathcal{C}}$. We have not indicated in detail which ball
corresponds to which, as this is irrelevant for the proof. Note however that
condition (3) in Lemma 5.10. There is a darker shaded ball, which is in two of
the collections ${\mathcal{B}}_{\textbf{B}}$ – an eventuality that we permit,
but that could be avoided by allowing collections of balls to be multi-sets.
Also note, that the balls can intersect each other in unlimited ways, and
there is also one solid lined ball in ${\mathcal{B}}\setminus{\mathcal{C}}$
which is not replaced.

###### Proof of Lemma 5.10.

We first show that
$\overline{\rho}\overline{\wedge}_{\tau,\overline{{\mathcal{B}}}}\Gamma$. Let
$\gamma\in\Gamma$. Since $\rho\overline{\wedge}_{\tau,{\mathcal{B}}}\Gamma$,
there exists a collection ${\mathcal{B}}_{\gamma}\subset\Gamma$ so that
$\tau{\mathcal{B}}_{\gamma}$ is pairwise disjoint, so that
$\textbf{B}\cap\gamma\neq\emptyset$ for every
$\textbf{B}\in{\mathcal{B}}_{\gamma}$ and

(5.11) $\sum_{\textbf{B}\in{\mathcal{B}}_{\gamma}}\rho(\textbf{B})\geq 1.$

We next define
$\overline{{\mathcal{B}}}_{\gamma}\subset\overline{{\mathcal{B}}}$. First, set
${\mathcal{B}}_{\gamma}^{1}={\mathcal{B}}_{\gamma}\setminus{\mathcal{C}}$.
Next, for each $\textbf{B}\in{\mathcal{B}}_{\gamma}\cap{\mathcal{C}}$ we have
$\rho_{\textbf{B}}\overline{\wedge}_{\tau,{\mathcal{B}}_{\textbf{B}}}\Gamma_{\textbf{B},(\tau-2)}$.
Since $\gamma\cap\textbf{B}\neq\emptyset$, and
$\operatorname{diam}(\gamma)\geq 2(\tau-2)\mathrm{rad}(\textbf{B})$, we have
that $\gamma\cap(X\setminus(\tau-2)\textbf{B})\neq\emptyset$. Thus, $\gamma$
contains a sub-arc in $\Gamma_{\textbf{B},(\tau-2)}$. Therefore, there exists
a collection
${\mathcal{B}}_{\gamma,\textbf{B}}\subset{\mathcal{B}}_{\textbf{B}}$ so that
$\tau{\mathcal{B}}_{\gamma,\textbf{B}}$ is pairwise disjoint, so that
$B\cap\gamma\neq\emptyset$ for every $B\in{\mathcal{B}}_{\gamma,\textbf{B}}$
and

$\sum_{B\in{\mathcal{B}}_{\gamma,\textbf{B}}}\rho_{\textbf{B}}(B)\geq 1.$

Since $\overline{\rho}(B)\geq\rho_{\textbf{B}}(B)\rho(\textbf{B})$ for every
$B\in{\mathcal{B}}_{\gamma,\textbf{B}}$, we have

(5.12)
$\sum_{B\in{\mathcal{B}}_{\gamma,\textbf{B}}}\overline{\rho}(B)\geq\rho(\textbf{B}).$

Set
${\mathcal{B}}_{\gamma}^{2}=\bigcup_{\textbf{B}\in{\mathcal{B}}_{\gamma}\cap{\mathcal{C}}}{\mathcal{B}}_{\gamma,\textbf{B}}$.
Finally, let
$\overline{{\mathcal{B}}}_{\gamma}={\mathcal{B}}_{\gamma}^{1}\cup{\mathcal{B}}_{\gamma}^{2}$.
Note that for every $\textbf{B}\in{\mathcal{B}}_{\gamma}\cap{\mathcal{C}}$, we
have
$\mathrm{rad}(\tau{\mathcal{B}}_{\gamma,\textbf{B}})\leq\tau\mathrm{rad}({\mathcal{B}}_{\textbf{B}})\leq\mathrm{rad}(B)$
and $B\cap(\tau-2)\textbf{B}\neq\emptyset$ for every
$B\in{\mathcal{B}}_{\gamma,\textbf{B}}$. Thus, every
$B\in\tau{\mathcal{B}}_{\gamma,\textbf{B}}$ satisfies $\tau
B\subset\tau\textbf{B}$. This inclusion implies that the collections
${\mathcal{B}}_{\gamma,\textbf{B}}$ are pairwise disjoint for distinct
$\textbf{B}\in{\mathcal{B}}_{\gamma}$, and each of these is disjoint from
${\mathcal{B}}_{\gamma}^{2}$. Thus, the collection
$\tau\overline{{\mathcal{B}}}_{\gamma}$ is pairwise disjoint.

