# Amorphic complexity of group actions with applications to quasicrystals

Gabriel Fuhrmann Department of Mathematical Sciences, Durham University, UK.
Email<EMAIL_ADDRESS>Maik Gröger Faculty of Mathematics,
University of Vienna, Austria & Faculty of Mathematics and Computer Science,
Jagiellonian University in Kraków, Poland. Email<EMAIL_ADDRESS>Tobias Jäger Department of Mathematics, University of Jena, Germany. Email:
<EMAIL_ADDRESS>Dominik Kwietniak Faculty of Mathematics and
Computer Science, Jagiellonian University in Kraków, Poland. Email:
<EMAIL_ADDRESS>
###### Abstract

In this article, we define amorphic complexity for actions of locally compact
$\sigma$-compact amenable groups on compact metric spaces. Amorphic
complexity, originally introduced for $\mathbb{Z}$-actions, is a topological
invariant which measures the complexity of dynamical systems in the regime of
zero entropy. We show that it is tailor-made to study strictly ergodic group
actions with discrete spectrum and continuous eigenfunctions. This class of
actions includes, in particular, Delone dynamical systems related to regular
model sets obtained via Meyer’s cut and project method. We provide sharp upper
bounds on amorphic complexity of such systems. In doing so, we observe an
intimate relationship between amorphic complexity and fractal geometry.

##  1 Introduction

The study of low-complexity notions for group actions is both a timely and a
classical topic. Its roots go back to Halmos, McKay, and von Neumann who
classified actions with discrete spectrum, as well as Auslander, Ellis,
Furstenberg, and Veech who set the foundations of the theory of equicontinuous
actions and their extensions. Recent years have seen plenty of progress in
illuminating the richness of possible dynamical behaviour of minimal actions
of general groups in the low complexity regime, see for example [Kri07, CP08,
CM16, ST17, Gla18, ŁS18, FK20]. As a matter of fact, the investigation of this
regime not only contributes to the understanding of group actions as such but
is of fundamental importance in the understanding of aperiodic order—with
further applications to geometry, number theory and harmonic analysis [Mey72,
BG13]—and the diffraction spectra of so-called Delone sets, that is,
mathematical models of physical quasicrystals. The latter results from the
observation that diffraction spectra of Delone sets can be studied by means of
certain associated Delone dynamical systems [LM06, BLM07, Len09], see also
[BG13] for further information and references. Analysing these Delone
dynamical systems, it is most natural to ask when two such systems are
conjugate [KS14]. The standard operating procedure to answer this question
clearly is to utilize dynamical invariants and one might be tempted to study
topological entropy of Delone dynamics. However, the physically most
interesting case of pure point diffraction turns out to necessarily come with
zero entropy [BLR07]. There is hence a need for finer topological invariants
which can distinguish zero entropy systems.

In this article, we propose amorphic complexity—a notion recently introduced
for $\mathbb{Z}$-actions [FGJ16]—as a promising candidate for this purpose. To
that end, we extend amorphic complexity to actions of locally compact,
$\sigma$-compact and amenable groups. We will see that amorphic complexity is
tailor-made to study strictly ergodic systems with discrete spectrum and
continuous eigenfunctions, that is, minimal mean equicontinuous systems
[FGL21, Corollary 1.6]. Most importantly, however, we show that amorphic
complexity is not only theoretically well-behaved but also well-computable in
specific examples. This is particularly true due to a neat connection to
fractal geometry. We elaborate on this in the last section of this article
where we apply our findings to model sets—particular Delone sets constructed
by means of Meyer’s cut and project method [Mey72].

Before we introduce amorphic complexity and discuss our main results in more
detail, let us briefly clarify some basic terminology. A triple $(X,G,\alpha)$
is called a _(topological) dynamical system_ if $X$ is a compact metric space
(endowed with a metric $d$), $G$ is a topological group and $\alpha$ is a
continuous action of $G$ on $X$ by homeomorphisms (continuity of $\alpha$ is
understood as continuity of the map $G\times X\ni(g,x)\mapsto\alpha(g)(x)\in
X$). In the following, we use the shorthand $gx$ instead of $\alpha(g)(x)$ for
the action of $g\in G$ on $x\in X$ via $\alpha$. Likewise, we may occasionally
keep the action $\alpha$ implicit and simply refer to $(X,G)$ as a dynamical
system.

As mentioned before, we throughout assume that $G$ is locally compact,
$\sigma$-compact and amenable. Recall that there is hence a _(left) Følner
sequence_ , that is, a sequence $(F_{n})_{n\in\mathbb{N}}$ of compact subsets
of $G$ having positive Haar measure such that

$\displaystyle\lim_{n\to\infty}\frac{m(KF_{n}\triangle
F_{n})}{m(F_{n})}=0\quad\textnormal{for every compact }K\subseteq G,$ (1)

where $\triangle$ denotes the symmetric difference and $m$ is a _(left) Haar
measure_ of $G$ (we may synonymously write $\left|F\right|$ for the Haar
measure $m(F)$ of a measurable set $F\subseteq G$) [EG67, Theorem 3.2.1]. We
will also make use of the existence of _right Følner sequences_ which fulfil a
condition analogous to (1) with the left Haar measure and the multiplication
from the left replaced by the right Haar measure and multiplication from the
right, respectively. However, we would like to stress that in the following,
each Følner sequence is assumed to be a left Følner sequence if not stated
otherwise. Given a (left or right) Følner sequence $\mathcal{F}=(F_{n})$, the
_(upper) asymptotic density_ of a measurable subset $E\subseteq G$ with
respect to $\mathcal{F}$ is defined as

$\mathrm{ad}_{\mathcal{F}}(E)=\varlimsup\limits_{n\to\infty}\frac{\left|E\cap
F_{n}\right|}{\left|F_{n}\right|}.$ (2)

Let us next turn to the definition of amorphic complexity of a dynamical
system $(X,G)$ with respect to a Følner sequence
$\mathcal{F}=(F_{n})_{n\in\mathbb{N}}$ in $G$. Given $x,y\in X$, $\delta>0$,
we set

$\Delta(X,G,\delta,x,y)=\left\\{t\in G\;|\;d(tx,ty)\geq\delta\right\\}.$

For $\nu\in(0,1]$, we say that $x$ and $y$ are _$(\delta,\nu)$ -separated_
with respect to $\mathcal{F}$ if

$\mathrm{ad}_{\mathcal{F}}(\Delta(X,G,\delta,x,y))=\varlimsup_{n\to\infty}\frac{\left|\Delta(X,G,\delta,x,y)\cap
F_{n}\right|}{\left|F_{n}\right|}\geq\nu.$

Accordingly, a subset $S\subseteq X$ is said to be _$(\delta,\nu)$ -separated_
with respect to $\mathcal{F}$ if all distinct points $x,y\in S$ are
$(\delta,\nu)$-separated. This already yields the first key notion in this
work: the (asymptotic) separation number of $(X,G)$ with respect to $\delta>0$
and $\nu\in(0,1]$, denoted by $\mathrm{Sep}_{\mathcal{F}}(X,G,\delta,\nu)$, is
the supremum over the cardinalities of all $(\delta,\nu)$-separated sets in
$X$.

In general, the asymptotic separation numbers do not have to be finite (even
though $X$ is compact) which immediately gives the following dichotomy: if
$\mathrm{Sep}_{\mathcal{F}}(X,G,\delta,\nu)$ is finite for all $\delta,\nu>0$,
we say $(X,G)$ has finite separation numbers with respect to $\mathcal{F}$
otherwise, we say it has infinite separation numbers. Our first main
result—consisting of the following two theorems whose proofs are given in
Section 3—identifies canonical classes of group actions with infinite and
finite separation numbers, respectively. First, we give two criteria for
infinite separation numbers.

######  Theorem 1.1.

If $(X,G)$ is weakly mixing with respect to a non-trivial $G$-invariant
probability measure, then $(X,G)$ has infinite separation numbers with respect
to every Følner sequence. Likewise, if $G$ allows for a uniform lattice and
$(X,G)$ has positive topological entropy, then $(X,G)$ has infinite separation
numbers with respect to every Følner sequence.

In the opposite direction, it turns out that in the minimal case finite
separation numbers can be used to characterize mean equicontinuity.

######  Theorem 1.2.

Let $G$ be a unimodular group, meaning it has a sequence which is a left and a
right Følner sequence (this holds, in particular, if $G$ is abelian). Further,
suppose $(X,G)$ is a minimal dynamical system. Then $(X,G)$ has finite
separation numbers with respect to every Følner sequence if and only if
$(X,G)$ is mean equicontinuous.

It is worth mentioning that the class of mean equicontinuous systems comprises
all Delone dynamical systems associated to regular model sets, see also
Section 5. For further examples of mean equicontinuous actions of groups
different from $\mathbb{Z}$, we refer the reader to the literature [Rob96,
Rob99, Cor06, Vor12, GR17, Gla18, ŁS18, FK20, GL20, FGL21].

If $(X,G)$ has finite separation numbers, we are in a position to obtain finer
information by studying the scaling behaviour of the separation numbers as the
separation frequency $\nu$ tends to zero. Here, we may in principle consider
arbitrary growth rates. So far, however, previous results indicate that
polynomial growth is the most relevant, see [FGJ16, GJ16, FG20] for
$G=\mathbb{Z}$. With this in mind, we define the _lower_ and _upper amorphic
complexity_ of $(X,G)$ with respect to $\mathcal{F}$ as

$\underline{\mathrm{ac}}_{\mathcal{F}}(X,G)=\adjustlimits{\sup}_{\delta>0}{\varliminf}_{\nu\to
0}\frac{\log\mathrm{Sep}_{\mathcal{F}}(X,G,\delta,\nu)}{-\log\nu}\quad\textnormal{and}\quad\overline{\mathrm{ac}}_{\mathcal{F}}(X,G)=\adjustlimits{\sup}_{\delta>0}{\varlimsup}_{\nu\to
0}\frac{\log\mathrm{Sep}_{\mathcal{F}}(X,G,\delta,\nu)}{-\log\nu}.$

In case that both values coincide, we call
$\mathrm{{ac}}_{\mathcal{F}}(X,G)=\underline{\mathrm{ac}}_{\mathcal{F}}(X,G)=\overline{\mathrm{ac}}_{\mathcal{F}}(X,G)$
the amorphic complexity of $(X,G)$ with respect to $\mathcal{F}$. It is
convenient to set $\mathrm{{ac}}_{\mathcal{F}}(X,G)=\infty$ if $(X,G)$ has
infinite separation numbers with respect to $\mathcal{F}$. We discuss the most
basic properties of amorphic complexity—including its invariance under
conjugacy—in Section 2.

Our second main result deals with the problem as to which extent the
asymptotic separation numbers and amorphic complexity depend on the particular
Følner sequence $\mathcal{F}$. In general, we cannot rule out different
amorphic complexities with respect to different Følner sequences. In fact,
this problem already occurs when $G=\mathbb{Z}$, see Section 4. With the next
theorem, however, we provide a sufficient criterion for the independence from
$\mathcal{F}$. Here, we say a dynamical system $(X,G)$ is _pointwise uniquely
ergodic_ if every orbit closure is uniquely ergodic. A strengthening of the
following statement and its proof can be found in Section 4.

