# A Szemerédi type theorem for sets of positive density in approximate
lattices

Michael Björklund  Department of Mathematics, Chalmers, Gothenburg, Sweden
<EMAIL_ADDRESS>and  Alexander Fish  School of Mathematics and Statistics
F07, University of Sydney, NSW 2006, Australia<EMAIL_ADDRESS>
###### Abstract.

An extension of Szemerédi’s Theorem is proved for sets of positive density in
cut-and-project sets in general locally compact and second countable abelian
groups. As a consequence, we establish a recent conjecture of Klick, Strungaru
and Tcaciuc. Furthermore, we show a Szemeredi type theorem for sets of
positive density in the set of fractional parts of $\mathbb{Q}_{p}$. Via a
novel version of Furstenberg’s Correspondence principle, which should be of
independent interest, we show that our Szemerédi Theorems can be deduced from
a general _transverse_ multiple recurrence theorem, which we establish using
recent works of Austin.

###### Key words and phrases:

Szemerédi’s Theorem, approximate lattice, cut-and-project sets

###### 2020 Mathematics Subject Classification:

Primary: 11B30, 22D40. Secondary: 05D10

##  1\. Introduction

###  1.1. Szemerédi’s Theorem and the aim of the paper

The _upper asymptotic density_ $\overline{d}(P_{o})$ of a subset
$P_{o}\subset\mathbb{Z}$ is defined by

$\overline{d}(P_{o})=\varlimsup_{n\rightarrow\infty}\frac{|P_{o}\cap[-n,n]|}{2n+1}.$

The celebrated theorem of Szemerédi [20], with the ergodic-theoretical proof
of Furstenberg [14, Theorem 11.13], asserts that for every
$P_{o}\subset\mathbb{Z}$ with positive upper asymptotic density, the set

$S=\\{\lambda\in\mathbb{Z}\,:\,\textrm{$\exists\,\lambda_{o}\in P_{o}$ such
that}\enskip\lambda_{o}+\lambda\cdot F\subset P_{o}\\}$

is syndetic in $\mathbb{Z}$, i.e. there is a finite set $K\subset\mathbb{Z}$
such that $S+K=\mathbb{Z}$. In this article we are interested in whether a
version of Szemerédi’s Theorem still holds if one replaces the ring
$\mathbb{Z}$ with an _approximate ring_.

More precisely, let $R$ be a locally compact ring, e.g.
$R=\mathbb{R},\mathbb{C},\mathbb{Q}_{p}$ (for some prime number $p$) or the
Hamilton quaternions $\mathbb{H}$, and suppose $\Lambda\subset R$ is a
symmetric, uniformly discrete and multiplicatively closed subset of $R$, which
is an approximate subgroup under addition, i.e. it contains $0$ and there is a
finite set $K\subset R$ such that $\Lambda+\Lambda\subset\Lambda+K$. Let
$\Lambda^{\infty}\subset R$ denote the additive group generated by $\Lambda$
inside $(R,+)$. The aim of this paper is to provide a partial (affirmative)
answer to the following question (see Corollary 1.7 below).

######  Question.

Let $P_{o}\subset\Lambda$ be a ”large” subset. Given a finite set
$F\subset\Lambda^{\infty}$, define

$S_{F}=\\{\lambda\in\Lambda\,:\,\textrm{$\exists\,\lambda_{o}\in P_{o}$ such
that}\enskip\lambda_{o}+\lambda\cdot F\subset P_{o}\\}.$

Is $S_{F}$ a syndetic subset of $\Lambda$ for every $F$? In other words, can
we find, for every finite set $F\subset\Lambda^{\infty}$, a finite subset
$K_{F}\subset R$ such that $\Lambda\subseteq S_{F}+K_{F}$?

######  Remark 1.1.

Note that we do not require the finite set $F$ to be contained in $\Lambda$,
but only in $\Lambda^{\infty}$, which is typically a _dense_ subgroup of $R$.

The exact definition of ”large” will be given below. However, we want to
emphasize that for the examples we have in mind, the set $\Lambda$ typically
has zero upper Banach density inside the group $\Lambda^{\infty}$, so it is in
general very difficult to deduce anything about patterns inside $\Lambda$ from
combinatorial arguments within $\Lambda^{\infty}$. In the next subsection, we
provide some explicit classes of examples in $R=\mathbb{R}$ and
$R=\mathbb{Q}_{p}$, for which we can answer the question above. We then
present the general framework for our investigations, and our main theorems,
along with a proof of a recent conjecture by Klick, Strungaru and Tcaciuc
[17].