Next,

$\displaystyle\sum_{B\in\overline{{\mathcal{B}}}_{\gamma}}\overline{\rho}(B)$
$\displaystyle=\sum_{B\in{\mathcal{B}}_{\gamma}^{1}}\overline{\rho}(B)+\sum_{B\in{\mathcal{B}}_{\gamma}^{2}}\overline{\rho}(B)$
$\displaystyle\geq\sum_{B\in{\mathcal{B}}_{\gamma}^{1}}\rho(B)+\sum_{\textbf{B}\in{\mathcal{B}}_{\gamma}\cap{\mathcal{C}}}\sum_{B\in\overline{{\mathcal{B}}}_{\gamma,\textbf{B}}}\overline{\rho}(B)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:rhoadmann}}}{{\geq}}\sum_{\textbf{B}\in{\mathcal{B}}_{\gamma}\setminus{\mathcal{C}}}\rho(\textbf{B})+\sum_{\textbf{B}\in{\mathcal{B}}_{\gamma}\cap{\mathcal{C}}}\rho(\textbf{B})$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:rhoadm}}}{{=}}\sum_{\textbf{B}\in{\mathcal{B}}_{\gamma}}\rho(\textbf{B})\geq
1.$

Thus, since $\gamma$ was arbitrary, we have
$\overline{\rho}\overline{\wedge}_{\tau,\overline{{\mathcal{B}}}}\Gamma.$

Finally, we compute the $p$-energy of $\overline{\rho}$. First, by
construction, for every $\textbf{B}\in{\mathcal{C}}$, we have

(5.13)
$\sum_{B\in{\mathcal{B}}_{\textbf{B}}}\rho_{\textbf{B}}(B)^{p}\leq\eta,$

and for every $\textbf{B}\in{\mathcal{B}}\setminus{\mathcal{C}}$, we have

(5.14)
$\sum_{B\in{\mathcal{B}}_{\textbf{B}}}\rho_{\textbf{B}}(B)^{p}=\rho_{\textbf{B}}(\textbf{B})=1,$

For every $B\in\overline{{\mathcal{B}}}$, there may be multiple
$\textbf{B}\in{\mathcal{B}}$ so that $B\in{\mathcal{B}}_{\textbf{B}}$.
However, for every $B\in\overline{{\mathcal{B}}}$, we have:

(5.15)
$\overline{\rho}(B)=\max\\{\rho(\textbf{B})\rho_{\textbf{B}}(B):\textbf{B}\in{\mathcal{B}}\
\text{s.t.}\
B\in{\mathcal{B}}_{\textbf{B}}\\}\leq\left(\sum_{\begin{subarray}{c}\textbf{B}\in{\mathcal{B}}\\\
\text{s.t.}\
B\in{\mathcal{B}}_{\textbf{B}}\end{subarray}}(\rho(\textbf{B})\rho_{\textbf{B}}(B))^{p}\right)^{\frac{1}{p}}.$

By combining these two we get:

