# Delone sets that are not rectifiable under Lipschitz co-uniformly continuous
bijections

Rodolfo Viera

Abstract. We prove that there exist Delone sets in $\mathbb{R}^{d}$, $d\geq
2$, which cannot be mapped onto the standard lattice $\mathbb{Z}^{d}$ by
Lipschitz co-uniformly continuous bijections satisfying an asymptotic control
on the lower distortion. The impossibility of the unrectifiability crucially
uses ideas of Lipschitz regular maps recently introduced by M. Dymond, V.
Kaluža and E. Kopecká.

##  1 Introduction

Motivated by problems in many branches of mathematics (e.g. metric embedding
theory [13, 16], geometric group theory [11], information theory [9],
mathematical physics of quasicrystals [1]), over the last years there has been
a lot of activity on Lipschitz embeddings of discrete sets. In this work, we
focus on a particular aspect of this wide theory, namely the Lipschitz
embeddability of Delone subsets of the Euclidean space into the standard
lattice. Recall that a Delone set $\mathcal{D}$ of a metric space $X$ is a
subset that is discrete and coarsely dense in a uniform way. This means that
there exist positive constants $\sigma,\Sigma$ such that $d(x,y)\geq\sigma$
for all $x\neq y$ in $\mathcal{D}$, and for each $z\in X$ there exists
$x\in\mathcal{D}$ for which $d(x,z)\leq\Sigma$.

Furstenberg and, independently, Gromov, asked whether for every Delone subset
of $\mathbb{R}^{d}$, $d\geq 2$, there exists a bi-Lipschitz bijection onto
$\mathbb{Z}^{d}$ (i.e. whether every Delone set in $\mathbb{R}^{d}$ is bi-
Lipschitz rectifiable). Furstenberg was interested in dynamical aspects of
this question (see [3] for a broader discussion), while Gromov was motivated
by instances of geometric group theory [11]. Their question was answered in
the negative by Burago and Kleiner [2] and, independently, by McMullen [14].
However, their results only yield existence of non bi-Lipschitz rectifiable
Delone sets. Concrete examples were produced by Cortez and Navas in [4]. These
examples can be constructed with supplementary properties of “dynamical type”.
In particular, they can be built so that they are repetitive, which means that
the translation action on the space of Delone sets (endowed with an
appropriate Chabauty topology) is minimal. Equivalently, for each $r>0$, there
exists $R>0$ such that every pattern that appears in a ball of radius $r$
actually appears in every ball of radius $R$. Besides the mathematical
relevance of this property, it is worth mentioning that all known examples of
real (physical) quasicrystals lead to repetitive Delone sets.

In order to answer in the negative the Furstenberg-Gromov’ question, Burago
and Kleiner in [2] and McMullen in [14] show that the existence of a non-
rectifiable Delone set is a consequence of the following result of analytical
nature: there exists a bounded away from zero continuous function
$\rho:I^{2}\to\mathbb{R}$, where $I^{2}:=[-1/2,1/2]^{2}$, for which the
prescribed Jacobian equation

$Jac(F)=\rho\hskip 28.45274pta.e,$ (1.1)

has no bi-Lipschitz solution $F:I^{2}\to\mathbb{R}^{2}$; such densities
$\rho:I^{2}\to\mathbb{R}$ which cannot be realizable as the Jacobian of a bi-
Lipschitz map are called non-bi-Lipschitz-realizable. For more details on the
prescribed Jacobian equation, we refer for instance to [5, 15, 17]. We point
out that “almost all” positive functions $\rho\in L^{\infty}$ can be used to
construct non-rectifiable Delone sets, as was shown in [18].

Motivated by a fundamental problem in discrete geometry and information theory
(see [9] and the references therein), Dymond, Kaluža and Kopecká recently
adapted the Burago-Kleiner / McMullen techniques to answer in the negative a
question raised by Feige in [13]. More precisely, in [8], they proved the
following remarkable fact:

Given $d\geq 2$, there is no constant $L$ such that, for all $n$, every subset
of $n^{d}$ points of $\mathbb{Z}^{d}$ can be bijectively mapped into
$\\{1,\ldots,n\\}^{d}$ by an $L$-Lipschitz map.

Their proof relies strongly on the existence of a positive continuous function
$\rho:I^{d}\to\mathbb{R}$, with $d\geq 2$, for which the generalized “push-
forward equation”

$F_{\\#}(\rho\lambda)=\lambda|_{F(I^{d})},$ (1.2)

has no Lipschitz (regular) solution $F:I^{d}\to\mathbb{R}^{d}$, where
$\lambda$ is the Lebesgue measure (for a definition of Lipschitz Regular maps,
see Section 3 below); observe that equation (1.2) coincides with (1.1)
whenever $F$ is bi-Lipschitz.

###  1.1 Notations and some basics on Lipschitz maps

Throughout this work we will denote by $||\cdot||$ the supremum-norm in
$\mathbb{R}^{d}$, and we denote by $B(x,r)$ the open ball with center
$x\in\mathbb{R}^{d}$ and radius $r>0$ with this norm; moreover, we write
$B(r)$ for the open ball with radius $r>0$ and centred at the origin. Given
$\varepsilon>0$ and a bounded set $A\subset\mathbb{R}^{d}$, let
$B(A,\varepsilon)$ be the $\varepsilon$-neighbourhood of $A$. Given two
subsets $A,B\subset\mathbb{R}^{d}$, we denote by $d(A,B)$ the distance between
$A$ and $B$ with respect to $||\cdot||$. For two positive numbers $r<R$ and a
point $x\in\mathbb{R}^{d}$, we denote by $\mathsf{Ann}(x,r,R)$ be the annulus
$\\{y\in\mathbb{R}^{d}:\ r<||y-x||\leq R\\}$. The set $C(I^{d})$ will denote
the space of real-valued continuous functions defined on the unit square
$I^{d}:=[-1/2,1/2]^{d}$, with the supremum-norm
$||f||_{\infty}:=\max\\{|f(x)|:\ x\in I^{d}\\}$. Finally, for $[\cdot]$ we
denote the integer part of a real number.

We say that a Delone set $\mathcal{D}\subset\mathbb{Z}^{2}$ satisfies the
$2\mathbb{Z}^{2}$-property if $2\mathbb{Z}\times\mathbb{Z}$ and
$\mathbb{Z}\times 2\mathbb{Z}$ are subsets of $\mathcal{D}$. Finally, we say
that a subset $R$ of an integer square
$T=([i,i+k]\times[j,j+k])\cap\mathbb{Z}^{2}$, where $i,j,k\in\mathbb{Z}$,
satisfies the $2\mathbb{Z}^{2}$-property, if
$T\cap(2\mathbb{Z}\times\mathbb{Z})$ and $T\cap(\mathbb{Z}\times 2\mathbb{Z})$
are contained in $R$.

Let $(X,d_{1})$ and $(Y,d_{2})$ be two metric spaces. A function $f:X\to Y$ is
said to be Lipschitz, if there exists $L>0$ such that for every $x,y\in X$:

$d_{2}(f(x),f(y))\leq Ld_{1}(x,y).$

Given two positive numbers $b\leq L$, we say that $f$ is $(b,L)$-bi-Lipschitz
if for every $x,y\in X$:

$bd_{1}(x,y)\leq d_{2}(f(x),f(y))\leq Ld_{1}(x,y);$

moreover $f$ is called $L$-bi-Lipschitz if $b=1/L$.

Given an increasing continuous function $\omega:(0,\infty)\to(0,\infty)$, we
say that a bijection $f:X\to Y$ is $\omega$-co-uniformly continuous if for
every $x\in X$ there holds

$f^{-1}(B(f(x),r))\subset B(x,\omega(r));$ (1.3)

a bijection $f:X\to Y$ is called co-uniformly continuous if is $\omega$-co-
uniformly continuous for some increasing continuous function
$\omega:(0,\infty)\to(0,\infty)$ (see [12] for further details). Additionally,
we say that a $\omega$-co-uniformly continuous map is of order $o(h(r))$ if
$\omega(r)=o(h(r))$ (where $o(\cdot)$ denotes the standard Landau’s notation).

Given a differentiable map $f:U\subset\mathbb{R}^{d}\to\mathbb{R}^{d}$, the
(determinant) Jacobian of $f$ is denoted by $Jac(f):=\det(Df)$. Recall that by
a classical theorem due to Rademacher (see Theorem 3.1.6 in [10]), every
Lipschitz map $f:U\subset\mathbb{R}^{d}\to\mathbb{R}^{n}$ is differentiable
almost everywhere. Thus, the expression “$Jac(f)$ a.e” make sense for
Lipschitz maps from $\mathbb{R}^{d}$ to itself.