######  Theorem 1.3.

Let $(X,G)$ be a dynamical system whose product $(X^{2},G)$ is pointwise
uniquely ergodic. Then $(X,G)$ has infinite separation numbers with respect to
some Følner sequence if and only if it has infinite separation numbers with
respect to all Følner sequences. Moreover,
$\overline{\mathrm{ac}}_{\mathcal{F}}(X,G)$ and
$\underline{\mathrm{ac}}_{\mathcal{F}}(X,G)$ are independent of the particular
Følner sequence $\mathcal{F}$.

It is worth mentioning that mean equicontinuous systems verify the assumptions
of the above theorem [FGL21, Theorem 1.2].

With our third main result, we apply amorphic complexity to the dynamics of
regular model sets. Before we come to the precise formulation, we need to
introduce some terminology. In doing so, we restrict to a rather brief
description of the most essential notions and refer the reader to Section 5
for the details. A _cut and project scheme_ is a triple $(G,H,\mathcal{L})$,
where $G$ and $H$ are locally compact abelian groups and $\mathcal{L}$ is an
irrational lattice in $G\times H$. Together with a compact subset
$W=\overline{\operatorname{int}(W)}\subseteq H$—referred to as a _window_
—$(G,H,\mathcal{L})$ defines a particular instance of a Delone set, a so-
called _model_ set

$\mbox{\Large$\curlywedge$}(W)=\pi_{G}((G\times W)\cap\mathcal{L}),$

where $\pi_{G}:G\times H\to G$ denotes the canonical projection. We call $W$
as well as $\mbox{\Large$\curlywedge$}(W)$ _regular_ if $\partial W$ is of
zero Haar measure and say $W$ is _irredundant_ if $\\{h\in
H\;|\;h+W=W\\}=\\{0\\}$. Now, as $\mbox{\Large$\curlywedge$}(W)$ is a subset
of $G$, $G$ naturally acts on $\mbox{\Large$\curlywedge$}(W)$ by translations.
It turns out that the closure of all translated copies of
$\mbox{\Large$\curlywedge$}(W)$ is compact (in a suitable topology on subsets
of $G$). Denoting this closure by $\Omega(\mbox{\Large$\curlywedge$}(W))$, we
arrive at the Delone dynamical system
$(\Omega(\mbox{\Large$\curlywedge$}(W)),G)$ associated to the model set
$\mbox{\Large$\curlywedge$}(W)$. We obtain

######  Theorem 1.4.

Let $(G,H,\mathcal{L})$ be a cut and project scheme with $W\subseteq H$ a
regular irredundant window and suppose $G$ and $H$ are second countable. Then
for every Følner sequence $\mathcal{F}$ in $G$, we get

$\overline{\mathrm{ac}}_{\mathcal{F}}(\Omega(\mbox{\Large$\curlywedge$}(W)),G)\leq\frac{\overline{\mathrm{dim}}_{\mathrm{B}}(H)}{\overline{\mathrm{dim}}_{\mathrm{B}}(H)-\overline{\mathrm{dim}}_{\mathrm{B}}(\partial
W)},$

assuming that $\overline{\mathrm{dim}}_{\mathrm{B}}(H)$ is finite.

Here, $\overline{\mathrm{dim}}_{\mathrm{B}}(\cdot)$ denotes the upper box
dimension, see Section 5 for the details. Let us remark that we further show
that the above estimates are sharp in that they are realised by particular
model sets. In conclusion, we obtain that every value in $[1,\infty)$ can be
attained by amorphic complexity of minimal systems.

Motivated by the above results, we finish with the following question.

Given a locally compact, $\sigma$-compact and amenable group acting minimally
on a compact metric space. Which values can amorphic complexity attain?

In particular, for minimal $\mathbb{Z}$\- or $\mathbb{R}$-actions, we
conjecture that amorphic complexity cannot take values in $(0,1)$. Indeed,
this complexity gap was recently established for subshifts associated to
primitive constant length substitutions [FG20] and is a classical phenomenon
which is well known to occur for polynomial entropy of minimal symbolic
subshifts. For non-minimal $\mathbb{Z}$-actions, however, it was recently
shown that all values in $(0,1)$ can be obtained by amorphic complexity, see
[Kul20, Kul].

#### Acknowledgments

This project has received funding from the European Union’s Horizon 2020
research and innovation program under the Marie Skłodowska-Curie grant
agreement No 750865. Furthermore, it received support by the DFG Emmy-Noether
grant Ja 1721/2-1, DFG Heisenberg grant Oe 538/6-1 and DFG Research Fellowship
grant GR 4899/1-1. DK was supported by the National Science Centre, Poland,
grant no. 2018/29/B/ST1/01340. GF, MG and DK would like to thank the
Mathematisches Forschungsinstitut Oberwolfach for its enormous hospitality
during a Research in Pairs stay (R1721) at the MFO in October 2017 where many
ideas of this work were developed. This work was finished during a visit of GF
and MG to the Jagiellonian University in Kraków in September 2020, which was
also supported by the National Science Centre, Poland, grant no.
2018/29/B/ST1/01340.

##  2 Basic properties of amorphic complexity

In this section, we collect the most basic properties of amorphic complexity.
In particular, given a group $G$ which allows for a lattice $\mathcal{L}$, we
discuss how amorphic complexity of a $G$-action relates to amorphic complexity
of the associated $\mathcal{L}$-action.

The proof of the following statement is verbatim as the proofs of [FGJ16,
Proposition 3.4 & Proposition 3.9].

######  Proposition 2.1.

Let $(X,G)$ and $(Y,G)$ be dynamical systems. We have:

  1. (a)

If $(Y,G)$ is a factor of $(X,G)$, then

$\underline{\mathrm{ac}}_{\mathcal{F}}(Y,G)\leq\underline{\mathrm{ac}}_{\mathcal{F}}(X,G)\quad\textnormal{and}\quad\overline{\mathrm{ac}}_{\mathcal{F}}(Y,G)\leq\overline{\mathrm{ac}}_{\mathcal{F}}(X,G).$

In particular, (upper and lower) amorphic complexity is a topological
invariant.

  2. (b)

We have that

$\displaystyle\underline{\mathrm{ac}}_{\mathcal{F}}(X\times
Y,G)\geq\underline{\mathrm{ac}}_{\mathcal{F}}(X,G)+\underline{\mathrm{ac}}_{\mathcal{F}}(Y,G),\quad\overline{\mathrm{ac}}_{\mathcal{F}}(X\times
Y,G)\leq\overline{\mathrm{ac}}_{\mathcal{F}}(X,G)+\overline{\mathrm{ac}}_{\mathcal{F}}(X,G).$

In particular, if $\mathrm{{ac}}_{\mathcal{F}}(X,G)$ and
$\mathrm{{ac}}_{\mathcal{F}}(Y,G)$ exist, then
$\mathrm{{ac}}_{\mathcal{F}}(X\times Y,G)$ exists as well.

Before we proceed with further properties of amorphic complexity, we take a
closer look at certain particularly well-behaved Følner sequences. Recall that
a _van Hove sequence_ $(A_{n})_{n\in\mathbb{N}}$ in $G$ is a sequence of
compacta $A_{n}\subseteq G$ of positive Haar measure such that

$\lim_{n\to\infty}\frac{m\big{(}\partial_{K}A_{n})}{m(A_{n})}=0,$

for every compact set $K\subseteq G$ with $e\in K$, where
$\partial_{K}A_{n}\coloneqq
KA_{n}\setminus\operatorname{int}\big{(}\bigcap_{g\in K}gA_{n}\big{)}$ (see
[Tem92, Appendix 3] and [Str05] for further reference). It is not hard to see
that every van Hove sequence is a Følner sequence. In fact, it holds

######  Proposition 2.2 ([Tem92, Appendix 3.K]).

Let $G$ be a locally compact $\sigma$-compact amenable topological group. A
sequence $(A_{n})$ of compact subsets of $G$ is a van Hove sequence if and
only if it is a Følner sequence and

$\displaystyle\lim_{n\to\infty}\frac{m(\partial_{U}A_{n})}{m(A_{n})}=0,$ (3)

for some open neighbourhood $U$ of the neutral element $e$ in $G$.

######  Remark 2.3.

In particular, if $G$ is discrete, then every Følner sequence in $G$ is, in
fact, a van Hove sequence.

It is well known that every locally compact $\sigma$-compact amenable group
allows for a van Hove sequence. For the convenience of the reader, we prove
the following (possibly well-known) refinement of this statement which we need
in the sequel.

######  Proposition 2.4.

Let $G$ be a locally compact $\sigma$-compact amenable topological group.
Suppose $(F_{n})$ is a Følner sequence in $G$ and $B$ is a compact
neighbourhood of $e$. Then $A_{n}\coloneqq BF_{n}$ defines a van Hove sequence
in $G$ with $\mathrm{ad}_{(A_{n})}(E)=\mathrm{ad}_{(F_{n})}(E)$ for every
measurable $E\subseteq G$.

###### Proof.

The last part follows from $E\cap A_{n}\subseteq(E\cap
F_{n})\cup(F_{n}\triangle A_{n})$ and

$\displaystyle 0\leq\lim_{n\to\infty}m(A_{n}\triangle
F_{n})/m(A_{n})\leq\lim_{n\to\infty}m(BF_{n}\triangle F_{n})/m(F_{n})=0,$ (4)

which is a consequence of the fact that $(F_{n})$ is a Følner sequence and
$F_{n}\subseteq BF_{n}=A_{n}$.

For the first part, we make use of Proposition 2.2. To that end, observe that
for every (compact) $K\subseteq G$ we have $KA_{n}\triangle
A_{n}\subseteq(KA_{n}\triangle F_{n})\cup(F_{n}\triangle
A_{n})=(KBF_{n}\triangle F_{n})\cup(F_{n}\triangle A_{n})$. Due to (4) and the
fact that $(F_{n})$ is a Følner sequence, this gives that $(A_{n})$ is a
Følner sequence, too. To see (3), we need the following

######  Claim 2.5.

There is a relatively compact open neighbourhood $U$ of $e$ such that
$F_{n}\subseteq\operatorname{int}\big{(}\bigcap_{g\in U}gA_{n}\big{)}$ for
each $n\in\mathbb{N}$.

###### Proof of the claim.

First, observe that $\operatorname{int}\big{(}\bigcap_{g\in
U}gBF_{n}\big{)}\supseteq\operatorname{int}\big{(}\bigcap_{g\in
U}gB\big{)}F_{n}$. To prove the claim, it hence suffices to show that there is
$U$ with $e\in\operatorname{int}\big{(}\bigcap_{g\in U}gB\big{)}$.