$\displaystyle\sum_{B\in\overline{{\mathcal{B}}}}\overline{\rho}(B)^{p}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:upperboundtildeB}}}{{\leq}}\sum_{\textbf{B}\in{\mathcal{B}}}\sum_{B\in{\mathcal{B}}_{\textbf{B}}}\rho(\textbf{B})^{p}\rho_{\textbf{B}}(B)^{p}$
$\displaystyle=\sum_{\textbf{B}\in{\mathcal{C}}}\sum_{B\in{\mathcal{B}}_{\textbf{B}}}\rho(\textbf{B})^{p}\rho_{\textbf{B}}(B)^{p}+\sum_{\textbf{B}\in{\mathcal{B}}\setminus{\mathcal{C}}}\sum_{B\in{\mathcal{B}}_{\textbf{B}}}\rho(\textbf{B})^{p}\rho_{\textbf{B}}(B)^{p}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:tilderhosum},\eqref{eq:tilderhosum2}}}{{\leq}}\sum_{\textbf{B}\in{\mathcal{C}}}\eta\rho(\textbf{B})^{p}+\sum_{\textbf{B}\in{\mathcal{B}}\setminus{\mathcal{C}}}\rho(\textbf{B})^{p}.$

This is the desired estimate and the proof is complete. ∎

If the space has uniformly small moduli, then we have readily available
collections ${\mathcal{B}}_{\textbf{B}}$ and functions $\rho_{\textbf{B}}$ as
in the statement of the previous lemma. Suppose now that $\Gamma$ is any
collection of curves and
$\rho\overline{\wedge}_{\tau,{\mathcal{B}}_{0}}\Gamma$ for some collection of
balls ${\mathcal{B}}_{0}$ with sufficiently small radii compared to the
diameter of the curves $\gamma\in\Gamma$. If the space has uniformly small
moduli for annuli, we can first use the push-down procedure on all of the
balls, in order to reduce the discrete modulus of $\Gamma$ below any threshold
we want. This process is called “weight reduction”. The balls in the resulting
collection can have arbitrarily large and small radii. However, any “large”
balls can be replaced by smaller ones with controlled size. This process can
be repeated until all the balls in a collection are roughly the same size.
This phase is called “size reduction”. As a consequence, the push-down
algorithm in conjunction with the uniformly small moduli of annuli leads to
the ability to convert the collection ${\mathcal{B}}$ into a collection of
balls $\overline{{\mathcal{B}}}$ which is roughly at the same scale, and so
that $\overline{{\rm Mod}}_{p,\tau}(\Gamma,\overline{{\mathcal{B}}})$ is as
small as we want. This process is called the “equalizing algorithm”, and is
depicted in Figure 2.

Figure 2. The equalizing algorithm: By using replacement and a uniform bound
on moduli of annuli, we can “uniformize” a wild cover ${\mathcal{B}}$. Let
${\mathcal{B}}$ be a covering using balls, where the size of the largest ball
is much bigger than the smallest. We take all the “large” balls, and form a
collection ${\mathcal{C}}$ of them. To them, we apply the push-down procedure
to reduce their size. We repeat this process until all large balls have been
pushed down to a size comparable to the smallest ball in our collection. In
the figure ${\mathcal{B}}$ consists of balls filled with white. The two large
balls have solid line boundaries, and are replaced by smaller light gray
filled balls. Two of these light gray balls are still too large, and are
replaced by even smaller dark gray filled balls.

The next lemma and its proof expresses formally the effect of the equalizing
algorithm.

######  Lemma 5.16.

Let $p\in(1,\infty),\tau\geq 4$. Let $X$ be an LLC compact metric space which
has uniformly small $p$-moduli of annuli, with constant $\delta_{-}>0$ as in
Definition 5.1. Then there exists a $\kappa\geq 1$ so that for every
$\epsilon_{0}>0$ and every $M\geq 1$ there exists a $N\geq 1$ with the
following property. Let ${\mathcal{B}}_{0}$ be a collection of balls for which
$\inf_{\gamma\in\Gamma}\operatorname{diam}(\gamma)\geq\tau\mathrm{rad}({\mathcal{B}}_{0})$,
and let $r>0$ be such that
$\delta_{-}^{N}\inf\\{\mathrm{rad}(B):B\in{\mathcal{B}}_{0}\\}\geq r$ and

$\overline{{\rm Mod}}_{p,\tau}(\Gamma,{\mathcal{B}}_{0})<M.$

Then, there exists a collection of balls ${\mathcal{V}}$ with

$\overline{{\rm Mod}}_{p,\tau}(\Gamma,{\mathcal{V}})\leq\epsilon.$

and $\kappa^{-1}r\leq r_{V}\leq r$ for all $V\in{\mathcal{V}}$.