Given an integrable function $\rho:I^{d}\to[0,+\infty)$, let $\rho\lambda$ be
the measure defined by

$\begin{split}\rho\lambda(A)&:=\displaystyle\int_{A}\rho\lambda,\hskip
14.22636pt\mbox{for every measurable set }A\subset I^{d}.\\\ \end{split}$

For a measurable map $F:A\to\mathbb{R}^{d}$ and a measure $\mu$ in $A$,
consider the pushforward measure $F_{\\#}\mu$ given by

$F_{\\#}\mu(B):=\mu(F^{-1}(B))\hskip 14.22636pt\mbox{for every measurable set
}B\subset F(A).$ (1.4)

###  1.2 Statement of the result

The aim of this work is to prove, following [2] and [8], that if we deal with
Lipschitz bijections which are co-uniform with a suitable asymptotic control
on the lower distortion, then there still exist Delone sets that fail to be
rectifiable.

Main Theorem. For each $d\geq 2$, there exist Delone subsets of $\
\mathbb{R}^{d}$ that admit no Lipschitz co-uniformly continuous bijection with
$\mathbb{Z}^{d}$, with co-uniformity of order $o(r^{d})$.

This result extends the well-known bi-Lipschitz case in [2, 14, 4]. In
particular the Main Theorem implies that there is a Delone set in
$\mathbb{R}^{d}$, $d\geq 2$, which cannot be mapped onto $\mathbb{Z}^{d}$ by
Lipschitz bijections with a Hölder co-uniformity. Closely related results have
been obtained recently in [7], where it is shown that there is a Delone set
$\mathcal{D}\subset\mathbb{R}^{d}$, $d\geq 2$, for which there is no bijection
$f:\mathcal{D}\to\mathbb{Z}^{d}$ which is $\omega$-homogeneous and
$\omega$-co-uniformly continuous for a particular modulus of continuity
$\omega:[0,+\infty)\to[0,+\infty)$.

It would be interesting to known if there are Delone sets on $\mathbb{R}^{d}$,
$d\geq 2$, that admit no Lipschitz bijections with the standard lattice
$\mathbb{Z}^{d}$, i.e, by avoiding any control in the lower distortion. In
addition, since non-rectifiable Delone sets do exist, it would be interesting
to know explicit examples of Delone sets that are Lipschitz rectifiable
without being bi-Lipschitz rectifiable.

Sketch of the proof. Our proof crucially follows the construction proposed in
[2], [8] and [7], with some mild though crucial changes along the way. Our
strategy is described below in the 2-dimensional setting:

  1. 1.-

Let $\rho:I^{2}\to\mathbb{R}$ be a positive continuous function such that
$8/9\leq\min\rho<\max\rho\leq 1$. As in [2], consider a Delone subset
$\mathcal{D}_{\rho}$ of $\mathbb{Z}^{2}$ satisfying the
$2\mathbb{Z}^{2}$-property and emulating the behaviour of $\rho$ at bigger and
bigger scales (see Section 2 for a precise construction).

  2. 2.-

If there is a Lipschitz bijection $f:\mathcal{D}_{\rho}\to\mathbb{Z}^{2}$,
then $f$ must be “regular”, which means that the preimage of a ball $B$ under
this map cannot contain a $2\\!\cdot\\!\mbox{radius}(B)$-separated set with
more than a certain prescribed number of elements (see Definition 3.1 and
Lemma 3.2 below).

  3. 3.-

By renormalizing $\mathcal{D}_{\rho}$ and $f$, and after passing to the limit,
we obtain a Lipschitz limit map $F:I^{2}\to\mathbb{R}^{2}$ defined on the unit
square, which can be also showed to be regular (see Sections 2 and 3 for more
details).

  4. 4.-

By one of the main results in [8], there is an open subset of $\mathbb{R}^{2}$
whose preimage under $F$ is made up of finitely many disjoint open sets
restricted to which the limit map is bi-Lipschitz; moreover, the number of
these sets is uniformly controlled.

  5. 5.-

Finally, from the control on the lower distortion, we show that there is a
closed ball $\mathcal{Q}\subset F(I^{2})$ such that $F$ satisfies the equation

$F_{\\#}(\rho\lambda)|_{\mathcal{Q}}=\lambda|_{\mathcal{Q}}.$ (1.5)

The proof of that $F$ satisfies equation (1.5) is made by means of a control
of the loss of mass in $\mathcal{Q}$ under the renormalization. More
precisely, as a consequence of the control of the co-uniformity in (1.3), we
prove that the escape of mass in $\mathcal{Q}$ occurs always close to its
boundary.

Therefore, if we choose a density map $\rho$ for which there is no a closed
ball $\mathcal{Q}$ such that (1.5) has Lipschitz regular solutions, we obtain
a Delone set $\mathcal{D}_{\rho}$ which cannot be Lipschitz rectifiable under
co-uniformly continuous bijections of order $o(r^{2})$. This function $\rho$
must exist as a consequence of Proposition 4.1 below (see Theorem 4.1 in [8]).

The preceding steps can be summarized as follows: every bijection
$f:\mathcal{D}_{\rho}\to\mathbb{Z}^{2}$ having certain Lipschitz-regularity
(e.g, Lipschitz co-uniformly continuous, bi-Lipschitz) must induce certain
type of regularity over the continuous density $\rho$. Thus, the desired bad-
behaved Delone set $\mathcal{D}_{\rho}$ can be found by choosing $\rho$ lying
outside of this regularity class.

##  2 Constructing a Delone set from an anomalous density.

In this section we explain how to produce a Delone set from a bounded away
from zero density, as in [2] and [8] (see also [7]). To simplify computations,
notations and figures, we will restrict ourself to the 2-dimensional case; the
higher dimensional case follows analogously.

For our purposes, we will consider a continuous density
$\rho:I^{2}\to\mathbb{R}$ such that $8/9\leq\min\rho<\max\rho\leq 1$. Let
$(l_{n})_{n\in\mathbb{N}}$ and $(m_{n})_{n\in\mathbb{N}}$ be two sequences of
even positive integers, where $l_{n}$ is a multiple of $m_{n}$ and such that
$l_{n},m_{n}$ and $(l_{n}/m_{n})\longrightarrow+\infty$. Moreover, assume that
$l_{n}$ divides $l_{n+1}$ for every $n\in\mathbb{N}$; this last hypothesis
will be used later in the proof of the Main Theorem. Let
$(S_{n})_{n\in\mathbb{N}}$ be a sequence of disjoint squares with sides
parallel to the coordinate axes, vertices having even integer coordinates and
with side-length equal to $l_{n}$. For each $n\in\mathbb{N}$, let
$(T_{n,i})_{i=1}^{m_{n}^{2}}$ be a subdivision of $S_{n}$ by $m_{n}^{2}$
squares with sides parallel to the coordinate axes and side-length equal to
$l_{n}/m_{n}$ (see Figure 1). Finally, let
$\phi_{n}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be an affine linear map sending the
square $S_{n}$ onto the unit square $I^{2}$.

$S_{n}$$S_{n+1}$$\dfrac{l_{n+1}}{m_{n+1}}$$\dfrac{l_{n}}{m_{n}}$$l_{n+1}$$l_{n}$$T_{n,i}$
Figure 1: The squares $S_{n}$ and $S_{n+1}$, and a square $T_{n,i}$.

We build a Delone set $\mathcal{D}_{\rho}\subset\mathbb{Z}^{2}$ which
“emulates” the behaviour of $\rho$ as follows: in each square $T_{n,i}$ we put
$\left[\int_{T_{n,i}}\rho\circ\phi_{n}d\lambda\right]$ points with integer
coordinates in such a way that each $T_{n,i}$ satisfies the
$2\mathbb{Z}^{2}$-property (see Figure 2) and such that $\partial
T_{n,i}\cap\mathbb{Z}^{2}\subset\mathcal{D}_{\rho}$; notice that this is
possible since $8/9\leq\min\rho<\max\rho\leq 1$. Outside of
$\cup_{n\in\mathbb{N}}S_{n}$, put one point in each integer coordinate. This
construction provides a set $\mathcal{D}_{\rho}\subset\mathbb{Z}^{2}$ which is
actually a Delone set satisfying the $2\mathbb{Z}^{2}$-property since
$8/9\leq\min\rho<\max\rho\leq 1$.

Figure 2: A possible configuration of $\mathcal{D}_{\rho}\cap T_{n,i}$.