###  1.2. Some motivating examples

####  1.2.1. Examples from number fields

To keep things simple, let $D>0$ be a square-free integer, and define
$\Lambda\subset\mathbb{R}$ by

$\Lambda=\\{m+n\sqrt{D}\,:\,|m-n\sqrt{D}|\leq 1\\}.$

It is not hard to check that $\Lambda$ is symmetric, uniformly discrete in
$\mathbb{R}$ and multiplicatively closed. Furthermore,

$\Lambda+\Lambda\subset\Lambda+\\{-1,0,1\\},$

so $\Lambda$ is an approximate subgroup of $\mathbb{R}$ and
$\Lambda^{\infty}=\mathbb{Z}[\sqrt{D}]$. We say that a subset
$P_{o}\subset\Lambda$ has _positive upper asymptotic density_ if

$\varlimsup_{n\rightarrow\infty}\frac{|P_{o}\cap[-n,n]|}{2n}>0,$

where $[-n,n]$ is viewed as an interval in $\mathbb{R}$. The following result
is a special case of Corollary 1.7 below.

######  Theorem 1.2.

Suppose $P_{o}\subset\Lambda$ has positive upper asymptotic density. Then, for
every finite set $F\subset\mathbb{Z}[\sqrt{D}]$, the set

$S_{F}=\\{\lambda\in\Lambda\,:\,\textrm{$\exists\,\lambda_{o}\in P_{o}$ such
that}\enskip\lambda_{o}+\lambda\cdot F\subset P_{o}\\}$

is syndetic in $\Lambda$.

######  Remark 1.3.

The non-emptiness of the set $S_{F}$ can also be proved using the IP-version
of Szemerédi’s Theorem due to Furstenberg and Katznelson [16]. We sketch the
alternative proof of Theorem 1.2 in the appendix. This proof can also be
extended to the more general setting of cut-and-project sets discussed below
(see the appendix for more details). However, for lcsc abelian groups like
$\mathbb{Q}_{p}$ without _dense finitely generated subgroups_ , it seems
difficult (but certainly not impossible) to establish syndeticity of sets like
$S_{F}$ using this approach, but we believe that such a proof would definitely
be interesting as it might provide a way to prove Conjecture 1.2 below. Our
proof of syndeticity of $S_{F}$ instead employs some recent results of Austin
[2].

####  1.2.2. $p$-adic examples

Let $p$ be a prime number and define the set
$\Lambda_{p}\subset\mathbb{Q}_{p}$ by

$\Lambda_{p}=\\{\gamma\in\mathbb{Z}[1/p]\,:\,|\gamma|_{\infty}\leq 1\\},$

where $|\cdot|_{\infty}$ denotes the (real) absolute value when
$\mathbb{Z}[1/p]$ is viewed as a sub-ring of $\mathbb{R}$. Even though
$\mathbb{Q}_{p}$ does not admit a lattice, the set $\Lambda_{p}$ is
nevertheless an approximate lattice, i.e. it is symmetric, uniformly discrete
and relatively dense _approximate subgroup_ of $\mathbb{Q}_{p}$. Furthermore,
$\Lambda_{p}$ is multiplicatively closed and
$\Lambda_{p}^{\infty}=\mathbb{Z}[1/p]$. The set $\Lambda_{p}$ is sometimes
referred to as the _fractional parts_ of $\mathbb{Q}_{p}$.

Note that the sequence $(F_{n})$ defined by
$F_{n}=p^{-n}\mathbb{Z}_{p}\subset\mathbb{Q}_{p}$ is a Følner sequence in
$\mathbb{Q}_{p}$ and $m_{\mathbb{Q}_{p}}(F_{n})=p^{n}$, where
$m_{\mathbb{Q}_{p}}$ denotes the Haar measure on $\mathbb{Q}_{p}$, normalized
so that $m_{\mathbb{Q}_{p}}(\mathbb{Z}_{p})=1$. We say that a subset
$P_{o}\subset\Lambda_{p}$ has _positive upper asymptotic density_ if

$\varlimsup_{n\rightarrow\infty}\frac{|P_{o}\cap F_{n}|}{p^{n}}>0.$

The following result is a special case of Corollary 1.7 below.