We briefly remark, that is is crucial for our later arguments, that
$\epsilon>0$ is independent of $\kappa$. That is, there exists a $\kappa$,
which controls the ratio of the largest ball in ${\mathcal{V}}$ to the
smallest one. Given this $\kappa$, we can choose any $\epsilon$ to obtain a
modulus smaller than that. The cost of decreasing $\epsilon$ is to increase
$N$ and thus reduce the scale $r$ of ${\mathcal{V}}$.

###### Proof of Lemma 5.16.

Let $\tau\geq 4$, and let $0<\delta_{-}<\delta_{+}<\tau^{-1}$ and
$\epsilon\in(0,1)$ be the parameters from the definition of having uniformly
small $p$-moduli of annuli, see Definition 5.1. Let $\kappa=2\delta_{-}^{-1}$,
$N=\lceil\log_{2}(\max\\{M/\epsilon_{0},1\\})/\log_{2}(\epsilon^{-1})\rceil+1$
and $r\leq\delta_{-}^{N}\inf\\{\mathrm{rad}(B):B\in{\mathcal{B}}_{0}\\}$.

Further, for any ball $\textbf{B}=B(x,s)$ in $X$, let
${\mathcal{B}}_{\textbf{B}}$ and $\rho_{B}$ denote the functions
${\mathcal{V}}_{x,s}$ and $\rho_{x,s}$ given in Definition 5.1 with

(5.17)
$\sum_{B\in{\mathcal{B}}_{\textbf{B}}}\rho_{\textbf{B}}(B)^{p}\leq\epsilon,$

and
$\rho\overline{\wedge}_{\tau,{\mathcal{B}}_{\textbf{B}}}\Gamma_{\textbf{B},\tau-2}.$

Let ${\mathcal{B}}_{0}$ be a finite cover of $X$ by balls with
$\inf_{\gamma\in\Gamma}\operatorname{diam}(\gamma)\geq\tau\mathrm{rad}({\mathcal{B}}_{0})$
for which $\overline{{\rm Mod}}_{p,\tau}(\Gamma,{\mathcal{B}}_{0})<M$. Then,
there exists a $\rho_{0}:{\mathcal{B}}_{0}\to[0,\infty)$ with
$\rho\overline{\wedge}_{\tau,{\mathcal{B}}_{0}}\Gamma$ and with mass

$\sum_{B\in{\mathcal{B}}_{0}}\rho(B)^{p}<M.$

First, we replace ${\mathcal{B}}_{0}$ through a finite sequence of
replacements by a collection of balls with respect to which $\Gamma$ has small
modulus. This we call the “weight reduction phase”. We construct a sequence of
covers ${\mathcal{B}}_{k}$, for $k\in{\mathbb{N}}$, as follows. Proceed
inductively and apply Lemma 5.10 for each $k\in{\mathbb{N}}$ with
${\mathcal{B}}={\mathcal{B}}_{k}$, $\rho=\rho_{k}$ and
${\mathcal{C}}={\mathcal{B}}_{k}$, and with
$\rho_{\textbf{B}},{\mathcal{B}}_{\textbf{B}}$ and $\epsilon=\eta$ satisfying
(5.17), to obtain a collection ${\mathcal{B}}_{k+1}=\overline{{\mathcal{B}}}$
and function $\rho_{k+1}=\overline{\rho}$ with
$\rho_{k+1}\overline{\wedge}_{\tau,{\mathcal{B}}_{k+1}}\Gamma$ and

$\sum_{B\in{\mathcal{B}}_{k+1}}\rho(B)^{p}\leq\epsilon^{k}M.$

We note that for each ball $B\in{\mathcal{B}}_{\textbf{B}}$ we have
$\mathrm{rad}(B)\geq\delta_{-}\mathrm{rad}(\textbf{B})$, by the assumption of
uniformly small moduli of annuli. Therefore, for all $k\in{\mathbb{N}}$ we get

(5.18)
$\inf\\{\mathrm{rad}(B):B\in{\mathcal{B}}_{k+1}\\}\geq\delta_{-}\inf\\{\mathrm{rad}(B):B\in{\mathcal{B}}_{k}\\}.$