##  3 Rescaling up to the limit and Lipschitz regularity

In this section we show, after renormalization and passing to the limit, that
a Lipschitz bijection $f:\mathcal{D}_{\rho}\to\mathbb{Z}^{2}$ induces a
Lipschitz regular map from the unit square. We start this section with some
basic background and recent results on Lipschitz regular maps. For additional
information about these mappings, we refer to [6].

######  Definition 3.1.

Let $X$ and $Y$ be two metric spaces. We say that a Lipschitz map $f:X\to Y$
is Lipschitz regular if there is a constant $C\in\mathbb{N}$ such that for
every ball $B\subset Y$ of radius $r>0$, the set $f^{-1}(B)$ can be covered by
at most $C$ balls of radii $Cr$. The smallest such $C$ (that works for every
$r>0$) is called the regularity constant of $f$, and is denoted by
$\mathsf{Reg}(f)$.

A useful equivalent interpretation of this definition is provided by the next
lemma. (The proof is straightforward and is left to the reader.)

######  Lemma 3.2.

A Lipschitz map $f:X\to Y$ is Lipschitz regular if and only if there is a
constant $C\in\mathbb{N}$ such that for every ball $B\subset Y$ of radius
$r>0$, the set $f^{-1}(B)$ does not contain a $Cr$-separated set with more
than $C$ elements. If this is the case, then $\mathsf{Reg}(f)\leq C$.
Conversely, if $f$ is Lipschitz regular, then $C$ can be taken as being equal
to $2\mathsf{Reg}(f)$.

One of the main results of [8] (namely, Theorem 2.10 therein) is that every
Lipschitz regular map defined on a bounded region of $\mathbb{R}^{d}$ can be
“densely decomposed” into bi-Lipschitz pieces, as stated below.

######  Theorem 3.3.

Let $U\subset\mathbb{R}^{d}$ be a nonempty open set. If
$f:\overline{U}\to\mathbb{R}^{d}$ is a Lipschitz regular map, then there exist
pairwise disjoint open sets $(A_{n})_{n\in\mathbb{N}}$ in $U$ such that
$\bigcup_{n\in\mathbb{N}}A_{n}$ is dense in $\overline{U}$ and, for each
$n\in\mathbb{N}$, the map $f|_{A_{n}}$ is bi-Lipschitz with lower bi-Lipschitz
constant $b=b\,(\mathsf{Reg}(f))$.

As a consequence of Theorem 3.3, it is showed in [8] that the image by a
Lipschitz regular map contains an open set whose preimage is made of a
controlled number of open subsets where the map is bi-Lipschitz, with lower
bi-Lipschitz constant depending only on the regularity constant. This is
stated below, and corresponds to Proposition 2.15 in [8].

######  Proposition 3.4.

Let $U\subset\mathbb{R}^{d}$ be a nonempty open set. If
$f:\overline{U}\to\mathbb{R}^{d}$ is a Lipschitz regular map, then there exist
a nonempty open set $T\subset f(\overline{U})$, an integer
$N\in\\{1,\ldots,\mathsf{Reg}(f)\\}$, and pairwise disjoint open sets
$W_{1},\ldots W_{N}\subset\overline{U}$, such that
$f^{-1}(T)=\displaystyle\bigcup_{i=1}^{N}W_{i}$ and, for each $1\leq i\leq N$,
the map $f|_{W_{i}}:W_{i}\to T$ is a bi-Lipschitz homeomorphism, with lower
bi-Lipschitz constant $b=b(\mathsf{Reg}(f))$. Actually, one may take
$b=\frac{1}{2\mathsf{Reg}(f)^{2}}$.

The following result deals with maps defined on discrete sets of points. It
asserts that every Lipschitz bijection defined on a Delone set that satisfies
the $2\mathbb{Z}^{2}$-property onto the integer lattice must be Lipschitz
regular. Actually, as we will see along the proof, the
$2\mathbb{Z}^{2}$-property may be replaced by any other property ensuring that
densities of points in large balls are everywhere bounded from below (away
from zero). Given $x\in\mathbb{R}^{2}$, $r>0$ and a subset $\mathcal{L}$ of
$\mathbb{R}^{2}$, we will denote the set $B(x,r)\cap\mathcal{L}$ by
$B_{\mathcal{L}}(x,r)$.

######  Proposition 3.5.

Let $\mathcal{D}\subset\mathbb{Z}^{2}$ be a Delone set satisfying the
$2\mathbb{Z}^{2}$-property. If $f\\!:\mathcal{D}\to\mathbb{Z}^{2}$ is an
$L$-Lipschitz bijection, then $f$ is Lipschitz regular, with
$\mathsf{Reg}(f)\leq\max\\{2,C\,(L+1)^{2}\\}$ for some universal constant
$C>0$.

###### Proof.

Let $y\in\mathbb{Z}^{2}$ and $r>0$. Consider $\Gamma\subset f^{-1}(B(y,r))$ a
maximal $2r$-separated set, and write $\Gamma=\\{x_{1},\ldots
x_{|\Gamma|}\\}$. Then, by the $L$-Lipschitz condition we have that

$f\left(\bigcup_{i=1}^{|\Gamma|}B_{\mathcal{D}}(x_{i},r)\right)\subset
B_{\mathbb{Z}^{2}}(y,r+rL).$

Observe that, for $i=1,\ldots,|\Gamma|$, the open balls $B(x_{i},r)$ are
pairwise disjoint. Since $f$ is a bijection we obtain that, for a certain
constant $C_{1}\geq 1$,

$\displaystyle\sum_{i=1}^{|\Gamma|}\left|B_{\mathcal{D}}(x_{i},r)\right|=\left|f\left(\bigcup_{i=1}^{|\Gamma|}B_{\mathcal{D}}(x_{i},r)\right)\right|\leq\big{|}B_{\mathbb{Z}^{2}}(y,r+rL)\big{|}\leq
C_{1}r^{2}(L+1)^{2}.$ (3.1)

Now, by the $2\mathbb{Z}^{2}$-property, the cardinality
$|B_{\mathcal{D}}(x_{i},r)|$ is at least $C_{2}r^{2}$ for another universal
constant $C_{2}>0$. (The value of $C_{2}$ can be taken as $8/9-\varepsilon$
provided $r$ is large enough.) Thus, by (3.1),

$C_{2}r^{2}|\Gamma|\leq C_{1}r^{2}(L+1)^{2}.$

We hence conclude that $|\Gamma|\leq C_{1}(L+1)^{2}/C_{2}$. Therefore, by
Lemma 3.2, $f$ is Lipschitz regular with regularity constant at most
$\max\\{2,C_{1}(L+1)^{2}/C_{2}\\}$. ∎

From now on, let $\mathcal{P}_{n}:=\mathcal{D}_{\rho}\cap S_{n}$, where
$\mathcal{D}_{\rho}$ and $(S_{n})_{n\in\mathbb{N}}$ are the Delone set and the
sequence of squares given in §2, respectively. Let
$\phi_{n}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be the homothety defined in Section
2, and define $\mathcal{R}_{n}:=\phi_{n}(\mathcal{P}_{n})$.

Assume there is an $L$-Lipschitz bijection
$f:\mathcal{D}_{\rho}\to\mathbb{Z}^{2}$. As in [2] and [8], we proceed to
normalize $f$ to each square $\mathcal{P}_{n}$, that is, to consider the map
$f_{n}:\mathcal{R}_{n}\to\frac{1}{l_{n}}\mathbb{Z}^{2}$ defined by

$f_{n}(x):=\frac{1}{l_{n}}(f\circ\phi_{n}^{-1}(x)-f\circ\phi_{n}^{-1}(\tilde{x}_{n})),$
(3.2)

where $\tilde{x}_{n}\in\mathcal{R}_{n}$ is some base point. Notice that for
each $n\geq 1$ the map $f_{n}$ is Lipschitz regular with
$\mathsf{Lip}(f_{n})\leq L$ and $\mathsf{Reg}(f_{n})=\mathsf{Reg}(f)$. By
Kirszbraun’s extension theorem111Actually, we do not really need to keep the
same Lipschitz constant $L$ for the extension map, but just another (larger)
constant that depends only on $L$, and a weaker form of Kirszbraum’s theorem
proving this is much easier to establish. (see, for instance, Theorem 2.10.43
in [10]), each function $f_{n}$ can be extended to an $L$-Lipschitz map

${}\widehat{f}_{n}:I^{2}\to\mathbb{R}^{2}.$ (3.3)

By the Arzelá-Ascoli’s theorem, there exists a subsequence
$(\widehat{f}_{n_{k}})_{k\in\mathbb{N}}$ of $(\widehat{f}_{n})_{n\geq 1}$,
converging to an $L$-Lipschitz map $F:I^{2}\to\mathbb{R}^{2}$; from now on,
the subsequence $(\widehat{f}_{n_{k}})_{k\geq 1}$ will be just denoted
$(\widehat{f}_{n})_{n\geq 1}$. As we next show, the Lipschitz regularity is
inherited from $f$ to $F$.