For a contradiction, suppose $e\in\overline{\bigcup_{g\in U}gB^{c}}$ for every
$U$ in the open neighbourhood filter $\mathcal{U}$ of $e$. In other words,
suppose there is a net $(g_{U})_{U\in\mathcal{U}}$ with $g_{U}\in U$ (so that
$g_{U}\to e$) and a net $(h_{U})_{U\in\mathcal{U}}$ in $B^{c}$ such that
$g_{U}h_{U}\to e$. This, however, implies $h_{U}\to e$ which contradicts
$e\in\operatorname{int}(B)$. Therefore, there is $U\in\mathcal{U}$ with
$e\in\operatorname{int}\big{(}\bigcap_{g\in U}gB\big{)}$. Clearly, $U$ can be
chosen open and relatively compact. $\circ$

Now, pick some $U$ as in the above claim. As $(F_{n})$ is a Følner sequence,
we have

$\displaystyle m(\partial_{U}A_{n})/m(A_{n})\leq m(UA_{n}\setminus
F_{n})/m(F_{n})\leq m(\overline{U}BF_{n}\setminus
F_{n})/m(F_{n})\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}0.$

Finally, it follows from Proposition 2.2 that $(A_{n})$ is a van Hove
sequence. ∎

For the next statement, recall that a _uniform lattice_ $\mathcal{L}$ in $G$
is a discrete subgroup of $G$ such that there exists a measurable precompact
subset $C\subseteq G$, referred to as _fundamental domain_ , with
$G=\bigsqcup_{\lambda\in\mathcal{L}}C\lambda$ and $m(C)>0$. With the lattice
$\mathcal{L}$ being a subgroup of $G$, we have a naturally defined dynamical
system $(X,\mathcal{L})$ and it turns out that amorphic complexity is well
behaved when going from $(X,G)$ over to $(X,\mathcal{L})$.

######  Lemma 2.6.

Assume $(X,G)$ is a dynamical system and $G$ allows for a uniform lattice
$\mathcal{L}$. Then for every Følner sequence $\mathcal{F}$ in $G$ there is a
Følner sequence $\mathcal{F}^{\prime}$ in $\mathcal{L}$ such that

$\displaystyle\underline{\mathrm{ac}}_{\mathcal{F}}(X,G)=\underline{\mathrm{ac}}_{\mathcal{F}^{\prime}}(X,\mathcal{L})\qquad\textnormal{and}\qquad\overline{\mathrm{ac}}_{\mathcal{F}}(X,G)=\overline{\mathrm{ac}}_{\mathcal{F}^{\prime}}(X,\mathcal{L}).$

Furthermore, $(X,G)$ has infinite separation numbers with respect to
$\mathcal{F}$ if and only if $(X,\mathcal{L})$ has infinite separation numbers
with respect to $\mathcal{F}^{\prime}$.

###### Proof.

We denote the Haar measure on $G$ by $m$ and that on $\mathcal{L}$ by
$|\cdot|$. Let $C\subseteq G$ be a fundamental domain as in the above
definition of a uniform lattice. First, observe that for all $\delta>0$ there
are $\delta^{-}_{\delta},\delta^{+}_{\delta}>0$ such that for all $x,y\in X$
and $c\in C$ we have $d(c^{-1}x,c^{-1}y)\geq\delta^{-}_{\delta}$ whenever
$d(x,y)\geq\delta$ and $d(cx,cy)\geq\delta^{+}_{\delta}$ whenever
$d(x,y)\geq\delta^{-}_{\delta}$. This straightforwardly follows from the
precompactness of $C$.

Further, due to Proposition 2.4, we may assume without loss of generality that
$\mathcal{F}$ is a van Hove sequence. Under this assumption, there are van
Hove sequences $\mathcal{F}^{\prime}=(F_{n}^{\prime})$ and
$\mathcal{F}^{\prime\prime}=(F_{n}^{\prime\prime})$ in $\mathcal{L}$ with
$\lim_{n\to\infty}|F_{n}^{\prime}|/|F_{n}^{\prime\prime}|=1$ such that
$CF_{n}^{\prime}$ and $CF_{n}^{\prime\prime}$ are von Hove sequences in $G$
and $CF_{n}^{\prime}\subseteq F_{n}\subseteq CF_{n}^{\prime\prime}$, see for
example [Hau20, Lemma 3.2]. We will show that for all $x,y\in X$ and
$\delta>0$ we have

$\displaystyle\mathrm{ad}_{\mathcal{F}}(\Delta(X,G,\delta,x,y))\leq\mathrm{ad}_{\mathcal{F}^{\prime}}(\Delta(X,\mathcal{L},\delta^{-}_{\delta},x,y))\leq\mathrm{ad}_{\mathcal{F}}(\Delta(X,G,\delta^{+}_{\delta},x,y)).$
(5)

Clearly, this implies that for all $\nu\in(0,1)$ and all $\delta>0$

$\mathrm{Sep}_{\mathcal{F}}(X,G,\delta,\nu)\leq\mathrm{Sep}_{\mathcal{F}^{\prime}}(X,\mathcal{L},\delta^{-}_{\delta},\nu)\leq\mathrm{Sep}_{\mathcal{F}}(X,G,\delta^{+}_{\delta},\nu)$

and thus proves the statement.

By definition of $\delta^{-}_{\delta}$ and $\delta^{+}_{\delta}$ and since $C$
is a fundamental domain, we have

$\displaystyle\Delta(X,G,\delta,x,y)\subseteq
C\Delta(X,\mathcal{L},\delta^{-}_{\delta},x,y)\subseteq\Delta(X,G,\delta^{+}_{\delta},x,y).$

Hence, utilizing the fact that for any subset $F\subseteq\mathcal{L}$ we have
$m(CF)=|F|\cdot m(C)$, we obtain (5) from the following computation

$\displaystyle\mathrm{ad}_{\mathcal{F}}(\Delta(X,G,\delta,x,y))$
$\displaystyle=\varlimsup_{n\to\infty}m(\Delta(X,G,\delta,x,y)\cap
F_{n})/m(F_{n})$
$\displaystyle\leq\varlimsup_{n\to\infty}m(C\Delta(X,\mathcal{L},\delta^{-}_{\delta},x,y)\cap
CF_{n}^{\prime\prime})/m(CF_{n}^{\prime})$
$\displaystyle=\varlimsup_{n\to\infty}m(C\Delta(X,\mathcal{L},\delta^{-}_{\delta},x,y)\cap
CF_{n}^{\prime\prime})/m(CF_{n}^{\prime\prime})\cdot|F_{n}^{\prime\prime}|/|F_{n}^{\prime}|$
$\displaystyle=\mathrm{ad}_{\mathcal{F}^{\prime\prime}}(\Delta(X,\mathcal{L},\delta^{-}_{\delta},x,y))=\mathrm{ad}_{\mathcal{F}^{\prime}}(\Delta(X,\mathcal{L},\delta^{-}_{\delta},x,y))$
$\displaystyle=\varlimsup_{n\to\infty}m(C\Delta(X,\mathcal{L},\delta^{-}_{\delta},x,y)\cap
CF_{n}^{\prime})/m(CF_{n}^{\prime\prime})$
$\displaystyle\leq\varlimsup_{n\to\infty}m(\Delta(X,G,\delta^{+}_{\delta},x,y)\cap
F_{n})/m(F_{n})$
$\displaystyle=\mathrm{ad}_{\mathcal{F}}(\Delta(X,G,\delta^{+}_{\delta},x,y)).\qed$

######  Remark 2.7.

  1. (a)

If $(F_{n})$ is a van Hove sequence, then the sets $F_{n}^{\prime}$ and
$F_{n}^{\prime\prime}$ in the above proof are explicitly given by
$F_{n}^{\prime}=\\{h\in\mathcal{L}\;|\;Ch\subseteq F_{n}\\}$ and
$F_{n}^{\prime\prime}=\\{h\in\mathcal{L}\;|\;Ch\cap F_{n}\neq\emptyset\\}$,
see the proof of [Hau20, Lemma 3.2].

  2. (b)

Let us briefly comment on the necessity of the passage through Proposition 2.4
in the above proof. As mentioned in Remark 2.3, a Følner sequence in a
discrete group is necessarily a van Hove sequence. Consequently, given a
Følner sequence $(F_{n}^{\prime})$ in the lattice $\mathcal{L}$ of $G$,
$(F_{n}^{\prime})$ is actually a van Hove sequence and therefore, one can show
that $(CF_{n}^{\prime})$ defines a van Hove sequence in $G$. Accordingly,
whenever we seek to bound a Følner sequence $(F_{n})$ in $G$ from below and
above by sequences $(CF_{n}^{\prime})$ and $(CF_{n}^{\prime\prime})$ similarly
as in the previous proof, we actually bound $(F_{n})$ by van Hove sequences.
It turns out that this implies that $(F_{n})$ itself must be van Hove. These
observations are straightforward (though slightly tedious) to check.

##  3 On finiteness of separation numbers

This section deals with the scope of amorphic complexity. In particular, we
identify mean equicontinuous systems as those systems where separation numbers
are finite with respect to every Følner sequence and amorphic complexity may
hence be finite itself. Moreover, we show that positive entropy as well as
weak mixing imply infinite separation numbers.

###  3.1 Mean equicontinuity and finite separation numbers

We next discuss a natural class of dynamical systems with finite separation
numbers: the class of mean equicontinuous systems, see [Aus59, Rob96, HJ97,
Rob99, Cor06, Vor12, DG16, Gla18, ŁS18, FG20, FK20, GL20, FGL21] for numerous
examples. In our discussion of mean equicontinuity, we follow the terminology
of [FGL21]. Given a left or right Følner sequence $\mathcal{F}$, a system
$(X,G)$ is _(Besicovitch) $\mathcal{F}$-mean equicontinuous_ if for all
$\varepsilon>0$ there is $\delta>0$ such that for all $x,y\in X$ with
$d(x,y)<\delta$ we have

$D_{\mathcal{F}}(x,y)\coloneqq\varlimsup\limits_{n\to\infty}1/m(F_{n})\int\limits_{F_{n}}d(tx,ty)\,dm(t)<\varepsilon.$

In this case, $D_{\mathcal{F}}$ clearly defines a continuous pseudometric on
$X$. Thus, by identifying points $x,y\in X$ with $D_{\mathcal{F}}(x,y)=0$, we
obtain a compact metric space which we denote by $X/D_{\mathcal{F}}$.

Before we proceed, let us briefly recall the concept of the (upper) box
dimension of a compact metric space $(M,d)$. Given $\varepsilon>0$, we call a
subset $S$ of $M$ _$\varepsilon$ -separated_ if for all $s\neq s^{\prime}\in
S$ we have $d(s,s^{\prime})\geq\varepsilon$ and denote by $M_{\varepsilon}$
the maximal cardinality of an $\varepsilon$-separated subset of $M$. It is
well known and easy to see that $M_{\varepsilon}<\infty$ due to compactness.
With this notation, the _upper box dimension_ of $M$ is defined as

$\displaystyle\overline{}\mathrm{dim}_{B}(M)=\varlimsup\limits_{\varepsilon\to
0}\frac{\log M_{\varepsilon}}{-\log\varepsilon}.$

Now, for $\mathcal{F}$-mean equicontinuous $(X,G)$, we have

$D_{\mathcal{F}}(x,y)\geq\varlimsup\limits_{n\to\infty}1/m(F_{n})\int\limits_{F_{n}}\mathbf{1}_{[\delta,\infty)}(d(tx,ty))\cdot
d(tx,ty)\,dm(t)\geq\delta\cdot\mathrm{ad}_{\mathcal{F}}(\Delta(X,G,\delta,x,y))$

for all $\delta>0$ and $x,y\in X$ and hence,
$(X/D_{\mathcal{F}})_{\delta\nu}\geq\mathrm{Sep}_{\mathcal{F}}(X,G,\delta,\nu)$.
It follows

######  Proposition 3.1.

If $(X,G)$ is $\mathcal{F}$-mean equicontinuous for some left or right Følner
sequence $\mathcal{F}$, then it has finite separation numbers with respect to
$\mathcal{F}$ and

$\overline{\mathrm{ac}}_{\mathcal{F}}(X,G)\leq\overline{\dim}_{B}(X/D_{\mathcal{F}}).$

It is important to note that if $\mathcal{F}$ is a left Følner sequence, then
$D_{\mathcal{F}}$ is not necessarily invariant. In particular, the equivalence
relation defined by $D_{\mathcal{F}}$ may not define a factor of $(X,G)$ even
if $D_{\mathcal{F}}$ is continuous. However, it is easy to see that
$D_{\mathcal{F}}$ is invariant if $\mathcal{F}$ is a right Følner sequence. We
utilize this observation below.