######  Theorem 1.4.

Suppose $P_{o}\subset\Lambda_{p}$ has positive upper asymptotic density. Then,
for every finite set $F\subset\mathbb{Z}[1/p]$, the set

$S_{F}=\\{\lambda\in\Lambda\,:\,\textrm{$\exists\,\lambda_{o}\in P_{o}$ such
that}\enskip\lambda_{o}+\lambda\cdot F\subset P_{o}\\}$

is syndetic in $\Lambda_{p}$.

###  1.3. General framework

Both of the examples above are special cases of the _cut-and-project
construction_ , which we now review in general. Let $G$ and $H$ be locally
compact and second countable (lcsc) abelian groups, and let $\Gamma<G\times H$
be a (uniform) lattice such that the projection of $\Gamma$ to $G$ is
injective and the projection of $\Gamma$ to $H$ is dense. Let $W\subset H$ be
a bounded and symmetric Borel set with non-empty interior, _containing $0$_,
and define the _cut-and-project set_ $\Lambda=\Lambda(G,H,\Gamma,W)$ by

$\Lambda=\operatorname{pr}_{G}(\Gamma\cap(G\times W))\subset G,$

where $\operatorname{pr}_{G}$ denotes the projection $(g,h)\mapsto g$ from
$G\times H$ to $G$. We stress that our assumptions that $W$ is symmetric and
its interior contains $0$ are not standard, and in most of the literature,
cut-and-project sets are defined without these assumptions. They are not
strictly necessary for our theorems, but do simplify their exposition.

The examples in the previous sub-section are of this form with

$G=H=\mathbb{R},\enskip\Gamma=\\{(m+n\sqrt{D},m-n\sqrt{D})\,:\,m,n\in\mathbb{Z}\\},\enskip
W=[-1,1],$

and

$G=\mathbb{Q}_{p},\enskip
H=\mathbb{R},\enskip\Gamma=\\{(\gamma,\gamma)\,:\,\gamma\in\mathbb{Z}[1/p]\\},\enskip
W=[-1,1]$

respectively. It is not difficult that every $\Lambda$ of this form is
uniformly discrete and relatively dense (i.e. there is a compact set $Q\subset
G$ such that $Q+\Lambda=G$). Furthermore, there is a finite $F\subset G$ such
that

$\Lambda-\Lambda\subset\Lambda+F.$

The study of cut-and-project sets was initiated by Yves Meyer [18].

We say that a uniformly discrete and relatively dense subset $\Lambda\subset
G$ is a (uniform) _approximate lattice_ if it is symmetric, $0\in\Lambda$ and
there is a finite set $F$ in $G$ such that $\Lambda+\Lambda\subset\Lambda+F$.
Note that if $\Lambda$ is an approximate lattice, then for every $q\geq 1$,
the $q$-iterated sum

$\Lambda^{q}=\underbrace{\Lambda+\ldots+\Lambda}$

is contained in a finite union of translates of $\Lambda$, and is thus again
uniformly discrete (and relatively dense) in $G$. Furthermore, every cut-and-
project set $\Lambda(G,H,\Gamma,W)$, for a _symmetric_ set $W\subset H$ set
whose interior contains $0$, is an approximate lattice.

Fix a Haar measure $m_{G}$ on $G$. Given a strong Følner sequence $(F_{n})$
(see Subsection 2.3 for definitions), we define the _upper asymptotic density
$\overline{d}_{(F_{n})}(P_{o})$_ of a uniformly discrete subset $P_{o}\subset
G$ by

$\overline{d}_{(F_{n})}(P_{o})=\varlimsup_{n\rightarrow\infty}\frac{|P_{o}\cap
F_{n}|}{m_{G}(F_{n})},$

and the _upper Banach density $d^{*}(P_{o})$_ by

$d^{*}(P_{o})=\sup\big{\\{}\overline{d}_{(F_{n})}(P_{o})\,:\,\textrm{$(F_{n})$
is a strong F\o lner sequence in $G$}\\}.$

We show below (Subsection 2.4) that $d^{*}(P_{o})$ is always finite.