By iteration of this inequality, we get

(5.19)
$\inf\\{\mathrm{rad}(B):B\in{\mathcal{B}}_{N}\\}\geq\delta_{-}^{N}\inf\\{\mathrm{rad}(B):B\in{\mathcal{B}}_{0}\\}.$

By the choice of $N$, we have $\epsilon^{N}M\leq\epsilon_{0}$. Thus,
$\rho_{N}$ satisfies the desired mass bound:

(5.20) $\sum_{B\in{\mathcal{B}}_{N}}\rho_{N}(B)^{p}\leq\epsilon_{0}.$

The balls in ${\mathcal{B}}_{N}$ have various different sizes. Next, we will
embark on a “size reduction phase”. Let
$\overline{{\mathcal{B}}}_{0}={\mathcal{B}}_{N}$, and
$\overline{\rho}_{0}=\rho_{N}$. Let
$s:=\min\\{\mathrm{rad}(B):B\in{\mathcal{B}}_{N}\\},$ and let
$S_{0}=\mathrm{rad}(\overline{{\mathcal{B}}}_{0})$. From the assumption and
(5.19), we obtain

(5.21) $s\geq\delta_{-}^{N}\inf\\{\mathrm{rad}(B):B\in{\mathcal{B}}_{0}\\}\geq
r.$

If $S_{0}\leq\kappa r$, then we do not do anything and we let $L=0$. If on the
other hand $S_{0}>\kappa r$, we start running the following algorithm.

Set $k=0$. While $S_{k}>\kappa r$, let
${\mathcal{C}}_{k}=\\{B\in\overline{{\mathcal{B}}}_{k}:\mathrm{rad}(B)>\kappa
r\\}$. Apply Lemma 5.10 with ${\mathcal{B}}=\overline{{\mathcal{B}}}_{k}$,
$\rho=\overline{\rho}_{k}$ and ${\mathcal{C}}={\mathcal{C}}_{k}$ and with
$\rho_{\textbf{B}},{\mathcal{B}}_{\textbf{B}}$ and with $\epsilon=\eta$
satisfying (5.17). This gives a collection $\overline{{\mathcal{B}}}_{k+1}$
and strongly admissible function $\overline{\rho}_{k+1}$. Set
$S_{k+1}=\mathrm{rad}(\overline{{\mathcal{B}}}_{k+1})$, and increment $k$ by
one. Once $S_{k}\leq\kappa r$, terminate the algorithm. We will soon show that
the algorithm terminates. Let $L=k$ be the time it terminates.

We have, as part of Lemma 5.10, that
$\overline{\rho}_{k}\overline{\wedge}_{\tau,\overline{{\mathcal{B}}}_{k}}\Gamma$
for every $k\geq 0$. Further, by noting that $\epsilon\in(0,1)$, we get

$\sum_{B\in\overline{{\mathcal{B}}}_{k+1}}\overline{\rho}_{k+1}(B)^{p}\leq\sum_{B\in\overline{{\mathcal{B}}}_{k+1}}\overline{\rho}_{k}(B)^{p}.$

By iterating this $k$ times, we get from (5.20) that

(5.22)
$\sum_{B\in\overline{{\mathcal{B}}}_{k}}\overline{\rho}_{k}(B)^{p}\leq\sum_{B\in\overline{{\mathcal{B}}}_{0}}\overline{\rho}_{0}(B)^{p}\leq\
\epsilon_{0}.$