In any case, it is certainly desirable to have an invariant pseudometric which
does not depend on a particular (right) Følner sequence. To that end, we may
consider

$D(x,y)\coloneqq\sup\\{D_{\mathcal{F}}(x,y)\;|\;\mathcal{F}\textnormal{ is a
left Følner sequence}\\}$

which is, in fact, invariant (see [FGL21, Proposition 3.12]). We say $(X,G)$
is _(Weyl) mean equicontinuous_ if $D$ is continuous.

######  Proposition 3.2 ([FGL21, Proposition 5.8]).

Suppose $(X,G)$ is $\mathcal{F}$-mean equicontinuous for some right Følner
sequence $\mathcal{F}$. Then $(X,G)$ is mean equicontinuous.

Given a left or right Følner sequence $\mathcal{F}$, a system $(X,G)$ is
called _$\mathcal{F}$ -mean sensitive_ if there exists $\eta>0$ such that for
every open set $U\subseteq X$ we can find $x,y\in U$ with
$D_{\mathcal{F}}(x,y)\geq\eta$. Moreover, we say $(X,G)$ is _(Weyl) mean
sensitive_ if there exists $\eta>0$ such that for every open set $U\subseteq
X$ we can find $x,y\in U$ with $D(x,y)\geq\eta$. We have the following direct
generalisation of the equivalence of (1) and (3) in [LTY15, Proposition 5.1]
whose proof extends almost literally to the current setting.

######  Proposition 3.3.

The system $(X,G)$ is $\mathcal{F}$-mean sensitive (with respect to a left or
right Følner sequence $\mathcal{F}$) if and only if there is $\eta>0$ such
that for every $x\in X$ we have that $\\{y\in
X\;|\;D_{\mathcal{F}}(x,y)\geq\eta\\}$ is residual in $X$.

Clearly, if $\mathrm{ad}_{\mathcal{F}}(\Delta(X,G,\eta/2,x,y))<\eta/2$, then
$D_{\mathcal{F}}(x,y)\leq\eta/2+(1-\eta/2)\cdot\eta/2<\eta$ (assuming, without
loss of generality, that the maximal distance of points in $X$ is $1$).

######  Corollary 3.4.

If a dynamical system $(X,G)$ is $\mathcal{F}$-mean sensitive (for a left or
right Følner sequence $\mathcal{F}$), then it has infinite separation numbers
with respect to $\mathcal{F}$.

In the following, we take a closer look at the relation between mean
equicontinuity and mean sensitivity in the minimal case. The proof of the next
statement is similar to the one for $\mathbb{Z}$-actions [LTY15, Proposition
4.3 & Theorem 5.4–5.5], see also [ZHL19, Corollary 5.6] for a similar
statement for countable amenable groups. For the convenience of the reader, we
provide a direct proof in the current setting.

######  Lemma 3.5.

Let $(X,G)$ be minimal. Then $(X,G)$ is either mean equicontinuous or mean
sensitive. Furthermore, if $(X,G)$ is mean sensitive, then it is
$\mathcal{F}$-mean sensitive for every right Følner sequence $\mathcal{F}$.

###### Proof.

Suppose $(X,G)$ is not mean equicontinuous. That is, there is $x\in X$ and
$\eta>0$ such that for all $\delta>0$ there is $y_{\delta}\in B_{\delta}(x)$
with $D(x,y_{\delta})>\eta$. Now, given any open set $U\subseteq X$, there is
$g\in G$ and $\delta_{0}>0$ such that $gB_{\delta_{0}}(x)\subseteq U$. Since
$D$ is invariant, we have $D(gx,gy_{\delta_{0}})=D(x,y_{\delta_{0}})>\eta$
which proves the first part.

For the second part, observe that Proposition 3.2 gives that for every right
Følner sequence $\mathcal{F}$ there exist $x\in X$ and $\eta>0$ such that for
all $\delta>0$ there is $y\in B_{\delta}(x)$ with $D_{\mathcal{F}}(x,y)>\eta$.
Since $\mathcal{F}$ is assumed to be a right Følner sequence,
$D_{\mathcal{F}}$ is invariant and we can argue similarly as for $D$ to obtain
$\mathcal{F}$-mean sensitivity. ∎

######  Remark 3.6.

Recall that $G$ acts _effectively_ on $X$ if for all $g\in G$ there is $x\in
X$ such that $gx\neq x$. According to [FGL21, Corollary 7.3], $G$ is
necessarily maximally almost periodic (see [FGL21] and references therein) if
$G$ allows for a minimal, mean equicontinuous and effective action on a
compact metric space $X$. Hence, Lemma 3.5 gives that every minimal effective
action by a group which is not maximally almost periodic (such as the
_continuous_ Heisenberg group $H_{3}(\mathbb{R})$) is mean sensitive.

Recall that a locally compact $\sigma$-compact amenable group $G$ is
_unimodular_ if and only if $G$ allows for a _two-sided Følner sequence_ ,
that is, a sequence $\mathcal{F}$ which is both a left and a right Følner
sequence. In conclusion to the above statements, we obtain

######  Corollary 3.7.

Suppose $G$ is unimodular and $(X,G)$ is minimal. Then $(X,G)$ is mean
equicontinuous if and only if the separation numbers of $(X,G)$ are finite
with respect to every left Følner sequence.

###### Proof.

By definition, mean equicontinuity implies $\mathcal{F}$-mean equicontinuity
with respect to every left Følner sequence. Hence, the “only if”-part follows
from Proposition 3.1.

For the other direction, let $\mathcal{F}$ be a two-sided Følner sequence.
Since we assume the separation numbers with respect to $\mathcal{F}$ to be
finite, we have that $(X,G)$ is not $\mathcal{F}$-mean sensitive. Since
$D_{\mathcal{F}}$ is invariant, we can argue similarly as in Lemma 3.5 to
obtain that $(X,G)$ is $\mathcal{F}$-mean equicontinuous. Utilizing
Proposition 3.2, we obtain the desired statement. ∎

###  3.2 Entropy, mixing and infinite separation numbers

In this section, we discuss how chaotic behaviour—more specifically: weak
mixing or positive entropy—implies infinite separation numbers. Here, we
occasionally have to assume that a Følner sequence we consider is _tempered_ ,
that is, there is $C>0$ such that for all $n$ we have
$m(\bigcup_{k<n}F_{k}^{-1}F_{n})<C\cdot m(F_{n})$. It is well known that every
Følner sequence allows for a tempered subsequence, see [Lin01, Proposition
1.4].

In line with [GW16], we call an invariant measure $\mu$ of $(X,G)$ _weakly
mixing_ if for every system $(Y,G)$ and all of its ergodic measures $\nu$ we
have that $\mu\times\nu$ is ergodic for $(X\times Y,G)$. Hence, if $\mu$ is
weakly mixing, $\mu^{m}=\bigtimes_{k=1}^{m}\mu$ is ergodic for $(X^{m},G)$ and
all $m\in\mathbb{N}$.

######  Theorem 3.8.

Let $(X,G)$ be a dynamical system with a weakly mixing measure $\mu$ and
suppose the support of $\mu$ is not a singleton. Then $(X,G)$ has infinite
separation numbers with respect to every Følner sequence.

###### Proof.

For a tempered Følner sequence, the proof is similar to that of the respective
statement for $\mathbb{Z}$-actions ([FGJ16, Theorem 2.2]) if we replace
Birkhoff’s Ergodic Theorem by Lindenstrauss’ Pointwise Ergodic Theorem [Lin01,
Theorem 1.2]. Here, we have to make use of the ergodicity of $\mu^{m}$ just as
in [FGJ16].

Now, given an arbitrary Følner sequence, we can always go over to a tempered
subsequence (see [Lin01, Proposition 1.4]). This gives infinite separation
numbers for a subsequence and hence, due to the $\limsup$ in (2), infinite
separation numbers for the original sequence. ∎

We next turn to systems with positive topological entropy. Our goal is to show

######  Theorem 3.9.

Suppose $G$ allows for a uniform lattice and the dynamical system $(X,G)$ has
positive topological entropy. Then $(X,G)$ has infinite separation numbers
with respect to every Følner sequence in $G$.

######  Remark 3.10.

Observe that the proof of a similar statement for $\mathbb{Z}$-actions (see
[FGJ16, Theorem 2.3]) utilised results that are only available for
$G=\mathbb{Z}$. The present approach provides an alternative to the somewhat
implicit argument in [FGJ16].

######  Remark 3.11.

We do not make explicit use of the actual definition of entropy in the
following and rather utilize results from the theory of topological
independence. Therefore, we refrain from discussing the basics of entropy
theory in the present work. Interested readers are referred to e.g. [OW87,
KL16, Bow20, Hau20] for a background and further references.

In order to prove Theorem 3.9, we first restrict to actions of countable
discrete (and, as throughout assumed, amenable) groups.

######  Definition 3.12 (cf. [KL16, Definition 8.7]).

Let $(X,G)$ be a dynamical system and suppose $G$ is countable and discrete.
Given a pair $\mathbf{A}=(A_{0},A_{1})$ of subsets of $X$, we say that a set
$J\subseteq G$ is an _independence set_ for $\mathbf{A}$ if for every non-
empty finite subset $I\subseteq J$ and every $(s_{g})_{g\in
I}\in\\{0,1\\}^{I}$ there exists $x\in X$ with $gx\in A_{s_{g}}$ for every
$g\in I$.

######  Theorem 3.13 ([KL16, Theorem 12.19 & Proposition 12.7]).

Suppose $G$ is discrete and countable and $(X,G)$ is a dynamical system. If
$(X,G)$ has positive topological entropy, then there is a pair
$\mathbf{A}=(A_{0},A_{1})$ of disjoint compact subsets of $X$ and $d>0$ such
that for every tempered Følner sequence $\mathcal{F}=(F_{n})$ in $G$ there is
an independence set $J$ of $\mathbf{A}$ with
$\mathrm{ad}_{\mathcal{F}}(J)=\lim_{n\to\infty}|F_{n}\cap J|/|F_{n}|\geq d>0$.

Let $\mathbf{A}$, $\mathcal{F}$ and $J\subseteq G$ be as in the above
statement. Observe that due to the compactness of $A_{0}$ and $A_{1}$ we
actually have that for every $s=(s_{j})_{j\in J}\in\\{0,1\\}^{J}$ there exists
$x\in X$ which _follows_ $s$, that is, $jx\in A_{s_{j}}$ for every $j\in J$.