###  1.4. Main combinatorial results

We can now state our first main result, whose proof will be given in Section
6. Let $G$ be a lcsc abelian group, and denote by $\operatorname{End}(G)$ the
space of continuous endomorphisms on $G$.

######  Theorem 1.5.

Let $\Lambda\subset G$ be a an approximate lattice and suppose there is a set
$\Lambda_{o}\subset\Lambda$ with positive upper Banach density such that
$\Lambda_{o}-\Lambda_{o}\subset\Lambda$. Let $P_{o}\subset\Lambda$ be subset
with positive Banach upper density. Suppose that there exist
$\alpha_{1},\ldots,\alpha_{r}\in\operatorname{End}(G)$ and $q\geq 1$ such that
$\alpha_{k}(\Lambda)\subset\Lambda^{q}$ for all $k$. Then there exists $c>0$
such that the set

$S=\Big{\\{}\lambda\in\Lambda\,:\,d^{*}\Big{(}P_{o}\cap\Big{(}\bigcap_{k=1}^{r}(P_{o}-\alpha_{k}(\lambda))\Big{)}\Big{)}\geq
c\Big{\\}}$

is syndetic in $\Lambda$. In particular, there exists a subset
$P_{o}^{\prime}\subset P_{o}$ with positive upper Banach density, such that

$\lambda_{o}+\\{\alpha_{1}(\lambda),\ldots,\alpha_{r}(\lambda)\\}\subset
P_{o},\quad\textrm{for all $\lambda_{o}\in P_{o}^{\prime}$ and $\lambda\in
S$}.$

######  Remark 1.6.

If $\Lambda$ is an approximate lattice in $G$ which is also a cut-and-project
set of the form $\Lambda(G,H,\Gamma,W)$ for some _symmetric_ set $0\in
W^{o}\subset W\subset H$ then we can choose
$\Lambda_{o}=\Lambda(G,H,\Gamma,W_{o})$ in Theorem 1.5, where $W_{o}$ is an
identity neighbourhood in $H$ such that $W_{o}-W_{o}\subset W^{o}$. Note that
in this case, $\Lambda_{o}$ is relatively dense. However, there are examples
of approximate lattices (already for $G=\mathbb{R}$) which do not contain the
difference set of any relatively dense set [9, Example, Subsection 3.1]. These
examples do however contain the difference set of a set of positive upper
Banach density, so the theorem above is applicable. We further stress that
these examples do _not_ contain any higher order difference sets, i.e. they do
not contain sets of the form
$\Lambda_{o}-\Lambda_{o}+\Lambda_{o}-\Lambda_{o}$, for some subset
$\Lambda_{o}$ of positive upper Banach density.

If $G=(R,+)$ for some locally compact ring $R$, we can deduce the following
corollary from Theorem 1.5.

######  Corollary 1.7.

Let $R$ be a locally compact and second countable ring, and let
$\Lambda\subset R$ be a multiplicatively closed cut-and-project set. Suppose
$P_{o}\subset\Lambda$ has positive upper Banach density in $(R,+)$. Then, for
every finite set $F\subset\Lambda^{\infty}$, the set

$S_{o}=\\{\lambda\in\Lambda\,:\lambda_{o}+\lambda\cdot F\subset
P_{o}\enskip\textrm{for some $\lambda_{o}\in P_{o}$}\\}$

is syndetic in $\Lambda$.

###### Proof.

Let $\Lambda=\Lambda(H,\Gamma,W)\subset R$ for some $H,\Gamma$ and $W$, and
suppose $\Lambda$ is multiplicatively closed, i.e.
$\Lambda\cdot\Lambda\subset\Lambda$. Since we assume that the set $W$ has non-
empty interior and contains $0$, we can find a symmetric and open set
$W_{o}\subset W$ which contains $0$ such that $W_{o}-W_{o}\subset W$. If we
define $\Lambda_{o}=\Lambda_{o}(R,H,\Gamma,W_{o})$, then
$\Lambda_{o}\subset\Lambda$ is a Meyer set in $R$ (and thus has positive upper
Banach density) and

$\Lambda_{o}-\Lambda_{o}\subset\Lambda.$

Let $F=\\{\lambda_{1},\ldots,\lambda_{r}\\}\subset\Lambda^{\infty}$ be a
finite set, and let $q\geq 1$ denote the smallest integer such that
$F\subset\Lambda^{q}$. Define
$\alpha_{1}\ldots,\alpha_{r}\in\operatorname{End}(R)$ by
$\alpha_{k}(g)=g\lambda_{k}$ for $k=1,\ldots,r$. Then, since
$\Lambda\cdot\Lambda\subset\Lambda$, we have