######  Proposition 3.6.

The map $F:I^{2}\to\mathbb{R}^{2}$ built above is Lipschitz regular, with
$\mathsf{Reg}(F)\leq 34\ \mathsf{Reg}(f)$.

###### Proof.

Let $y\in F(I^{2})$ and $r>0$. Consider a maximal
$2\mathsf{Reg}(f)r$-separated set $\Gamma=\\{x_{1},\ldots,x_{|\Gamma|}\\}$
contained in $F^{-1}(B(y,r))$. Given

$0<\varepsilon<\min\left\\{2\mathsf{Reg}(f)r\left(1-\sqrt{\frac{2-\sqrt{2}}{2+\sqrt{2}}}\right),d(\Gamma,\partial
F^{-1}(B(y,r)))\right\\},$

by the convergence of $\widehat{f}_{n}$ to $F$, there is a positive integer
$n_{0}=n_{0}(\varepsilon)$ such that, for every $n\geq n_{0}$, there exist
$p_{1},\ldots,p_{|\Gamma|}\in\mathcal{R}_{n}$ for which the following hold:

  * •

for every $i=1,\ldots,|\Gamma|$, we have that $||p_{i}-x_{i}||<\varepsilon/2$,

  * •

the set $\Gamma_{n}:=\\{p_{1},\ldots,p_{|\Gamma|}\\}$ is contained in
$F^{-1}(B(y,r))$ and,

  * •

$f_{n}(\Gamma_{n})\subset B(y,r)$.

Observe that $\Gamma_{n}$ is $(2\mathsf{Reg}(f)r-\varepsilon)$-separated,
since $\Gamma$ is $2\mathsf{Reg}(f)r$-separated.

We will delete some points in $\Gamma_{n}$ in an appropriate way in order to
obtain a set $\Gamma_{n}^{\prime}\subset f_{n}^{-1}(B(y,r))$ that is
$2\mathsf{Reg}(f)r$-separated and such that
$|\Gamma_{n}^{\prime}|\geq|\Gamma|/17$. By Lemma 3.2, this will imply that

$|\Gamma|\leq 17\,|\Gamma_{n}^{\prime}|\leq
34\,\mathsf{Reg}(f_{n})=34\,\mathsf{Reg}(f)$

hence $F$ is Lipschitz-regular with $\mathsf{Reg}(F)\leq 42\,\mathsf{Reg}(f)$.

To build the set $\Gamma_{n}^{\prime}$, we consider the angle

$\alpha=\arctan\left(\frac{2\mathsf{Reg}(f)r-\varepsilon}{2\mathsf{Reg}(f)r}\right)\geq\arctan\left(\sqrt{\frac{2-\sqrt{2}}{2+\sqrt{2}}}\right)=\frac{\pi}{8},$

where the inequality follows from the condition

$\varepsilon<2\mathsf{Reg}(f)r\left(1-\sqrt{\frac{2-\sqrt{2}}{2+\sqrt{2}}}\right).$

This is the angle that appears in Figure 3 below. In the area depicted in
black, no pair of points in $\Gamma_{n}$ is at distance
$>2\mathsf{Reg}(f)r-\varepsilon$. The same happens in a similar region with
angle $\pi/8$. Since 16 of these polygonal regions cover exactly the anular
region between a square of radius $2\mathsf{Reg}(f)r-\varepsilon$ and another
of radius $2\mathsf{Reg}(f)r$ (with the same center), we deduce -by the
pigeonhole principle- that no more than $16$ points of $\Gamma_{n}$ in this
annular region can be $(2\mathsf{Reg}(f)r-\epsilon)$-separated.

$\alpha$$A$$B$$C$$2\mathsf{Reg}(f)r-\varepsilon$$D$ Figure 3: In the figure,
$\overline{AC}=2\mathsf{Reg}(f)r,\ \overline{BC}=\varepsilon$ and
$\overline{CD}=2\mathsf{Reg}(f)r-\varepsilon$. In the black region there is no
a pair of points in $\Gamma_{n}$ at distance $>2\mathsf{Reg}(f)r-\varepsilon$.

Now, for each $i\in\\{1,\ldots,|\Gamma|\\}$, let $\Gamma_{n}^{i}$ be the set
of all points $p\in\Gamma_{n}$ such that
$2\mathsf{Reg}(f)r-\varepsilon\leq||p_{i}-p||\leq 2\mathsf{Reg}(f)r$. We have
shown that this set contains at most $16$ points. We erase those corresponding
to $p_{1}$, then those corresponding to the $p_{i}$ with minimal index that
survive after the first deletion ($i\geq 2$), and so on. At the end, we get
the subset $\Gamma_{n}^{\prime}$ with the desired properties. ∎

##  4 Proof of the Main Theorem

This section is dedicated to the proof of the Main Theorem. Exploiting the bi-
Lipschitz decomposition of Lipschitz regular maps recently introduced by
Dymond, Kaluža and Kopecká in [8] together the co-uniformity of
$f:\mathcal{D}_{\rho}\to\mathbb{Z}^{2}$, in this section we show that the
Lipschitz-rectifiability of $\mathcal{D}_{\rho}$ under co-uniformly continuous
bijections of order $o(r^{2})$ induces certain regularity over the positive
continuous density $\rho:I^{2}\to\mathbb{R}$. Thus, the existence of a non-
Lipschitz-rectifiable Delone set $\mathcal{D}_{\rho}$ will be a consequence of
the existence of a continuous function $\rho$ which does not belong to this
regularity class.

Let $(A_{n})_{n\in\mathbb{N}}$ be a basis for the topology of the unit square
$I^{2}$. As in [8], let $\mathcal{E}_{C,L,n}$ be the set of positive
continuous functions $\rho:I^{2}\to\mathbb{R}$ for which the following holds:
there are pairwise disjoint open sets $Y_{1},\ldots,Y_{N}\subset I^{2}$,
$Y_{1}=A_{n}$, where $1\leq N\leq C$, an open set $V\subset\mathbb{R}^{2}$,
and a family of $(b(C),L)$-bi-Lipschitz homeomorphisms $F_{i}:Y_{i}\to V$ such
that

$\rho(y)=|Jac(F_{1})(y)|-\displaystyle\sum_{i=2}^{n}\rho(F_{i}^{-1}\circ
F_{1})(y)|Jac(F_{i}^{-1}\circ F_{1})(y)|\hskip 28.45274pt\mbox{a.e in }Y_{1}.$
(4.1)

In [8] it is shown that “almost all” positive continuous functions do not have
a bi-Lipschitz decomposition as in (4.1). This corresponds to Theorem 4.1 in
[8].

######  Proposition 4.1 (Dymond, Kaluža and Kopecká, 2018).

A generic positive function $\rho:I^{2}\to\mathbb{R}$ does not belong to
$\bigcup_{C,L,n\in\mathbb{N}}\mathcal{E}_{C,L,n}$.

From now on let $\rho$ be a positive continuous function such that
$8/9\leq\min\rho<\max\rho\leq 1$ as in Proposition 4.1 and let
$\mathcal{D}_{\rho}\subset\mathbb{Z}^{2}$ be the corresponding Delone set
constructed as in Section 2. Assume that there is an $L$-Lipschitz,
$\omega$-co-uniformly continuous bijection
$f:\mathcal{D}_{\rho}\to\mathbb{Z}^{2}$ for some increasing continuous
function $\omega:[0,\infty)\to[0,\infty)$ such that $\omega(r)=o(r^{2})$ and
let $F:I^{2}\to\mathbb{R}^{2}$ be the (limit) Lipschitz regular map obtained
as in Section 3. By Proposition 3.4 there exist a non-empty open set $W\subset
F(I^{2})$ and open disjoint subsets $V_{1},\ldots,V_{N}\subset I^{2}$, where
$N\leq\mathsf{Reg}(F)$, such that $\bigcup_{i=1}^{N}V_{i}=F^{-1}(W)$ and, for
each $i\leq i\leq N$, the map $F|_{V_{i}}:V_{i}\to W$ is bi-Lipschitz, with
lower bi-Lipschitz constant $b=\frac{1}{2\mathsf{Reg}(F)^{2}}$. Thereupon, the
Main Theorem is a consequence of the next proposition.