######  Lemma 3.14.

Let $G$ be a countable discrete group and suppose $(X,G)$ has positive
topological entropy. Then $(X,G)$ has infinite separation numbers with respect
to every Følner sequence in $G$. In fact, there are $\delta>0$ and
$\nu\in(0,1]$ such that for every Følner sequence there is an uncountable
$(\delta,\nu)$-separated set.

###### Proof.

Let $\mathbf{A}=(A_{0},A_{1})$ and $d>0$ be as in Theorem 3.13. Given a Følner
sequence $\mathcal{F}$, we may assume without loss of generality that
$\mathcal{F}$ is tempered. By Theorem 3.13, we have an associated independence
set $J\subseteq G$ for $\mathbf{A}$ with $\mathrm{ad}_{\mathcal{F}}(J)\geq d$.
Set $\delta=\mathop{\textrm{dist}}(A_{0},A_{1})$ and
$\nu=d/2\leq\mathrm{ad}_{\mathcal{F}}(J)/2$. Our goal is to show that there is
an infinite subset $S\subseteq\\{0,1\\}^{J}$ such that whenever $x,y\in X$
follow distinct elements in $S$, then
$\mathrm{ad}_{\mathcal{F}}(\Delta(X,G,\delta,x,y))\geq\nu$.

To that end, we first define a sequence $(G_{n})_{n\in\mathbb{N}}$ of pairwise
disjoint non-empty finite subsets of $G$ such that for every infinite set
$M\subseteq\mathbb{N}$ we have

$\displaystyle\mathrm{ad}_{\mathcal{F}}(\bigcup_{n\in M}G_{n})\geq 1-\nu.$ (6)

We may do so by starting with $G_{1}=F_{1}$. Assuming we have already chosen
$G_{1},\ldots,G_{n}$ for some $n\in\mathbb{N}$, let $N=N(n)\in\mathbb{N}$ be
large enough to guarantee that

$|F_{N}\setminus(G_{1}\cup\ldots\cup G_{n})|\geq(1-\nu)|F_{N}|$

and set $G_{n+1}=F_{N}\setminus(G_{1}\cup\ldots\cup G_{n})$. Note that this
gives that $(G_{n})$ satisfies (6) for every infinite $M\subseteq\mathbb{N}$
because

$\mathrm{ad}_{\mathcal{F}}(\bigcup_{n\in
M}G_{n})\geq\varlimsup_{\begin{subarray}{c}n\to\infty\\\ n\in
M\end{subarray}}\frac{|F_{N(n-1)}\cap G_{n}|}{|F_{N(n-1)}|}\geq 1-\nu,$

for any infinite $M\subseteq\mathbb{N}$.

Now, let $E$ be an uncountable family of subsets of $\mathbb{N}$ such that
$M\triangle M^{\prime}$ is infinite for distinct $M,M^{\prime}\in E$. Given
$M\in E$, we define $s^{M}\in\\{0,1\\}^{J}$ by

$s^{M}_{j}=\begin{cases}1&\text{ if }j\in G_{n}\text{ and }n\in M,\\\ 0&\text{
otherwise.}\end{cases}$

Set $S=\\{s^{M}\in\\{0,1\\}^{J}\;|\;M\in E\\}$. Given $s\in S$, choose some
$x(s)\in X$ which follows $s$ (recall the discussion before the statement). It
is straightforward to see that for distinct $M,M^{\prime}\in E$, we have for
$x=x(s^{M})$ and $x^{\prime}=x(s^{M^{\prime}})$ that

$\displaystyle\Delta(X,G,\delta,x,x^{\prime})$ $\displaystyle=\\{g\in
G\;|\;d(gx,gx^{\prime})\geq\delta\\}\supseteq\\{g\in J\;|\;\;s^{M}_{g}\neq
s^{M^{\prime}}_{g}\\}$ $\displaystyle=J\cap\big{(}\bigcup_{n\in M\triangle
M^{\prime}}G_{n}\big{)}.$

Using (6), we obtain

$\mathrm{ad}_{\mathcal{F}}\big{(}J\cap\bigcup_{n\in M\triangle
M^{\prime}}G_{n}\big{)}\geq\mathrm{ad}_{\mathcal{F}}(J)/2\geq\nu.$

Hence, $\\{x(s)\in X\;|\;s\in S\\}$ is the uncountable
$(\delta,\nu)$-separated set we sought. ∎

###### Proof of Theorem 3.9.

Let us denote by $\mathcal{L}$ a lattice (as provided by the assumptions) in
$G$. Note that since $G$ is $\sigma$-compact, we have that $\mathcal{L}$ is
countable.

Due to [Hau20, Theorem 5.2], positive topological entropy of $(X,G)$ implies
positive topological entropy of $(X,\mathcal{L})$. Hence, Lemma 3.14 gives
that $(X,\mathcal{L})$ has infinite separation numbers with respect to every
Følner sequence. Due to Lemma 2.6, this implies infinite separation numbers of
$(X,G)$ with respect to every Følner sequence. ∎

##  4 Independence of Følner sequences

In general, amorphic complexity might depend on the particular Følner sequence
with respect to which we compute the separation numbers. For $G=\mathbb{Z}$,
this can be seen by considering the example in [FGJ16, page 541]. There,
$\mathrm{{ac}}_{\mathcal{F}}(X,\mathbb{Z})=\infty$ for
$\mathcal{F}=([0,n))_{n\in\mathbb{N}}$ while
$\mathrm{{ac}}_{\mathcal{F}^{\prime}}(X,\mathbb{Z})=0$ for
$\mathcal{F}^{\prime}=((-n,0])_{n\in\mathbb{N}}$.

The goal of this section is to show

######  Theorem 4.1.

Let $(X,G)$ be a dynamical system whose product $(X^{2},G)$ is pointwise
uniquely ergodic. Then $\overline{\mathrm{ac}}_{\mathcal{F}}(X,G)$ and
$\underline{\mathrm{ac}}_{\mathcal{F}}(X,G)$ are independent of the particular
(left) Følner sequence $\mathcal{F}$.

######  Remark 4.2.

Notice that due to [FGL21, Theorem 1.2], the above gives that amorphic
complexity of mean equicontinuous systems is independent of the particular
Følner sequence.

In fact, we have the following stronger statement which immediately yields
Theorem 4.1.

######  Theorem 4.3.

Let $(X,G)$ be a dynamical system whose product $(X^{2},G)$ is pointwise
uniquely ergodic. The following holds.

  1. (i)

Suppose there is a Følner sequence $\mathcal{F}$ such that
$\mathrm{Sep}_{\mathcal{F}}(X,G,\delta,\nu)=\infty$ for some
$\delta,\nu\in(0,1)$. Then there exists $\delta_{0}>0$ such that
$\mathrm{Sep}_{\mathcal{F}^{\prime}}(X,G,\delta^{\prime},\nu)=\infty$ for
every Følner sequence $\mathcal{F}^{\prime}$ and every
$\delta^{\prime}\in(0,\delta_{0}]$.

  2. (ii)

Let $\mathcal{F}^{0}$ and $\mathcal{F}^{1}$ be Følner sequences and suppose
$\mathrm{Sep}_{\mathcal{F}^{0}}(X,G,\delta,\nu)<\infty$ for all
$\nu,\delta\in(0,1)$. Then there is a cocountable set $A\in(0,1)$ such that
for all $\delta\in A$ we have
$\mathrm{Sep}_{\mathcal{F}^{0}}(X,G,\delta,\nu)=\mathrm{Sep}_{\mathcal{F}^{1}}(X,G,\delta,\nu)$
for all but countably many $\nu$.

###### Proof.

Without loss of generality, we may assume that $\mathrm{diam}(X)=1$. We start
by providing some general observations. Given $\delta\in(0,1)$, let
$(h_{\ell})$ and $(H_{\ell})$ be sequences of non-decreasing continuous self-
maps on $[0,1]$. For large enough $\ell\in\mathbb{N}$, assume that
$h_{\ell}(z)=0$ for $z\in[0,\delta]$ and $h_{\ell}(z)=1$ for
$z\in[\delta+1/\ell,1]$ as well as $H_{\ell}=0$ on $[0,\delta-1/\ell]$ and
$H_{\ell}=1$ on $[\delta,1]$. Clearly,
$h_{\ell}(z)\leq\mathbf{1}_{[\delta,1]}(z)\leq H_{\ell}(z)$ for all
$z\in[0,1]$ and large enough $\ell\in\mathbb{N}$. Hence, for all $x,y\in X$,
every Følner sequence $\mathcal{F}=(F_{n})$, and sufficiently large $\ell$, we
have

$\displaystyle\begin{split}&\int\limits_{X^{2}}h_{\ell}(d(v,w))d\mu_{(x,y)}(v,w)=\lim_{n\to\infty}1/|F_{n}|\cdot\int\limits_{F_{n}}h_{\ell}(d(sx,sy))dm(s)\\\
&\leq\lim_{n\to\infty}1/|F_{n}|\cdot\int\limits_{F_{n}}\mathbf{1}_{[\delta,1]}(d(sx,sy))dm(s)=\mathrm{ad}_{\mathcal{F}}(\Delta(X,G,\delta,x,y))\\\
&\leq\lim_{n\to\infty}1/|F_{n}|\cdot\int\limits_{F_{n}}H_{\ell}(d(sx,sy))dm(s)=\int\limits_{X^{2}}H_{\ell}(d(v,w))d\mu_{(x,y)}(v,w),\end{split}$
(7)

where we used the pointwise unique ergodicity of $(X^{2},G)$ and where
$\mu_{(x,y)}$ denotes the unique invariant measure on the orbit closure of
$(x,y)\in X^{2}$. Sending $\ell\to\infty$, we obtain equality in (7) unless

$\mu_{(x,y)}(\\{(v,w)\in X^{2}\;|\;d(v,w)=\delta\\})>0.$ (8)

In other words, if (8) does not hold, then
$\mathrm{ad}_{\mathcal{F}}(\Delta(X,G,\delta,x,y))$ is actually independent of
the Følner sequence $\mathcal{F}$. Notice that given $(x,y)$, there can be at
most countably many $\delta$ which verify (8).

Let us prove statement (i). Suppose $\mathcal{F}$ is a Følner sequence and
$\mathrm{Sep}_{\mathcal{F}}(X,G,\delta,\nu)=\infty$ for some
$\delta,\nu\in(0,1)$. Let $\mathcal{S}$ be a countable family of finite
$(X,G,\delta,\nu)$-separated sets (with respect to $\mathcal{F}$) such that
$\sup_{S\in\mathcal{S}}\\#S=\infty$. Further, let $C\subseteq(0,1)$ be the
collection of all $\delta\in(0,1)$ such that for some $S\in\mathcal{S}$ there
are $(x,y)\in S^{2}$ such that (8) holds. As $C$ is at most countable, there
exists $\delta_{0}\in(0,\delta]$ such that for any $S\in\mathcal{S}$ we have

$\mathrm{ad}_{\mathcal{F}^{\prime}}(\Delta(X,G,\delta_{0},x,y))=\mathrm{ad}_{\mathcal{F}}(\Delta(X,G,\delta_{0},x,y))\geq\mathrm{ad}_{\mathcal{F}}(\Delta(X,G,\delta,x,y))\geq\nu$

for all $x\neq y\in S$ and any Følner sequence $\mathcal{F}^{\prime}$ where we
used that $\left|\Delta(X,G,\cdot,x,y)\right|$ is non-increasing. It
straightforwardly follows that each $S$ is
$(X,G,\delta^{\prime},\nu)$-separated with respect to any Følner sequence
$\mathcal{F}$ and every $\delta^{\prime}\in(0,\delta_{0}]$. As $S$ can be
chosen arbitrarily large, this proves the first assertion.