$\alpha_{k}(\Lambda)\subset\Lambda\cdot\Lambda^{q}\subset\Lambda^{q},\quad\textrm{for
all $k=1,\ldots,q$}.$

Hence all conditions of Theorem 1.5 are satisfied, and there is a constant
$c>0$ such that

$S=\Big{\\{}\lambda\in\Lambda\,:\,d^{*}\Big{(}P_{o}\cap\Big{(}\bigcap_{k=1}^{r}\big{(}P_{o}-\lambda\lambda_{k}\big{)}\Big{)}\Big{)}\geq
c\\}$

is syndetic in $\Lambda$. Clearly, $S\subset S_{o}$, so we are done. ∎

###  1.5. Solution to a recent conjecture of Klick, Strungaru and Tcaciuc

Theorem 1.5 also provides a solution to the following conjecture, recently
made by Klick, Strungaru and Tcaciuc [17, Conjecture 6.7]. Recall that a
subset $\\{\lambda_{1},\ldots,\lambda_{r}\\}$ in a lcsc group $G$ is an
_arithmetic progression of length $r$_ if
$\lambda=\lambda_{k+1}-\lambda_{k}\neq 0$ for all $k=1,\ldots,r-1$, i.e. if
$\lambda_{k}=\lambda_{o}+k\cdot\lambda$ for all $k=1,\ldots,r$, for some
$\lambda_{o}$ and $\lambda\in G$. We refer to $\lambda$ as the _gap_ of the
arithmetic progression.

######  Conjecture 1.1.

Let $G$ be a lcsc group and let $\Lambda\subset G$ be a cut-and-project set.
Then every subset $P_{o}\subset\Lambda$ with positive upper Banach density
contains arbitrary long arithmetic progressions.

Let us now briefly show how Theorem 1.5 establishes the following
strengthening of this conjecture.

######  Theorem 1.8.

Let $G$ be a lcsc group, let $\Lambda\subset G$ be a cut-and-project set and
suppose that $P_{o}\subset\Lambda$ has positive upper Banach density. Then,
for every $r\geq 1$, there exist a syndetic set $S_{r}\subset\Lambda$ and a
set $P_{r}\subset P_{o}$ of positive upper Banach density, such that

$\lambda_{o}+\\{\lambda,2\lambda,\ldots,r\lambda\\}\subset P_{o},$

for all $\lambda_{o}\in P_{r}$ and $\lambda\in S_{r}$. In particular, $P_{o}$
contains arbitrary long arithmetic progressions, and the set of all possible
gaps of these arithmetic progressions contains a syndetic subset of $\Lambda$.

###### Proof.

Let $\Lambda\subset G$ be a cut-and-project set and let $P_{o}\subset\Lambda$
be a set of positive upper Banach density. We may without loss of generality
assume that $0\in\Lambda$. Fix $r\geq 1$, and define
$\alpha_{1},\ldots,\alpha_{r}\in\operatorname{End}(G)$ by

$\alpha_{k}(g)=k\cdot g,\quad\textrm{for $k=1,\ldots,r$}.$

Then, since $\Lambda\subset\Lambda^{2}\ldots\subset\Lambda^{r}$, we have
$\alpha_{k}(\Lambda)\subset\Lambda^{r}$ for all $k=1,\ldots,r$, so the
conditions of Theorem 1.5 are satisfied with $q=r$, so we conclude that there
exists a syndetic set $S_{r}\subset\Lambda$ and a set $P_{r}\subset P_{o}$
with positive upper Banach density such that

$\lambda_{o}+\\{\alpha_{1}(\lambda),\ldots,\alpha_{r}(\lambda)\\}=\lambda_{o}+\\{\lambda,2\lambda,\ldots,r\lambda\\}\subset
P_{o}$

for all $\lambda\in P_{r}$ and $\lambda\in S_{r}$. ∎

###  1.6. Main dynamical result

In Section 3 we establish a version of Furstenberg’s famous Correspondence
Principle for uniformly discrete subsets of a lcsc abelian group $G$. Using
this Correspondence Principle, it is not hard to deduce Theorem 1.5 from the
following dynamical statement, whose proof will be given in Section 5.