######  Proposition 4.2.

For $\rho,f,F$ and $W$ as in the previous paragraph, let $\mathcal{Q}\subset
W$ be a closed ball centred at a point $y_{m}\in(1/l_{m})\mathbb{Z}^{2}$, for
some positive integer $m$. Then we have

$F_{\\#}(\rho\lambda)|_{\mathcal{Q}}=\lambda|_{\mathcal{Q}}.$ (4.2)

###### Proof of Main Theorem from Proposition 4.2.

For every $i=1,\ldots,N$ write $F_{i}:=F|_{V_{i}}$ and let $n$ be a natural
number such that $A_{n}\subset V_{1}\cap F^{-1}(\mathcal{Q})$; besides, for
each $i=2,\ldots,N$ denote by $A_{i,n}:=F^{-1}(F(A_{n}))\cap V_{i}$ and
$A_{1,n}:=A_{n}$. From the equation (4.2) we have that

$\displaystyle\sum_{i=1}^{N}\int_{A_{i,n}}\rho d\lambda=\lambda(F(A_{n})).$

By a change of variable and the Euclidean Area formula for bi-Lipschitz maps,
the previous equation can be rewritten as

$\displaystyle\int_{A_{n}}\sum_{i=1}^{N}\rho(F_{i}^{-1}\circ
F_{1})|Jac(F_{i}^{-1}\circ
F_{1})|d\lambda=\displaystyle\int_{A_{n}}|Jac(F_{1})|d\lambda,$

which is equivalent to the equation

$\rho(y)=|Jac(F_{1})(y)|-\displaystyle\sum_{i=2}^{n}\rho(F_{i}^{-1}\circ
F_{1})(y)|Jac(F_{i}^{-1}\circ F_{1})(y)|\hskip 28.45274pt\mbox{a.e in }A_{n}.$

Thus, we conclude that $\rho\in\mathcal{E}_{C,L,n\in\mathbb{N}}$ for
$C=1/2\mathsf{Reg}(F)^{2}$, which contradicts the choice of $\rho$. Therefore,
$\mathcal{D}_{\rho}$ cannot be mapped onto $\mathbb{Z}^{2}$ by Lipschitz
bijections, as announced. ∎

###  4.1 Mass-loss control under renormalization

In what follows we prove Proposition 4.2. Let $\mathcal{Q}\subset W$ be a
closed ball such that $d(\mathcal{Q},\partial W)>0$ and define
$\mathcal{H}_{i}:=F^{-1}(\mathcal{Q})\cap V_{i}$ (recall that we consider
“balls” for the sup-norm in $\mathbb{R}^{2}$) and
$\mathcal{H}:=\cup_{i=1}^{N}\mathcal{H}_{i}$; in addition, choose
$\varepsilon>0$ such that the closure of the $\varepsilon$-neighbourhood
$B(\mathcal{Q},\varepsilon)$ of $\mathcal{Q}$ is contained in $W$. We start by
showing that from a large-enough $n\in\mathbb{N}$ the points in
$\mathcal{R}_{n}\cap\mathcal{H}$ are mapped under $f_{n}$ into
$B(\mathcal{Q},\varepsilon)$, and that
$f_{n}^{-1}(\mathcal{Q}\cap(1/l_{n})\mathbb{Z}^{2})$ is completely contained
in the $\varepsilon$-neighbourhood of $\mathcal{H}$ (with $f_{n}$ defined as
in (3.2)); nevertheless, there may exist some points in
$\mathcal{Q}\cap(1/l_{n})\mathbb{Z}^{2}$ which do not have a pre-image under
$f_{n}$, producing loss of mass under the renormalization (see Figure 4).

$f_{n}$$\mathcal{H}_{i}\cap\mathcal{R}_{n}$$I^{2}$$B(\mathcal{Q},\varepsilon)$$\mathcal{Q}$
Figure 4: The set $\mathcal{H}\cap\mathcal{R}_{n}$ is mapped under $f_{n}$
into $B(\mathcal{Q},\varepsilon)$ and
$f_{n}^{-1}(\mathcal{Q}\cap(1/l_{n})\mathbb{Z}^{2})$ is contained in the
$\varepsilon$-neighbourhood of $\mathcal{H}$.

######  Lemma 4.3.

Given $0<\varepsilon<\min\\{d(\mathcal{H},\partial
F^{-1}(W));d(\mathcal{Q},\partial
W);\displaystyle\min_{i<j}d(\mathcal{H}_{i},\mathcal{H}_{j})\\}$, there is a
positive integer $n_{0}=n_{0}(\varepsilon)$ such that for every $n\geq n_{0}$
the following holds:

  * i)

$f_{n}(\mathcal{H}\cap\mathcal{R}_{n})$ is contained in
$B(\mathcal{Q},\varepsilon)$;

  * ii)

$f_{n}^{-1}\left(\mathcal{Q}\cap(1/l_{n})\mathbb{Z}^{2}\right)\subset
B(\mathcal{H},\varepsilon)$.

###### Proof.

Assume that i) does not hold. Then there must exist an increasing sequence of
integers $k_{n}$ and a sequence of points
$x_{n}\in\mathcal{H}\cap\mathcal{R}_{k_{n}}$ such that $f_{k_{n}}(x_{n})$ does
not belong to $B(\mathcal{Q},\varepsilon)$. From the compactness of
$\mathcal{H}$ and after passing to a subsequence, we may assume that
$(x_{n})_{n\geq 1}$ converges to a point $x\in\mathcal{H}$. By the uniform
convergence of $\widehat{f}_{n}$ to $F$ (recall that $\widehat{f}_{n}$ was
defined in (3.3)), we have that $f_{k_{n}}(x_{n})\longrightarrow F(x)$. Since
$B(\mathcal{Q},\varepsilon)$ is an open set, $F(x)$ cannot belong to
$B(\mathcal{Q},\varepsilon)$. However, this contradicts the fact that
$\mathcal{H}\subset F^{-1}(B(\mathcal{Q},\varepsilon))$.

To prove ii) we proceed also by contradiction. Suppose that there exist an
increasing sequence of positive integers $(i_{n})_{n\in\mathbb{N}}$ and a
sequence of points $(u_{i_{n}})_{n\in\mathbb{N}}$ such that for every $n\geq
1$ we have that $u_{i_{n}}$ belongs to
$f_{i_{n}}^{-1}(\mathcal{Q}\cap(1/l_{i_{n}})\mathbb{Z}^{2})\setminus
B(\mathcal{H},\varepsilon)$. Observe that $(u_{i_{n}})_{n\in\mathbb{N}}$
converges, up to a subsequence, to an element $u\in I^{2}\setminus
B(\mathcal{H},\varepsilon)$. On the other hand, by the compactness of
$\mathcal{Q}$ and since $\widehat{f}_{n}$ converges uniformly to $F$, we have
that $f_{i_{n}}(u_{i_{n}})$ converges to $F(u)\in\mathcal{Q}$. However this
would imply that $u\in\mathcal{H}$, which is impossible since $u\in
I^{2}\setminus B(\mathcal{H},\varepsilon)$. ∎

From now on, for each natural number $n$ we consider $f_{n}$ as a map from
$\phi_{n}(\mathcal{D}_{\rho})$ to $(1/l_{n})\mathbb{Z}^{2}$; then the
extension of the restriction of $f_{n}$ over $\mathcal{R}_{n}$, namely
$\widehat{f_{n}|_{\mathcal{R}_{n}}}$, converges uniformly to $F$. Consider
$k\in\mathbb{N}$ satisfying that

$0<\frac{1}{k}<\min\left\\{\frac{d(\mathcal{H},\partial
F^{-1}(W))}{2\mathsf{Reg}(F)^{2}+1},\
\min_{i<j}d(\mathcal{H}_{i},\mathcal{H}_{j}),\ d(\mathcal{Q},\partial
W)\right\\}.$ (4.3)

For every non-negative integer $j$, let $\mathcal{Q}_{k,j}$ be the set points
$y\in B(\mathcal{Q},1/k)$ for which $d(y,\partial
B(\mathcal{Q},1/k))>jL/l_{n}$; notice that $\mathcal{Q}_{k,j}$ can be empty
for a sufficiently large positive integer $j$. The key result to prove
Proposition 4.2 is given by Lemma 4.7 below, which claims that for
sufficiently large positive integers $k,n$, there is an annulus $A_{k,n}$
centred at the origin and with external radius equal to the diameter of
$f_{n}^{-1}(\mathcal{Q}_{k,0})$, such that
$f_{n}^{-1}(\mathcal{Q}\cap(1/l_{n})\mathbb{Z}^{2})\cap A_{k,n}$ is mapped
under $f_{n}$ into a neighbourhood of $\partial B(\mathcal{Q},1/k)$.
Concretely, Lemma 4.7 is a consequence of the fact that points in the pre-
image under $f_{n}$ of a closed ball $B$ contained in $W$ which are distant
from the origin, must be mapped by $f_{n}$ “close” to the boundary of $B$;
this last fact (Lemma 4.4 below) was inspired by the proof of Lemma 6 in [4]
and relies strongly on the geometry of $\mathcal{D}_{\rho}$, namely, on the
$2\mathbb{Z}^{2}$-property.