Let us consider (ii). First, observe that due to (i), we have
$\mathrm{Sep}_{\mathcal{F}^{1}}(X,G,\delta,\nu)<\infty$ for all
$\delta,\nu\in(0,1)$. Given $\delta\in(0,1)$, we call $\nu\in(0,1)$ _$\delta$
-singular_ if
$\mathrm{Sep}_{\mathcal{F}^{i}}(X,G,\delta,\nu)<\mathrm{Sep}_{\mathcal{F}^{i}}(X,G,\delta-\varepsilon,\nu)$
for all $\varepsilon>0$ and some $i\in\\{0,1\\}$. Otherwise, we say $\nu$ is
_$\delta$ -regular_. The collection of all $\delta$-singular elements of
$(0,1)$ is denoted by $B_{\delta}$. We say $\delta$ is _singular_ if
$B_{\delta}$ is uncountable. Otherwise, we call $\delta\in(0,1)$ _regular_.
The collection of all singular $\delta$ in $(0,1)$ is denoted by $B$. We set
$A=(0,1)\setminus B$.

Next, we show that for all $\delta\in(0,1)$ and each $\nu\in B_{\delta}^{c}$
we have
$\mathrm{Sep}_{\mathcal{F}^{0}}(X,G,\delta,\nu)=\mathrm{Sep}_{\mathcal{F}^{1}}(X,G,\delta,\nu)$.
To prove (ii), it then remains to show that $B$ is countable.

Given $\delta\in(0,1)$, let $\nu\in(0,1)$ be $\delta$-regular. By definition,
there is $\varepsilon>0$ such that
$\mathrm{Sep}_{\mathcal{F}^{i}}(X,G,\delta,\nu)=\mathrm{Sep}_{\mathcal{F}^{i}}(X,G,\delta^{\prime},\nu)$
for all $\delta^{\prime}\in(\delta-\varepsilon,\delta)$ and $i=0,1$. Let
$S\subseteq X$ be $\delta$-$\nu$-separated w.r.t. $\mathcal{F}^{0}$ and
suppose $S$ is of maximal cardinality. As $S$ is finite, the collection of all
$\delta\in(0,1)$ which verify (8) for some pair $(x,y)\in S^{2}$ is countable.
There is hence $\delta^{\prime}\in(\delta-\varepsilon,\delta)$ which does not
verify (8) for any $(x,y)\in S^{2}$. Clearly, $S$ is
$\delta^{\prime}$-$\nu$-separated for $\mathcal{F}^{0}$. By the above, $S$ is
also $\delta^{\prime}$-$\nu$-separated for $\mathcal{F}^{1}$. Hence,

$\mathrm{Sep}_{\mathcal{F}^{1}}(X,G,\delta,\nu)=\mathrm{Sep}_{\mathcal{F}^{1}}(X,G,\delta^{\prime},\nu)\geq\mathrm{Sep}_{\mathcal{F}^{0}}(X,G,\delta^{\prime},\nu)=\mathrm{Sep}_{\mathcal{F}^{0}}(X,G,\delta,\nu).$

By changing the roles of $\mathcal{F}^{0}$ and $\mathcal{F}^{1}$, we obtain
the converse inequality and accordingly
$\mathrm{Sep}_{\mathcal{F}^{0}}(X,G,\delta,\nu)=\mathrm{Sep}_{\mathcal{F}^{1}}(X,G,\delta,\nu)$
for all $\delta$-regular $\nu$.

It remains to show that $B$ is countable. To that end, we need the following

######  Claim 4.4.

If $\delta\in(0,1)$ is singular, then $B_{\delta}$ has non-empty interior.

###### Proof of the claim.

Let $\nu\in(0,1)$ be $\delta$-singular and $\nu^{\prime}\in(0,\nu)$ be
$\delta$-regular. Observe that due to the monotonicity in both arguments of
$\mathrm{Sep}_{\mathcal{F}^{i}}(X,G,\cdot,\cdot)$, there has to be a _jump
point_ $\nu_{0}$ between $\nu$ and $\nu^{\prime}$ (possibly coinciding with
$\nu$ or $\nu^{\prime}$), i.e., a point $\nu_{0}$ such that for $i=0$ or $i=1$
we have
$\mathrm{Sep}_{\mathcal{F}^{i}}(X,G,\delta,\nu_{0}-\varepsilon)>\mathrm{Sep}_{\mathcal{F}^{i}}(X,G,\delta,\nu_{0})$
for all $\varepsilon>0$. As $\mathrm{Sep}_{\mathcal{F}^{i}}(X,G,\delta,\cdot)$
is non-increasing and integer-valued, each compact subinterval of $(0,1)$ can
contain at most finitely many such jump points. Therefore, the set of
$\delta$-singular points is a union of isolated points and intervals. Since a
subset of $(0,1)$ with only isolated points is at most countable, this proves
the claim. $\circ$

Now, for a contradiction, assume that $B$ is uncountable. By the above claim,
$B_{\delta}$ contains an interval $I_{\delta}$ whenever $\delta\in B$. Thus,
there must be an uncountable set $B^{\prime}\subseteq B$ with
$\bigcap_{\delta\in B^{\prime}}I_{\delta}\neq\emptyset$. Accordingly, there is
$\nu\in(0,1)$ such that $\nu$ is $\delta$-singular for all $\delta\in
B^{\prime}$. As $\mathrm{Sep}_{\mathcal{F}^{i}}(X,G,\cdot,\nu)$ is non-
increasing, there can be at most countably many $\delta$ with
$\mathrm{Sep}_{\mathcal{F}^{i}}(X,G,\delta-\varepsilon,\nu)>\mathrm{Sep}_{\mathcal{F}^{i}}(X,G,\delta,\nu)$
for all $\varepsilon>0$. This contradicts the uncountability of $B^{\prime}$.
Hence, $B$ is at most countable. This finishes the proof. ∎

##  5 Application to regular model sets

In this section, we study amorphic complexity of (the dynamical hull of) model
sets. Given a model set, our third main result provides an upper bound for its
amorphic complexity which may be understood as a measure of its amorphicity.
We start by collecting a number of preliminary facts concerning Delone sets,
cut and project schemes and their associated dynamics.

###  5.1 Delone dynamical systems and model sets

From now on, in what follows, $G$ is a locally compact second countable
abelian group with Haar measure $m_{G}$. Further, in all of the following, we
switch to additive notation for the group operation in $G$, accounting for its
commutativity. By the Birkhoff-Kakutani Theorem, $G$ is metrizable and the
metric $d_{G}$ can be chosen to be invariant under $G$. In fact, open balls
with respect to $d_{G}$ are relatively compact [Str74] so that $G$ is
automatically $\sigma$-compact.

A set $\Gamma\subseteq G$ is called $r$-uniformly discrete if there exists
$r>0$ such that $d_{G}(g,g^{\prime})>r$ for all $g\neq g^{\prime}\in\Gamma$.
Moreover, $\Gamma$ is called $R$-relatively dense (or $R$-syndetic) if there
exists $R>0$ such that $\Gamma\cap B_{G}(g,R)\neq\emptyset$ for all $g\in G$,
where $B_{G}(g,R)$ denotes the open $d_{G}$-ball of radius $R$ centred at $g$.
We call $\Gamma$ a Delone set if it is uniformly discrete and relatively
dense. The collection of all Delone sets in $G$ will be denoted by
$\mathcal{D}(G)$.

Given $\rho>0$ and $g\in\Gamma$, the tuple
$(B_{G}(0,\rho)\cap(\Gamma-g),\rho)$ is called a ($\rho$-)patch of $\Gamma$.
The set of all patches of $\Gamma$ is denoted by $\mathcal{P}(\Gamma)$. A
Delone set $\Gamma$ is said to have finite local complexity (FLC) if for all
$\rho>0$ the number of its $\rho$-patches is finite. For
$\Gamma,\Gamma^{\prime}\in\mathcal{D}(G)$, set

$\mathop{\textrm{dist}}(\Gamma,\Gamma^{\prime})=\inf\left\\{\varepsilon>0\;|\;\exists
g\in B_{G}(0,\varepsilon):(\Gamma-g)\cap
B_{G}(0,1/\varepsilon)=\Gamma^{\prime}\cap B_{G}(0,1/\varepsilon)\right\\}.$

Then
$d(\Gamma,\Gamma^{\prime})=\min\\{1/\sqrt{2},\mathop{\textrm{dist}}(\Gamma,\Gamma^{\prime})\\}$
defines a metric on $\mathcal{D}(G)$ (see [LMS02, Section 2]). Moreover, for
any Delone set $\Gamma\subseteq G$ with FLC the dynamical hull of $\Gamma$,
defined as

$\Omega(\Gamma)=\overline{\left\\{\Gamma-g\;|\;g\in G\right\\}},$

where the closure is taken with respect to $d$, is compact [Sch99, Proposition
2.3]. The dynamical system $(\Omega(\Gamma),G)$, given by the translation
action of $G$ on the hull $\Omega(\Gamma)$, is called a Delone dynamical
system.

The method of choice to construct Delone sets is to utilize a _cut and project
scheme_ (CPS). A CPS consists of a triple $(G,H,\mathcal{L})$ of two locally
compact abelian groups $G$ (external group) and $H$ (internal group) and a
uniform lattice $\mathcal{L}\subseteq G\times H$ which is irrational, that is,
the natural projections $\pi_{G}:G\times H\to G$ and $\pi_{H}:G\times H\to H$
satisfy

  * (i)

the restriction $\pi_{G}|_{\mathcal{L}}$ is injective;

  * (ii)

the image $\pi_{H}(\mathcal{L})$ is dense.

If not stated otherwise, we throughout assume that $G$ and $H$ are second
countable. As a consequence of (i), if we let $L=\pi_{G}(\mathcal{L})$ and
$L^{*}=\pi_{H}(\mathcal{L})$, the star map

$*:L\rightarrow L^{*}:l\mapsto
l^{*}=\pi_{H}\circ\left.\pi_{G}\right|_{\mathcal{L}}^{-1}(l)$

is well defined and surjective. Given a precompact set $W\subseteq H$
(referred to as window), we define the point set

$\mbox{\Large$\curlywedge$}(W)=\pi_{G}\left(\mathcal{L}\cap(G\times
W)\right)=\\{l\in L\;|\;l^{*}\in W\\}.$

If $W$ is compact and proper (that is, $\overline{\mathrm{int}(W)}=W$), then
$\mbox{\Large$\curlywedge$}(W)$ is a Delone set and has FLC [Rob07]. In this
case, we call $\mbox{\Large$\curlywedge$}(W)$ a model set. If further
$m_{H}(\partial W)=0$, then we call the window, as well as the resulting model
set, regular. Otherwise, we refer to $W$ and $\mbox{\Large$\curlywedge$}(W)$
as irregular. Delone dynamical systems associated to regular model sets are
mean equicontinuous, see [FGL21, Remark 6.2 & Corollary 6.3].