Let $(X,\mathscr{B}_{X})$ be a standard Borel space, equipped with a Borel
measurable $G$-action $G\times X\rightarrow X,\enskip(g,x)\mapsto g.x$. We say
that a Borel set $Y\subset X$ is a _separated cross-section_ if $G.Y=X$ and
$\Xi_{Y}\cap U=\\{0\\}$, for some open identity neighbourhood $U$ in $G$,
where $\Xi_{Y}$ denotes the set of return times, defined by

$\Xi_{Y}=\\{g\in G\,:\,g.Y\cap Y\neq\emptyset\\}\subset G.$

For every $\mu\in\operatorname{Prob}_{G}(X)$, there exists a unique non-
negative Borel measure $\mu_{Y}$ on $Y$ (called the _transverse measure_
associated to $\mu$) such that $\mu(V.B)=m_{G}(V)\cdot\mu_{Y}(B)$ for every
Borel set $V\subset G$ such that $V-V\subset U$ and for every Borel set
$B\subset Y$. It is well-known that a separated cross-section always exists.
We refer to Subsection 2.1 for more details.

######  Theorem 1.9 (Multiple recurrence for cross-sections).

Let $\Lambda\subset G$ be an approximate lattice and suppose that there exists
a set $\Lambda_{o}\subset\Lambda$ with positive upper Banach density such that
$\Lambda_{o}-\Lambda_{o}\subset\Lambda$. Let
$\alpha_{1},\ldots,\alpha_{r}\in\operatorname{End}(G)$ and suppose that there
is $q\geq 1$ such that $\alpha_{k}(\Lambda)\subset\Lambda^{q}$ for all $k$. If
there is an open identity neighbourhood $V\subset U$ such that
$(\Xi_{Y}-\Lambda^{q})\cap V=\\{0\\}$, then for every
$\mu\in\operatorname{Prob}_{G}(X)$ and for every Borel set $B\subset Y$ with
positive $\mu_{Y}$-measure, there exists $c>0$ such that the set

$S=\Big{\\{}\lambda\in\Lambda\,:\,\mu_{Y}\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).B\Big{)}\geq
c\Big{\\}}$

is syndetic in $\Lambda$.

###  1.7. Potential polynomial extensions

A series of impressive extensions of Szemerédi’s Theorem have been proved over
the years. Polynomial versions have been particularly popular, starting with
the ground-breaking works of Bergelson, Leibman and McCutcheon [4, 5, 6]. In
view of these works, specifically [6, Theorem 0.3], we feel that it is natural
to formulate the following conjectural polynomial strengthening of Corollary
1.7. If $R$ is a locally compact ring, a _polynomial_ is a map $p:R\rightarrow
R$ of the form

$p(x)=\sum_{k=0}^{r}a_{k}x^{k},\quad\textrm{for some
$a_{o},a_{1},\ldots,a_{r}\in R$}.$

######  Conjecture 1.2.

Let $R$ be a locally compact and second countable ring and let $\Lambda\subset
R$ be a multiplicatively closed cut-and-project set. Suppose that
$P_{o}\subset\Lambda$ has positive upper Banach density in $(R,+)$. Then, for
every $r\geq 1$, for all polynomials $p_{1},\ldots,p_{r}:R\rightarrow R$ such
that $p_{k}(0)=0$ and $p_{k}(\Lambda)\subset\Lambda^{q}$ for some $q\geq 1$
and for all $k$, and for every finite set $F\subset\Lambda^{\infty}$, the set

$S_{F}=\\{\lambda\in\Lambda\,:\,\textrm{$\exists\,\lambda_{o}\in P_{o}$ such
that}\enskip\lambda_{o}+\\{p_{1}(\lambda),\ldots,p_{r}(\lambda)\\}\subset
P_{o}\\}$

is syndetic.

The arguments in this paper, more specifically the ones to prove Proposition
5.3 below, are currently not flexible enough to prove this conjecture.

Similarly, given that much of the ergodic-theoretical machinery can be
extended to actions of nilpotent groups, it is not be unreasonable to expect
that some version of Theorem 1.5 (and Theorem 1.9) could be extended to cut-
and-project sets in lcsc _nilpotent_ groups. For more details about such sets,
we refer the reader to [8, 9].