For every $j\geq 0$ such that $\mathcal{Q}_{k,j}\neq\emptyset$, let
$x_{k,j}^{(n)}$ be a point in $f_{n}^{-1}(\mathcal{Q}_{k,j})$ with the
property that $||x_{k,j}^{(n)}||$ is maximal. Let
$i(n):=\max\\{i\in\mathbb{N}:\ \mathcal{Q}_{k,i}\neq\emptyset\\}$, i.e, $i(n)$
satisfies that the side-length of $\mathcal{Q}_{k,i(n)}$ is less or equal than
$2L/l_{n}$. We claim that the distance from $f(x_{k,j}^{(n)})$ to
$\partial\mathcal{Q}_{k,j}$ cannot be larger than $L/l_{n}$.

######  Lemma 4.4.

For every $j<i(n)$ there holds
$d(f_{n}(x_{k,j}^{(n)}),\partial\mathcal{Q}_{k,j})\leq\,L/l_{n}$.

###### Proof.

Indeed, if $d(f_{n}(x_{k,j}^{(n)}),\partial\mathcal{Q}_{k,j})>L/l_{n}$, then
$B(f_{n}(x_{k,j}^{(n)}),L/l_{n})$ is strictly contained in
$\mathcal{Q}_{k,j}$. On the other hand, the $2\mathbb{Z}^{2}$-property implies
that at least one of the points $x_{k,j}^{(n)}\pm e_{1,n},x_{k,j}^{(n)}\pm
e_{2,n},x_{k,j}^{(n)}\pm e_{1,n}\pm e_{2,n}$ (where $e_{1}=(1/l_{n},0)$ and
$e_{2}=(0,1/l_{n})$) belongs to $\phi_{n}(\mathcal{D}_{\rho})$ and lies at
distance from the origin larger than that $||x_{k,j}^{(n)}||$; see Figure 5
for a picture of this situation when $x_{k,j}^{(n)}$ lies to the left-hand
side of the square $B(||x_{k,j}^{(n)}||)$. Then, from the $L$-Lipschitz
condition, this point is mapped by $f_{n}$ into
$B(f(x_{k,j}^{(n)}),L/l_{n})\subset\mathcal{Q}_{k,j}$, contradicting the
choice of $x_{k,j}^{(n)}$.

$x_{k,j}^{(n)}$$B(||x_{k,j}^{(n)}||)$ Figure 5: A possible configuration of
$\phi_{n}(\mathcal{D}_{\rho})$ around $x_{k,j}^{(n)}$.

∎

######  Lemma 4.5.

For every positive integer $j<i(n)$ there holds
$||x_{k,j}^{(n)}||<||x_{k,j-1}^{(n)}||$. In particular the set
$\mathsf{Ann}((0,0),||x_{k,j}^{(n)}||,||x_{k,j-1}^{(n)}||)$ is non-empty.

###### Proof.

We proceed by contradiction. By construction of $x_{k,j}^{(n)}$, it is
sufficient to consider the case

$||x_{k,j}^{(n)}||=||x_{k,j-1}^{(n)}||.$ (4.4)

In this case by Lemma 4.4, the preceding equality (4.4) and the definition of
$x_{k,j}^{(n)}$, we get that

$d(f_{n}(x_{k,j-1}^{(n)}),\partial\mathcal{Q}_{k,j-1})\leq
L/l_{n}\qquad\text{and}\qquad
d(f_{n}(x_{k,j-1}^{(n)}),\partial\mathcal{Q}_{k,j})\leq L/l_{n}$

This implies that
$f_{n}(x_{k,j-1}^{(n)})\in\partial\mathcal{Q}_{k,j-1}\cap\mathcal{Q}_{k,j-1}$,
which is impossible since $\mathcal{Q}_{k,j-1}$ is an open set. ∎

Now we shall prove that $x_{k,i(n)}^{(n)}$ belongs to $I^{2}$ for a
sufficiently large $n\in\mathbb{N}$.

######  Lemma 4.6.

Given $k$ as in (4.3), there must exist a positive integer $n_{k}$ depending
on $k$ such that for every $n\geq n_{k}$ the point $x_{k,i(n)}^{(n)}$ belongs
to $\mathcal{H}$.

###### Proof.

By the bi-Lipschitz decomposition of $F$, there is a large-enough
$n_{k}\in\mathbb{N}$ such that for every $n\geq n_{k}$ there holds

$d(\partial F^{-1}(\mathcal{Q}_{k,i(n)}),\partial\mathcal{H})>\frac{1}{k}.$

Hence the claim follows by applying part ii) of Lemma 4.3 to
$\mathcal{Q}_{k,i(n_{k})}$, $\varepsilon=1/k$ and by observing that
$\mathcal{Q}_{k,i(n)}\subset\mathcal{Q}_{k,i(n_{k})}$ for every $n\geq n_{k}$.
∎

######  Lemma 4.7.

For every $0\leq j\leq i(n)$, the set
$\mathsf{Ann}((0,0),||x_{k,j}^{(n)}||,||x_{k,j-1}^{(n)}||)\cap
f_{n}^{-1}(\mathcal{Q}_{k,0})$ is mapped into
$\mathcal{Q}_{k,j-1}\setminus\mathcal{Q}_{k,j}$ under $f_{n}$.

###### Proof.

From the maximality of $||x_{k,1}^{(n)}||$, we get

$\mathsf{Ann}((0,0),||x_{k,1}^{(n)}||,||x_{k,0}^{(n)}||)\cap
f_{n}^{-1}(\mathcal{Q}_{k,0})\subset\mathcal{Q}_{k,0}\setminus\mathcal{Q}_{k,1}.$

Otherwise, there must exists $u\in f_{n}^{-1}(\mathcal{Q}_{k,0})$ with
$||u||>||x_{k,1}^{(n)}||$ such that $f_{n}(u)\in\mathcal{Q}_{k,1}$,
contradicting the choice of $x_{k,1}^{(n)}\in f_{n}^{-1}(\mathcal{Q}_{k,1})$.
Analogously, for every non-negative integer $j$ such that
$\mathcal{Q}_{k,j}\neq\emptyset$ there holds:

$\mathsf{Ann}((0,0),||x_{k,j}^{(n)}||,||x_{k,j-1}^{(n)}||)\cap
f_{n}^{-1}(\mathcal{Q}_{k,j-1})\subset\mathcal{Q}_{k,j-1}\setminus\mathcal{Q}_{k,j}.$

∎

In particular, from Lemma 4.7 we have that there are no points in
$\mathsf{Ann}((0,0),||x_{k,j}^{(n)}||,||x_{k,j-1}^{(n)}||)\cap
f_{n}^{-1}(\mathcal{Q}_{k,j-1})$ which are sent into
$\mathcal{Q}_{k,0}\setminus\mathcal{Q}_{k,j-1}$.

Henceforth, we consider the points $x_{k,j}:=x_{k,j}^{(n)}$ where the
$x_{k,j}^{(n)}$’s are the points as in the previous lemmas. For every
$n\in\mathbb{N}$ let $j(n)$ be the minimum positive integer with the property
that $x_{k,j}\in I^{2}$, which in fact must exists as a consequence of Lemma
4.6.

######  Remark 4.8.

If $\mathcal{Q}$ is centred at a point $y_{m}\in(1/l_{m})\mathbb{Z}^{2}$, then
from Lemma 4.7 we conclude that for a sufficiently large positive integer $n$,
the pre-image of the center of $\mathcal{Q}$ under $f_{n}$, namely
$f_{n}^{-1}(y_{m})$, belongs to $I^{2}$.