We say that a subset $A\subseteq H$ is irredundant if $\\{h\in
H\;|\;h+A=A\\}=\\{0\\}$. Clearly, if $\partial W$ is irredundant, then so is
$W$. A CPS is called Euclidean if $G=\mathbb{R}^{N}$ and $H=\mathbb{R}^{M}$
for some $M,N\in\mathbb{N}$, and planar if $N=M=1$. Note that in the Euclidean
case, any compact window is irredundant. Further, observe that if $W$ is not
irredundant, it is possible to construct a CPS
$(G,H^{\prime},\mathcal{L}^{\prime})$ with irredundant window
$W^{\prime}\subseteq H^{\prime}$ such that for each
$\Lambda\in\Omega(\mbox{\Large$\curlywedge$}(W))$ with
$\mbox{\Large$\curlywedge$}(\mathrm{int}(W))\subseteq\Lambda\subseteq\mbox{\Large$\curlywedge$}(W)$
we have
$\mbox{\Large$\curlywedge$}(\mathrm{int}(W^{\prime}))\subseteq\Lambda\subseteq\mbox{\Large$\curlywedge$}(W^{\prime})$
(compare [LM06, Section 5] and [BLM07, Lemma 7]).

As $\mathcal{L}$ is a uniform lattice in $G\times H$, the quotient
$\mathbb{T}\coloneqq(G\times H)/\mathcal{L}$ is a compact abelian group. A
natural action of $G$ on $\mathbb{T}$ is given by
$(u,[s,t]_{\mathcal{L}})\mapsto[s+u,t]_{\mathcal{L}}$. Here,
$[s,t]_{\mathcal{L}}$ denotes the equivalence class of $(s,t)\in G\times H$ in
$\mathbb{T}$. Observe that due to the assumptions on $(G,H,\mathcal{L})$, this
action is equicontinuous, minimal and has hence a unique invariant measure
$\mu_{\mathbb{T}}$. Furthermore, if $W\subseteq H$ is irredundant,
$(\mathbb{T},G)$ is the maximal equicontinuous factor of
$(\Omega(\mbox{\Large$\curlywedge$}(W)),G)$ [BLM07]. The respective factor map
$\beta$ is also referred to as torus parametrization.

Given an irredundant window $W$, the fibres of the torus parametrization are
characterized as follows: for
$\Gamma\in\Omega(\mbox{\Large$\curlywedge$}(W))$, we have

$\Gamma\in\beta^{-1}([s,t]_{\mathcal{L}})\quad\Leftrightarrow\quad\mbox{\Large$\curlywedge$}(\mathrm{int}(W)+t)-s\subseteq\Gamma\subseteq\mbox{\Large$\curlywedge$}(W+t)-s$
(9)

as well as

$\Gamma\in\beta^{-1}([0,t]_{\mathcal{L}})\quad\Leftrightarrow\quad\exists\,(t_{j})\in{L^{*}}^{\mathbb{N}}\text{
with }\lim_{j\rightarrow\infty}t_{j}=t\text{ and
}\lim_{j\rightarrow\infty}\mbox{\Large$\curlywedge$}(W+t_{j})=\Gamma.$

In the following, we denote by $\mathrm{Vol}(\mathcal{L})$ the volume of a
fundamental domain of $\mathcal{L}$. Note that $\mathrm{Vol}(\mathcal{L})$ is
well defined.

######  Proposition 5.1 ([HR15, Proposition 3.4]).

Let $(G,H,\mathcal{L})$ be a CPS and $W\subseteq H$ be precompact. Then for
every van Hove sequence $\mathcal{F}=(F_{n})$ in $G$ we have

$\frac{m_{H}(\mathrm{int}(W))}{\mathrm{Vol}(\mathcal{L})}\leq\varliminf_{n\to\infty}\frac{\sharp(\mbox{\Large$\curlywedge$}(W)\cap
F_{n})}{m_{G}(F_{n})}\leq\varlimsup_{t\to\infty}\frac{\sharp(\mbox{\Large$\curlywedge$}(W)\cap
F_{n})}{m_{G}(F_{n})}\leq\frac{m_{H}(W)}{\mathrm{Vol}(\mathcal{L})}.$

Let us collect three more statements which follow easily from the definition
of the metric $d$ on $\mathcal{D}(G)$. Similarly to the notion of
$(\delta,\nu)$-separation of elements of a dynamical system (see Section 1),
given a van Hove sequence $\mathcal{F}$ in $G$, we set

$\nu_{\mathcal{F}}(\delta,\Gamma,\Gamma^{\prime})=\mathrm{ad}_{\mathcal{F}}(\\{g\in
G\;|\;d(g\Gamma,g\Gamma^{\prime})\geq\delta\\}),$

where $\delta>0$ and $\Gamma,\Gamma^{\prime}\in\mathcal{D}(G)$.

######  Proposition 5.2.

For every van Hove sequence $\mathcal{F}=(F_{n})$ in $G$ we have

$\nu_{\mathcal{F}}(\delta,\Gamma,\Gamma^{\prime})\leq
m_{G}(B_{G}(0,1/\delta))\varlimsup_{n\to\infty}\frac{\sharp((\Gamma\Delta\Gamma^{\prime})\cap
F_{n})}{m_{G}(F_{n})},$

with $\delta>0$ and $\Gamma,\Gamma^{\prime}\in\mathcal{D}(G)$.

Accordingly, together with Proposition 5.1, we get

######  Corollary 5.3.

If $m_{H}(\partial W)=0$ and
$\mbox{\Large$\curlywedge$}(\mathrm{int}(W))\subseteq\Gamma\subseteq\mbox{\Large$\curlywedge$}(W)$,
then
$\nu_{\mathcal{F}}(\delta,\Gamma,\Gamma^{\prime})=\nu_{\mathcal{F}}(\delta,\mbox{\Large$\curlywedge$}(W),\Gamma^{\prime})$
for all van Hove sequences $\mathcal{F}$, $\delta>0$ and
$\Gamma^{\prime}\in\mathcal{D}(G)$.

Finally, observe that

######  Proposition 5.4.

Suppose $\delta>0$, $\Gamma,\Gamma^{\prime}\in\mathcal{D}(G)$ and $g\in
B_{G}(0,{\delta/2})$. If $d(\Gamma,\Gamma^{\prime})\geq\delta$, then
$d(\Gamma,\Gamma^{\prime}+g)\geq\delta/2$.

###  5.2 Upper bound on the amorphic complexity of regular model sets

We next come to our third main result. First, recall that for a locally
compact $\sigma$-compact group $H$, the upper box dimension is given by

$\overline{\mathrm{dim}}_{\mathrm{B}}(H)=\varlimsup\limits_{\varepsilon\to
0}\frac{\log
m_{H}\big{(}\overline{B_{H}(h,\varepsilon)}\big{)}}{\log\varepsilon},$

where $h\in H$ is arbitrary. Observe that
$\overline{\mathrm{dim}}_{\mathrm{B}}(H)$ is well defined because of the
invariance of the metric $d_{H}$ and the Haar measure $m_{H}$. Note further
that the above definition, as well as the definition of the (upper) box
dimension of compact sets in Section 3.1, are special cases of a more general
concept of box dimension. We refrain from reproducing the slightly technical
(and standard) general definition here and refer the interested reader to
[Edg98, Section 1.4] instead.

We will also make use of _Minkowski’s characterisation_ of the box dimension
of a given compact set $M\subseteq H$ by

$\overline{\mathrm{dim}}_{\mathrm{B}}(M)=\overline{\mathrm{dim}}_{\mathrm{B}}(H)-\varliminf\limits_{\varepsilon\to
0}\frac{\log
m_{H}\big{(}\overline{B_{H}(M,\varepsilon)}\big{)}}{\log\varepsilon}.$

The proof of this fact in our setting is similar to the one in the Euclidean
space, see for instance [Fal03, Proposition 3.2].

Finally, in order to derive upper bounds on amorphic complexity, it is
convenient to make use of an alternative characterisation which utilises
spanning sets instead of separating sets—similar as in the derivation of upper
bounds for topological entropy (or box dimension). Given $\delta>0$ and
$\nu\in(0,1]$, we say a subset $S\subseteq X$ is _$(\delta,\nu)$ -spanning_
with respect to a Følner sequence $\mathcal{F}$ if for all $x\in X$ there
exists $s\in S$ such that
$\mathrm{ad}_{\mathcal{F}}(\Delta(X,G,\delta,x,s))<\nu$. We denote by
$\mathrm{Span}_{\mathcal{F}}(X,G,\delta,\nu)$ the smallest cardinality among
the $(\delta,\nu)$-spanning sets with respect to $\mathcal{F}$. It is not
difficult to see that $\mathrm{Span}_{\mathcal{F}}(X,G,\delta,\nu)$ instead of
$\mathrm{Sep}_{\mathcal{F}}(X,G,\delta,\nu)$ can equivalently be used in
defining amorphic complexity, see also [FGJ16, Lemma 3.1 & Corollary 3.2].

######  Theorem 5.5.

Suppose $(G,H,\mathcal{L})$ is a cut and project scheme, where $G$ and $H$ are
locally compact second countable abelian groups. Furthermore, let $W\subseteq
H$ be compact, proper, regular and irredundant and assume that
$\overline{\mathrm{dim}}_{\mathrm{B}}(H)$ is finite. Then

$\overline{\mathrm{ac}}_{\mathcal{F}}(\Omega(\mbox{\Large$\curlywedge$}(W)),G)\
\leq\
\frac{\overline{\mathrm{dim}}_{\mathrm{B}}(H)}{\overline{\mathrm{dim}}_{\mathrm{B}}(H)-\overline{\mathrm{dim}}_{\mathrm{B}}(\partial
W)},$ (10)

for any Følner sequence $\mathcal{F}$.

###### Proof.

As $W$ is regular and hence $(\Omega(\mbox{\Large$\curlywedge$}(W)),G)$ mean
equicontinuous, we may assume without loss of generality that $\mathcal{F}$ is
van Hove, see Remark 4.2 and Theorem 4.3. We first choose compact sets
$A\subseteq G$ and $B\subseteq H$ such that $W\subseteq B$ and $\pi(A\times
B)=\mathbb{T}$, where $\pi:G\times H\to\mathbb{T}=(G\times H)/\mathcal{L}$ is
the canonical projection.

Given $(g,h)\in A\times B$, let
$\hat{\Gamma}_{g,h}=\mbox{\Large$\curlywedge$}(W+h)-g$. Observe that
$\hat{\Gamma}_{g,h}$ may not be an element of
$\Omega(\mbox{\Large$\curlywedge$}(W))$. While for our asymptotic estimates
this will be of no problem (due to Corollary 5.3), its explicit definition
makes $\hat{\Gamma}_{g,h}$ more convenient to deal with in computations.