Let us analyse the effect of the algorithm on the radii of the collections
$\overline{{\mathcal{B}}}_{k}$, and the termination of the algorithm. Assume
that $k\geq 0$. At each step, a ball $B$ in ${\mathcal{B}}_{k+1}$ is either
equal to a ball $\textbf{B}\in\overline{{\mathcal{B}}}_{k}$ with
$\mathrm{rad}(\textbf{B})\leq\kappa r$, or $B\in{\mathcal{B}}_{\textbf{B}}$
for some $\textbf{B}\in\overline{{\mathcal{B}}}_{k}$ with $\kappa
s<\mathrm{rad}(\textbf{B})\leq S_{k}$. By construction, in either case
$\mathrm{rad}(B)\leq\max\\{\delta_{+}\mathrm{rad}(\textbf{B}),s\kappa\\}$.
Thus, by taking a supremum over all balls
$B\in\overline{{\mathcal{B}}}_{k+1}$, we get that $S_{k+1}\leq\max\\{\kappa
r,\delta_{+}S_{k}\\}$. In particular, while $S_{k}>\kappa r$, then the values
$S_{k}$ form a geometrically decreasing sequence. This can only last for
finitely many steps. Therefore, there must exist some $L\geq 0$, when the
algorithm terminates with $S_{L}\leq\kappa r$.

We show now that by induction each ball $B\in\overline{{\mathcal{B}}}_{k}$,
for $k=0,\dots,L$ satisfies $\mathrm{rad}(B)\in[r,S_{k}]$. The upper bound is
obvious, so we focus on the lower bound. The case of $k=0$ is also obvious.
So, we focus on the induction step. During the algorithm, for $k=0,\dots,L-1$,
each ball $B\in\overline{{\mathcal{B}}}_{k+1}$ is either equal to a ball
$\textbf{B}\in\overline{{\mathcal{B}}}_{k}$ or for some
$\textbf{B}\in\overline{{\mathcal{B}}}_{k}$ we have
$B\in\overline{{\mathcal{B}}}_{\textbf{B}}$ and
$\mathrm{rad}(\textbf{B})>\kappa r$. In the first case,
$\mathrm{rad}(B)\in[r,\kappa r]$. In the second case
$\mathrm{rad}(B)\in[\delta_{-}\mathrm{rad}(\textbf{B}),\delta_{+}\mathrm{rad}(\textbf{B})]$,
and thus $\mathrm{rad}(B)\geq\delta_{-}\kappa r>r$ since
$\delta_{-}>\kappa^{-1}$ by choice of $\kappa$ at the beginning of the proof.
In either case $r\leq\mathrm{rad}(B)\leq S_{k}$. Therefore, for all
$B\in\overline{{\mathcal{B}}}_{k}$, for $k=0,\dots,L$ we have
$\mathrm{rad}(B)\in[r,S_{k}]$.

Now, for $k=L$, we have $\mathrm{rad}(B)\in[r,\kappa r]$, since
$S_{L}\leq\kappa r$. Now set ${\mathcal{V}}=\overline{{\mathcal{B}}}_{L}$. We
thus get the desired claim, since (5.22) gives the desired mass bound for
$\overline{{\rm Mod}}_{p,\tau}(\Gamma,{\mathcal{V}})$, since
$\overline{\rho}_{L}\overline{\wedge}_{\tau,\overline{{\mathcal{B}}}_{L}}\Gamma$,
and we have already observed that for all $V\in{\mathcal{V}}$ we have
$\kappa^{-1}r\leq r_{V}\leq r$. ∎

###  5.3. Estimate for Bourdon-Kleiner modulus

In this section, we use the algorithm in the previous subsection to give an
explicit relationship between the Bourdon-Kleiner modulus from Subsection 3.1
and our new discrete modulus from Subsection 3.4. The basic idea is to use the
uniformly small moduli property to give one collection ${\mathcal{V}}$, and
then to use Lemma 5.16 to push the collection down to roughly uniform size.
This push-down operation is quantitative. Once the collection consists of
balls of roughly the same size, we can apply Proposition 3.7 to compare the
modulus to the Bourdon-Kleiner modulus of the same collection.