Notice that for every $k\in\mathbb{N}$ satisfying (4.3), the point
$x_{k,j(n_{k})}$ belongs to $B(\mathcal{H},(2\mathsf{Reg}(F)^{2}+1)/k)$ (where
$n_{k}$ is the positive integer given in the proof of Lemma 4.6), and hence
the sequence $(x_{k,j(n_{k})})_{k\geq 1}$ converges (under a subsequence) to
an element $x_{lim}\in\mathcal{H}$. In addition, the sequence of balls
$(\mathcal{Q}_{k,j(n_{k})})_{k\geq 0}$ converges in the Hausdorff distance
(under a subsequence) to a closed ball $\mathcal{M}\subset\mathcal{Q}$. We use
this fact to prove the next claim.

######  Lemma 4.9.

The set $\mathcal{H}$ is contained in $\overline{B}(||x_{lim}||)$. In
particular $x_{lim}$ belongs to $\partial\mathcal{H}$.

###### Proof.

Suppose there is a point $u\in
int(\mathcal{H}_{i})\setminus\overline{B}(||x_{lim}||)$ for some $1\leq i\leq
N$. Thereupon there must exist a positive number $\alpha$ such that
$\overline{B}(u,\alpha)$ is contained
$int(\mathcal{H}_{i})\setminus\overline{B}(||x_{lim}||)$. Thus, from Lemma 4.7
and by the definition of $j(n_{k})$, there is a sufficiently large
$k_{0}\in\mathbb{N}$ such that
$f_{n_{k}}(\overline{B}(u,\alpha)\cap\mathcal{R}_{n_{k}})$ is contained in
$\mathcal{Q}_{k,j(n_{k})-1}\setminus\mathcal{Q}_{k,j(n_{k})}$ for every $k\geq
k_{0}$. Hence, after passing to the limit, we obtain that
$F(\overline{B}(u,\alpha))\subset\partial\mathcal{M}$, i.e, $F$ maps a set
with positive Lebesgue measure into a set with zero-Lebesgue measure, which is
impossible since $F|_{\mathcal{H}_{i}}$ is a bi-Lipschitz homeomorphism. Thus,
$int(\mathcal{H})\subset\overline{B}(||x_{lim}||)$ and since each connected
component of $\mathcal{H}$ is homeomorphic to a closed ball, we conclude that
$\mathcal{H}\subset\overline{B}(||x_{lim}||)$. ∎

###  4.2 Proof of Proposition 3.2

In order to prove Proposition 4.2 we need to estimate the cardinality of the
set of points in $\mathcal{Q}\cap(1/l_{n})\mathbb{Z}^{2}$ which have no pre-
image under $f_{n}:\mathcal{R}_{n}\to(1/l_{n})\mathbb{Z}^{2}$. To treat these
points, we require the following result which corresponds to a slightly
different version of Lemma 3.1 in [7] and whose proof is analogous.

######  Lemma 4.10.

Let $U\subset\mathbb{R}^{d}$ be a closed set which is the image of $I^{d}$
under a bi-Lipschitz map and let $g:U\to\mathbb{R}^{d}$ be a homeomorphism and
$h:U\to\mathbb{R}^{d}$ be continuous. Then $h(U)\Delta g(U)$ is contained in
$\overline{B}(\partial g(U),||g-h||_{\infty})$, where $\Delta$ denotes the
symmetric difference.

Before showing Proposition 4.2, we introduce some terminology. For every
$n\in\mathbb{N}$ consider the normalized counting measures $\mu_{n}$ and
$\nu_{n}$ defined by

$\begin{split}\mu_{n}(A):=\dfrac{|A\cap\mathcal{R}_{n}|}{l_{n}^{2}},\hskip
28.45274pt\nu_{n}(C):=\dfrac{|C\cap\frac{1}{l_{n}}\mathbb{Z}^{2}|}{l_{n}^{2}}\end{split}$

Notice that $(\nu_{n}|_{F(I^{2})})_{n\in\mathbb{N}}$ converges weakly to the
Lebesgue measure in $F(I^{2})$. Moreover, it can be shown that $\mu_{n}$
converges weakly to the measure $\rho\lambda$ in $I^{2}$; the proof of these
facts follows the very same lines as Claims 5.3.1 and 5.3.2 in [8].

######  Lemma 4.11.

The sequences of measures $(\mu_{n})_{n\in\mathbb{N}}$ and
$(\nu_{n})_{n\in\mathbb{N}}$ converge weakly to $\rho\lambda$ and $\lambda$,
respectively. In particular, $(\widehat{f}_{n})_{\\#}(\mu_{n})$ converges
weakly to $F_{\\#}(\rho\lambda)$.

Now we are in the position to prove Proposition 4.2. We prove this by showing
that the loss of mass in $\mathcal{Q}$ only happens close to its boundary.
Recall that $\mathcal{Q}$ is centred at a point
$y=y_{m}\in(1/l_{m})\mathbb{Z}^{2}$ for some $m\in\mathbb{N}$; since
$(l_{n})_{n\geq 1}$ satisfy that $l_{n}$ divides to $l_{n+1}$, then
$x\in(1/l_{n})\mathbb{Z}^{2}$ for every $n\geq m$.

###### Proof of Proposition 4.2.

To verify that $F_{\\#}(\rho\lambda)|_{\mathcal{Q}}=\lambda|_{\mathcal{Q}}$,
by Lemma 4.11 it is sufficient to show that the sequence
$(\hat{f}_{n_{k}}|_{F^{-1}(\mathcal{Q})})_{\\#}\mu_{n_{k}}$ converges weakly
to $\lambda|_{\mathcal{Q}}$. We proceed in a similar way as in the proof of
Lemma 3.4 in [7], but with some variations along our demonstration. By
definition of weak convergence of measures it is sufficient to prove that for
a given function $\varphi\in C_{0}(I^{2},\mathbb{R})$, the expression

$\left|\displaystyle\int_{\mathcal{Q}}\varphi
d\nu_{n_{k}}-\displaystyle\int_{\widehat{f}_{n_{k}}(F^{-1}(\mathcal{Q}))}\varphi
d(\widehat{f}_{n_{k}}|_{F^{-1}(\mathcal{Q})})_{\\#}\mu_{n_{k}}\right|$ (4.5)

tends to $0$ when $k$ goes to $+\infty$. Observe that by the triangle
inequality, the expression (4.5) can be bounded from above by

$\underbrace{\left|\displaystyle\int_{\mathcal{Q}}\varphi
d\nu_{n_{k}}-\displaystyle\int_{\widehat{f}_{n_{k}}(F^{-1}(\mathcal{Q}))}\varphi
d\nu_{n_{k}}\right|}_{=T_{1,n_{k}}}+\underbrace{\left|\displaystyle\int_{\widehat{f}_{n_{k}}(F^{-1}(\mathcal{Q}))}\varphi
d\nu_{n_{k}}-\displaystyle\int_{\widehat{f}_{n_{k}}(F^{-1}(\mathcal{Q}))}\varphi
d(\widehat{f}_{n_{k}}|_{F^{-1}(\mathcal{Q})})_{\\#}\mu_{n_{k}}\right|}_{=T_{2,n_{k}}};$
(4.6)

we denote by $T_{1,n_{k}}$ and $T_{2,n_{k}}$ the first and the second term in
(4.6), respectively.

Notice that $T_{1,n_{k}}$ is at most
$||\varphi||_{\infty}\nu_{n_{k}}(\mathcal{Q}\Delta\hat{f}_{n_{k}}(F^{-1}(\mathcal{Q})))$.
To show $T_{1,n_{k}}\longrightarrow 0$, observe that

$\begin{split}\mathcal{Q}\Delta\widehat{f}_{n_{k}}(F^{-1}(\mathcal{Q}))&\subset\bigcup_{i=1}^{N}F(\mathcal{H}_{i})\Delta\widehat{f}_{n_{k}}(\mathcal{H}_{i})\\\
&\subset\bigcup_{i=1}^{N}\overline{B}(\partial
F(\mathcal{H}_{i}),||F-\widehat{f}_{n_{k}}||_{\infty}),\end{split}$

where in the second step we use Lemma 4.10 for $U=\mathcal{H}_{i}$. Thus, by
using the weak convergence of $\nu_{n_{k}}$ to $\lambda$, it follows that

$\nu_{n_{k}}(\mathcal{Q}\Delta\widehat{f}_{n_{k}}(F^{-1}(\mathcal{Q})))\leq\nu_{n_{k}}(\overline{B}(\partial\mathcal{Q},||F-\widehat{f}_{n_{k}}||_{\infty}))\longrightarrow
0.$

Therefore, the first term in (4.6) converges to 0.