######  Claim 5.6.

Let $\delta>0$. If $d_{G}(g,g^{\prime})\leq\delta/2$ and
$d(\hat{\Gamma}_{g,h},\hat{\Gamma}_{g^{\prime},h^{\prime}})\geq\delta$, then

$[-g,-h]_{\mathcal{L}}\in\pi\left(B_{G}(0,2/\delta)\times(W\Delta(W+h^{\prime}-h))\right)=:D(\delta,h^{\prime}-h).$

###### Proof of the claim.

By Proposition 5.4, we know that
$d(\hat{\Gamma}_{g,h},\hat{\Gamma}_{g,h^{\prime}})\geq\delta/2$. Hence, there
exists $(\ell,\ell^{*})\in\mathcal{L}$ with $\ell\in B_{G}(g,2/\delta)$ and
$\ell\in\hat{\Gamma}_{g,h}\Delta\hat{\Gamma}_{g,h^{\prime}}$. The latter
implies that $\ell^{*}\in(W+h)\Delta(W+h^{\prime})$.

Equivalently, this means that $\ell-g\in B_{G}(0,2/\delta)$ and $\ell^{*}-h\in
W\Delta(W+h^{\prime}-h)$, so that

$[-g,-h]_{\mathcal{L}}=[\ell-g,\ell^{*}-h]_{\mathcal{L}}\in\pi\left(B_{G}(0,2/\delta)\times
W\Delta(W+h^{\prime}-h)\right).$

This proves the claim. $\circ$

We can now apply the claim to estimate the separation frequency of a pair
$\hat{\Gamma}_{g,h}$ and $\hat{\Gamma}_{g^{\prime},h^{\prime}}$.

$\displaystyle\nu_{\mathcal{F}}(\delta,\hat{\Gamma}_{g,h},\hat{\Gamma}_{g^{\prime},h^{\prime}})$
$\displaystyle=$
$\displaystyle\varlimsup_{n\to\infty}\frac{1}{m_{G}(F_{n})}\int_{F_{n}}\mathbf{1}_{[\delta,\infty)}(d(\hat{\Gamma}_{g,h}-t,\hat{\Gamma}_{g^{\prime},h^{\prime}}-t))dt$
$\displaystyle=$
$\displaystyle\varlimsup_{n\to\infty}\frac{1}{m_{G}(F_{n})}\int_{F_{n}}\mathbf{1}_{[\delta,\infty)}(d(\hat{\Gamma}_{g+t,h},\hat{\Gamma}_{g^{\prime}+t,h^{\prime}}))dt$
$\displaystyle\leq$
$\displaystyle\varlimsup_{n\to\infty}\frac{1}{m_{G}(F_{n})}\int_{F_{n}}\mathbf{1}_{D(\delta,h^{\prime}-h)}([-g-t,h]_{\mathcal{L}})dt$
$\displaystyle\stackrel{{\scriptstyle(*)}}{{=}}$
$\displaystyle\mu_{\mathbb{T}}(D(\delta,h^{\prime}-h))$ $\displaystyle\leq$
$\displaystyle m_{G}(B_{G}(0,2/\delta))\cdot m_{H}(W\Delta(W+h^{\prime}-h))$
$\displaystyle\leq$ $\displaystyle m_{G}(B_{G}(0,2/\delta))\cdot
m_{H}(\overline{B_{H}(\partial W,d(0,h^{\prime}-h))}),$

where the equality $(*)$ follows from the unique ergodicity of
$(\mathbb{T},G)$ and the fact that $\mu_{\mathbb{T}}(\partial
D(\delta,h^{\prime}-h))=0$.

Now, suppose that $\delta>0$ and $\nu>0$ are given. Let

$\varepsilon=\inf\left\\{\eta>0\;|\;m_{H}\left(B_{H}\left(\partial
W,\eta\right)\right)\geq\nu/m_{G}\left(B_{G}(0,2/\delta)\right)\right\\}.$

Then we have $m_{H}(B_{H}(\partial
W,\varepsilon))\leq\nu/m_{G}(B_{G}(0,2/\delta))$ but at the same time
$m_{H}\big{(}\overline{B_{H}(\partial
W,\varepsilon)}\big{)}\geq\nu/m_{G}(B_{G}(0,2/\delta))$ due to the regularity
of Haar measure. Consequently, if $d_{G}(g,g^{\prime})<\delta/2$ and
$d_{H}(h,h^{\prime})<\varepsilon$, then the first inequality combined with the
above estimate yields that $\hat{\Gamma}_{g,h}$ and
$\hat{\Gamma}_{g^{\prime},h^{\prime}}$ cannot be $(\delta,\nu)$-separated.

For $g\in G$ and $h\in H$, let $\Gamma_{g,h}$ denote some element of
$\Omega(\mbox{\Large$\curlywedge$}(W))$ with
$\mbox{\Large$\curlywedge$}(\mathrm{int}(W)+h)-g\subseteq\Gamma_{g,h}\subseteq\hat{\Gamma}_{g,h}$,
see (9). We cover $A$ by $N=N_{\delta/2}(A)$ balls of radius $\delta/2$ and
$B$ by $M=N_{\varepsilon}(B)$ balls of radius $\varepsilon$ and denote by
$(g_{n})_{n=1}^{N}$ and $(h_{m})_{m=1}^{M}$ the midpoints of these balls. Then
the set $\\{\Gamma_{g_{n},h_{m}}\;|\;n=1,\ldots,N,\ m=1,\ldots,M\\}$ is
$(\delta,\nu)$-spanning due to the above and Corollary 5.3. We obtain the
estimate

$\displaystyle\overline{\mathrm{ac}}_{\mathcal{F}}(\Omega(\mbox{\Large$\curlywedge$}(W)),G)$
$\displaystyle=$
$\displaystyle\adjustlimits{\sup}_{\delta>0}{\varlimsup}_{\nu\to
0}\frac{\log\mathrm{Span}_{\mathcal{F}}(\Omega(\mbox{\Large$\curlywedge$}(W)),G,\delta,\nu)}{-\log\nu}$
$\displaystyle\leq$
$\displaystyle\adjustlimits{\sup}_{\delta>0}{\varlimsup}_{\varepsilon\to
0}\frac{\log(N_{\delta/2}(A)\cdot N_{\varepsilon}(B))}{-\log
m_{H}\big{(}\overline{B_{H}(\partial W,\varepsilon)}\big{)}}$ $\displaystyle=$
$\displaystyle\varlimsup_{\varepsilon\to 0}\frac{\log
N_{\varepsilon}(B)/-\log\varepsilon}{\log m_{H}\big{(}\overline{B_{H}(\partial
W,\varepsilon})\big{)}/\log\varepsilon}$ $\displaystyle\leq$
$\displaystyle\frac{\overline{\mathrm{dim}}_{\mathrm{B}}(H)}{\overline{\mathrm{dim}}_{\mathrm{B}}(H)-\overline{\mathrm{dim}}_{\mathrm{B}}(\partial
W)},$

where we used Minkowski’s characterisation in the last step. This completes
the proof. ∎

######  Remark 5.7.

It is not too difficult to see that the above result is optimal in the sense
that equality is attained for some examples while at the same time, it cannot
hold in general.

  * (a)

In order to see that amorphic complexity can be smaller than the bound
provided by (10), let $H=\mathbb{R}$ and suppose $C\subseteq\mathbb{R}$ is an
arbitrary Cantor set of dimension $d\in[0,1)$. Let $W$ be a window given by
the union of $C$ with a countable number of gaps (that is, bounded connected
components of $\mathbb{R}\setminus C$) such that $\partial W=C$. Clearly, this
can be done such that for each $n$, we have that $W$ contains less than $n$
intervals of size $2^{-n}$ or bigger. If $\varepsilon\in(2^{-n},2^{-n+1}]$,
then each of these intervals contributes at most $2\varepsilon$ to
$m_{H}(W\Delta(W+\varepsilon))$, whereas the union of the other intervals
contributes at most $\varepsilon$ in total (and $\partial W$ does not
contribute since it is of zero measure). Hence, we obtain
$m_{H}(W\Delta(W+\varepsilon))\leq 2\varepsilon n\leq
2\varepsilon(-\log\varepsilon/\log 2+1)$. Accordingly, the computation in the
proof of Theorem 5.5 yields
$\overline{\mathrm{ac}}_{\mathcal{F}}(\Omega(\mbox{\Large$\curlywedge$}(W)),G)\leq
1<\frac{1}{1-d}$.

  * (b)

The most straightforward examples in which equality is attained in (10) are
given by CPS with $H=\mathbb{R}$. We refrain from discussing the
technicalities (which are in spirit similar to those in the proof of the above
theorem) and simply sketch the main ingredients of the construction. For
$\gamma>2$, consider a middle segment Cantor set $C_{\gamma}$ which is
constructed by always removing the middle $(1-2/\gamma)$-th part of intervals
in the canonical construction of Cantor sets. Observe that $C_{\gamma}$ is of
dimension $\overline{\mathrm{dim}}_{\mathrm{B}}(C_{\gamma})=\log 2/\log\gamma$
with gaps of size $(1-2/\gamma)\cdot\gamma^{-n}$. If $W$ is the window that is
obtained by including all gaps of size $(1-2/\gamma)\cdot\gamma^{-n}$ with $n$
odd, it can be readily checked that

$\lim_{\varepsilon\to 0}\frac{\log
m_{H}(W\Delta(W+\varepsilon))}{\log\varepsilon}=(1-\log 2/\log\gamma).$

We may assume without loss of generality to be given an element $(u,v)$ of
some set of generators of $\mathcal{L}$ with $C_{\gamma}\subseteq[0,v]$. Let
$h_{1},\ldots,h_{\lfloor 1/\varepsilon\rfloor}\in H$ be equidistributed in
$[0,v]\subseteq H$. Similarly to the estimates in the proof of Theorem 5.5, it
can be checked that for small enough $\delta$, we have that
$\\{\Gamma_{0,h_{1}},\ldots\Gamma_{0,h_{\lfloor 1/\varepsilon\rfloor}}\\}$ is
$(\delta,\nu)$-separated with
$\nu=m_{G}(B_{G}(0,1/\delta))m_{H}(W\Delta(W+\varepsilon))$ as $\varepsilon$
(and hence $\nu$) tends to zero. Then one obtains
$\overline{\mathrm{ac}}_{\mathcal{F}}(\Omega(\mbox{\Large$\curlywedge$}(W)),G)=\sup_{\delta>0}\varlimsup_{\nu\to
0}\frac{\log\mathrm{Sep}_{\mathcal{F}}(\Omega(\mbox{\Large$\curlywedge$}(W)),G,\delta,\nu)}{-\log\nu}\geq\frac{1}{1-\overline{\mathrm{dim}}_{\mathrm{B}}(C_{\gamma})}$.

  * (c)

Note that the construction sketched in (b) yields uncountably many regular
model sets that lie in different conjugacy classes. In fact, it shows that any
value in $[1,\infty)$ can be realised as the amorphic complexity of a regular
model set.

  * (d)

The above considerations indicate that while the structure of the boundary of
the window inflicts some upper bound on the complexity of the dynamics of the
resulting model set, it greatly depends on the interior of the window whether
this bound is actually attained or not. This coincides with similar
observations concerning the topological entropy of irregular model sets
[JLO19].

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