######  Proposition 5.23.

Fix $\kappa\geq 1,p\in(1,\infty)$. For each $k\in{\mathbb{N}}$, let
$\mathcal{U}_{k}$ be a $\kappa$-approximations at scale $2^{-k}$ for a compact
LLC space $X$. If $X$ has uniformly small $p$-moduli of annuli, then for every
$\epsilon>0$, there exists a $l\in{\mathbb{N}}$ for which for all $z\in X$ and
all $k\geq 0$, we have

${\rm Mod}_{p,\mathcal{U}_{l+k}}(\Gamma_{B(z,2^{-k}),2})\leq\epsilon$

###### Proof.

Fix $k\in{\mathbb{N}}$ and $\epsilon>0$. Let $\tau=4$ and let
$l_{0}=\lceil\log_{2}(\tau)\rceil+4$. Let $X$ have uniformly small $p$-moduli
of annuli with constant $\delta_{-}\in(0,\tau^{-1})$. Choose
$\kappa^{\prime}\geq\kappa$ be the constant from Lemma 5.16, and let $C$ be
the constant associated to $\kappa^{\prime},\tau$ and the space $X$ which
comes from Proposition 3.7. Set $\epsilon_{0}=C^{-1}\epsilon$.

By doubling, we have that there is a constant $D$ independent of $k$ so that
there are at most $D$ many sets in $\mathcal{U}_{k+l_{0}}$ which intersect
$B(z,2^{1-k})$. Set

${\mathcal{B}}_{0}=\\{B(x_{U},r_{U}):U\in\mathcal{U}_{k+l_{0}},U\cap
B(z,2^{1-k})\neq\emptyset\\}$

and set $\rho_{0}(B)=1$ for all $B\in{\mathcal{B}}_{0}$. Then, by applying the
definition, and since ${\mathcal{B}}_{0}$ covers $B(z,2^{1-k})$, we see
$\rho_{0}\overline{\wedge}_{\tau,{\mathcal{B}}_{0}}\Gamma_{B(z,2^{-k}),2}$. By
the size bound for ${\mathcal{B}}_{0}$, we get

$\sum_{B\in{\mathcal{B}}_{0}}\rho_{0}(B)^{p}\leq D.$

By Lemma 5.16, there exists an integer $N\in{\mathbb{N}}$ (which depend on
$\epsilon$, $D$ and the constants in the uniformly small moduli condition)
with the following properties. For any $r>0$ with
$\delta_{-}^{N}\inf\\{\mathrm{rad}(B):B\in{\mathcal{B}}_{0}\\}\geq r$ there is
a collection of balls ${\mathcal{V}}$ with

$\overline{{\rm
Mod}}_{p,\tau}(\Gamma,{\mathcal{V}})\leq\epsilon_{0}=C^{-1}\epsilon.$

and $r_{V}\in[\kappa^{\prime-1}r,r]$ for all $V\in{\mathcal{V}}$.

Now, if $l\geq l_{0}+N\lceil\log_{2}(\delta_{-}^{-1})\rceil+1$, then we can
choose $r=2^{-k-l}$. Then, by Proposition 3.7, we get for the
$\kappa$-approximation $\mathcal{U}_{l+k}$ at level $r$ that

${\rm Mod}_{p,\mathcal{U}_{l+k}}(\Gamma_{B(z,2^{-k}),2})\leq C\overline{{\rm
Mod}}_{p,\tau}(\Gamma,{\mathcal{V}})\leq\epsilon.$

∎

###  5.4. Proof of main theorem

###### Proof of Theorem 1.6.

Because the Ahlfors regular conformal dimension is always greater than the
conformal Hausdorff dimension, we have ${\rm dim}_{CH}(X)\leq{\rm
dim}_{CAR}(X)$. We are left to prove the converse inequality. Since $X$ is
connected, compact, locally connected and quasiself-similar, by Lemma 2.5 $X$
is LLC.

Suppose that ${\rm dim}_{CH}(X)<p$. Fix any sequence of
$\kappa$-approximations $\\{\mathcal{U}_{k}\\}_{k\in{\mathbb{N}}}$, where
$\mathcal{U}_{k}$ is at scale $2^{-k}$. By Lemma 5.2, we have that $X$ has
uniformly small moduli of annuli. Then, by Proposition 5.23, we have that

$\liminf_{m\to\infty}\sup_{x\in X,k\in{\mathbb{N}}}\\{{\rm
Mod}_{p}(\Gamma_{B(x,2^{-k}),2},\mathcal{U}):\mathcal{U}\text{ is a
}\kappa-\text{ approximation at level }2^{-k-m}\\}=0.$

Then, Theorem 3.5 implies that ${\rm dim}_{CAR}(X)\leq p$. This completes the
proof.

∎

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