To prove that $T_{2,n_{k}}$ tends to 0 we firstly note that this expression
can be bounded from above by

$\dfrac{||\varphi||_{\infty}}{l_{n_{k}}^{2}}|A_{n_{k}}|,$ (4.7)

where
$A_{n}:=\widehat{f}_{n}(\mathcal{H})\cap\frac{1}{l_{n}}\mathbb{Z}^{2}\setminus
f_{n}(\mathcal{H})$. Observe that $\widehat{f}_{n_{k}}(\mathcal{H})$ is
contained in $B(\mathcal{Q},||F-\widehat{f}_{n_{k}}||_{\infty})$ and thus

$A_{n_{k}}\subset\left(B(\mathcal{Q},||F-\widehat{f}_{n_{k}}||_{\infty})\cap\frac{1}{l_{n_{k}}}\mathbb{Z}^{2}\right)\setminus
f_{n_{k}}(\mathcal{H}\cap B(||x_{k,j(n_{k})}||)),$ (4.8)

where the set $\mathcal{H}\cap\mathcal{R}_{n_{k}}\cap B(||x_{k,j(n_{k})}||)$
is non-empty for a large-enough positive integer $k$, as a consequence of
Lemmas 4.7 and 4.9. Thus by the injectivity of $f_{n_{k}}$ and by the part i)
of Lemma 4.3 which says that $\mathcal{H}\cap\mathcal{R}_{n_{k}}$ is contained
in $f_{n_{k}}^{-1}(\mathcal{Q}_{k,0})\cap I^{2}$ for a sufficiently large $k$,
we get

$\begin{split}f_{n_{k}}(\mathcal{H}\cap
B(||x_{k,j(n_{k})}||))=f_{n_{k}}(f_{n_{k}}^{-1}(\mathcal{Q}_{k,0})\cap
B(||x_{k,j(n_{k})}||))\setminus
f_{n_{k}}(f_{n_{k}}^{-1}(\mathcal{Q}_{k,0})\cap
B(||x_{k,j(n_{k})}||)\setminus\mathcal{H})\end{split}$ (4.9)

(recall that by the choice of $k$ in (4.3), the set
$f_{n_{k}}^{-1}(\mathcal{Q}_{k,0})\cap\mathcal{R}_{n_{k}}$ is contained
$F^{-1}(W)$). Thus, from (4.8), (4.9) and Lemma 4.7, after taking cardinality
and dividing by $l_{n_{k}}^{2}$ we get:

$\begin{split}\frac{|A_{n_{k}}|}{l_{n_{k}}^{2}}\leq&\frac{1}{l_{n_{k}}^{2}}|f_{n_{k}}(f_{n_{k}}^{-1}(\mathcal{Q}_{k,0})\cap
B(||x_{k,j(n_{k})}||)\setminus\mathcal{H})|+\nu_{n_{k}}(B(\mathcal{Q},||F-\widehat{f}_{n_{k}}||_{\infty})\setminus\mathcal{Q}_{k,0})\\\
&+\nu_{n_{k}}(\mathcal{Q}_{k,0}\setminus\mathcal{Q}_{k,j(n_{k})}),\end{split}$
(4.10)

where
$B(\mathcal{Q},||F-\widehat{f}_{n_{k}}||_{\infty})\setminus\mathcal{Q}_{k,0}$
can be empty or not. By Lemma 4.3 and from the choice of $k$ in (4.3), the
first term in the right-hand side of (4.10) can be estimated as below:

$\begin{split}\frac{1}{l_{n_{k}}^{2}}|f_{n_{k}}(f_{n_{k}}^{-1}(\mathcal{Q}_{k,0})\cap
B(||x_{k,j(n_{k})}||)\setminus\mathcal{H})|&=\frac{1}{l_{n_{k}}^{2}}|f_{n_{k}}^{-1}(\mathcal{Q}_{k,0})\cap
B(||x_{k,j(n_{k})}||)\cap\mathcal{R}_{n_{k}}\setminus\mathcal{H}|\\\
&\leq\mu_{n_{k}}\left(B\left(\partial\mathcal{H},\frac{1}{k}\right)\right)\\\
&\longrightarrow\rho\lambda(\partial\mathcal{H})=0\end{split}$

If
$B(\mathcal{Q},||F-\widehat{f}_{n_{k}}||_{\infty})\subset\mathcal{Q}_{k,0}$,
then the second term in (4.10) is zero. In the other possible case, i.e if
$\mathcal{Q}_{k,0}\subset B(\mathcal{Q},||F-\widehat{f}_{n_{k}}||_{\infty})$,
we get

$\nu_{n_{k}}(B(\mathcal{Q},||F-\widehat{f}_{n_{k}}||_{\infty})\setminus\mathcal{Q}_{k,0})\leq\nu_{n_{k}}\left(B\left(\mathcal{Q},||F-\widehat{f}_{n_{k}}||_{\infty}+\frac{1}{k}\right)\right)\longrightarrow
0$

Finally, to estimate the most delicate term in (4.10), we observe that:

$\begin{split}\nu_{n_{k}}(\mathcal{Q}_{k,0}\setminus\mathcal{Q}_{k,j(n_{k})})&\leq\lambda\left(B\left(\mathcal{Q}_{k,0}\setminus\mathcal{Q}_{k,j(n_{k})},\frac{1}{l_{n_{k}}}\right)\right)\\\
&\leq\sum_{j=1}^{j(n_{k})}\lambda(\mathcal{Q}_{k,j-1}\setminus\mathcal{Q}_{k,j})+\lambda\left(B\left(\partial\mathcal{Q}_{k,0},\frac{1}{l_{n_{k}}}\right)\right)+\lambda\left(B\left(\partial\mathcal{Q}_{k,j(n_{k})},\frac{1}{l_{n_{k}}}\right)\right)\\\
&\leq\frac{L}{l_{n_{k}}}\left(2l_{\mathcal{Q}}+\frac{2}{k}-\frac{L}{l_{n_{k}}}\right)j(n_{k})+\lambda\left(B\left(\partial\mathcal{Q}_{k,0},\frac{1}{l_{n_{k}}}\right)\right)\\\
&\
+\lambda\left(B\left(\partial\mathcal{Q}_{k,j(n_{k})},\frac{1}{l_{n_{k}}}\right)\right),\end{split}$

where the bound $l_{\mathcal{Q}}$ denotes the side-length of $\mathcal{Q}$. On
one hand, it is direct that

$\lim_{k\to\infty}\lambda\left(B\left(\partial\mathcal{Q}_{k,0},\frac{1}{l_{n_{k}}}\right)\right)=\lim_{k\to\infty}\lambda\left(B\left(\partial\mathcal{Q}_{k,j(n_{k})},\frac{1}{l_{n_{k}}}\right)\right)=0.$

On the other hand, by the construction of $x_{k,j(n_{k})}$ and by Lemma 4.6 we
have that $j(n_{k})\leq||x_{k,0}||$. Moreover, by Remark 4.8 and since $f$ is
$\omega$-co-uniformly continuous, there holds that

$||x_{k,0}||\leq\frac{\omega\left(\left(\frac{l_{\mathcal{Q}}}{2}+\frac{1}{k}\right)l_{n_{k}}\right)}{l_{n_{k}}}+1.$

Hence, since the co-uniformity is of order $o(r^{2})$, we have that

$\frac{L}{l_{n_{k}}}\left(2l_{\mathcal{Q}}+\frac{2}{k}-\frac{L}{l_{n_{k}}}\right)j(n_{k})\leq
L\left(2l_{\mathcal{Q}}+\frac{2}{k}-\frac{L}{l_{n_{k}}}\right)\left(\frac{\omega\left(\left(\frac{l_{\mathcal{Q}}}{2}+\frac{1}{k}\right)l_{n_{k}}\right)}{l_{n_{k}}^{2}}+\frac{1}{l_{n_{k}}}\right)\longrightarrow
0.$

Thus (4.7) converges to 0 and therefore $T_{2,n_{k}}\longrightarrow 0$ when
$k\to\infty$, as desired. ∎

Acknowledgments. This work was funded by the ANID/FONDECYT Postdoctoral Grant
3210109. I would like to thank Andrés Navas for the enlightening discussions
and Mircea Petrache for his comments. I also thank the anonymous referee for
carefully reading this paper and helpful comments.

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Rodolfo Viera

Facultad de Matemáticas

Pontificia Universidad Católica de Chile

Av. Vicuña Mackenna 4860, Macul, Santiago, Chile

Email<EMAIL_ADDRESS>