###  1.8. Organization of the paper

In Section 2 we introduce notation and some fundamental concepts like cross-
sections, transverse measures, Banach densities and strong Følner sequences.
The most important result in this section is a transverse version of
Furstenberg’s Correspondence Principle (Theorem 3.1). In Section 4 we extend
Furstenberg and Katznelson’s multiple recurrence theorem to general locally
compact and second countable abelian groups, using recent works of Austin. In
Section 5, we prove our main dynamical theorem (Theorem 1.9), and in Section 6
we prove our main combinatorial theorem (Theorem 1.5). Finally, in the
appendix we show how a weaker version of Theorem 1.5 for cut-and-project sets
in $\mathbb{R}^{d}$ can be deduced from Furstenberg and Katznelson’s IP-theory
developed in the seminal paper [16].

###  1.9. Acknowledgements

This work was completed during the M.B’s research visit to SMRI (Sydney
Mathematical Research Institute), and he wishes to express his gratitude to
this organization for its hospitality. M.B is grateful to John Griesmer for
useful discussions on the topic of the paper. M.B. was supported by the grants
11253320 and 2023-03803 from the Swedish Research Council. A.F. was supported
by the ARC via grants DP210100162 and DP240100472.

##  2\. Preliminaries

Let $G$ be a locally compact and second countable (lcsc) abelian group. In
particular, as a topological space, $G$ is $\sigma$-compact [11, Theorem
2.B.4]. Let us also fix a Haar measure $m_{G}$ on $G$ for the rest of our
discussions.

###  2.1. Borel $G$-spaces, separated cross-sections and transverse measures

Let $(X,\mathscr{B}_{X})$ be a standard Borel space, equipped with a Borel
measurable action $a:G\times X\rightarrow X$. We refer to
$(X,\mathscr{B}_{X})$ as a _Borel $G$-space_. To make the notation less heavy,
we often write $a(g,x)=g.x$. We denote by $\operatorname{Prob}_{G}(X)$ the
space of $G$-invariant Borel probability measures on $X$. A Borel set
$Y\subset X$ is a _cross-section_ (or _transversal_) if $G.Y=X$, and we denote
by $\Xi_{Y}\subset G$ the set of _return times_ , defined by

$\Xi_{Y}=\\{g\in G\,:\,g.Y\cap Y\\}.$

If $U$ is an open identity neighbourhood in $G$, we say that a cross-section
$Y$ is _$U$ -separated_ if $U\cap\Xi_{Y}=\\{0\\}$. It is well-known (see e.g.
[19, Theorem 2.4]) that every Borel $G$-space admits $U$-separated cross-
sections for some open identity neighbourhood.

In what follows, we fix an open and bounded identity neighbourhood $U$ in $G$,
and a $U$-separated cross-section $Y\subset X$. The following result is
standard (see e.g. [10, Proposition 4.3]).

######  Lemma 2.1.

For all Borel sets $V\subset G$ and $B\subset Y$, the action set $V.B\subset
X$ is Borel measurable. Furthermore, for every
$\mu\in\operatorname{Prob}_{G}(X)$, there exists a _unique_ non-negative
finite Borel measure $\mu_{Y}$ on $Y$ such that

$\mu(V.B)=m_{G}(V)\cdot\mu_{Y}(B),$

for all Borel sets $V\subset G$ and $B\subset Y$ such that $V-V\subset U$.

######  Remark 2.2.

Note that we can always view $\mu_{Y}$ as a Borel measure on $X$ by defining
$\mu_{Y}(B)=\mu_{Y}(B\cap Y)$ for $B\in\mathscr{B}_{X}$.

Given $\mu\in\operatorname{Prob}_{G}(X)$, we refer to the Borel measure
$\mu_{Y}$ (which is finite, but not a probability measure in general) as the
_transverse measure associated to $\mu$_. Note that

$\mu_{Y}(B)>0\iff\mu(V.B)>0,$

for some (any) identity neighbourhood $V$ in $G$. The following lemma will be
used in several places in the proofs below.

######  Lemma 2.3.

Let $\Delta\subset G$ and suppose that there is an open identity neighbourhood
$W$ in $G$ such that $(\Xi_{Y}-\Delta)\cap W=\\{0\\}$. For every Borel set
$V_{o}$ in $G$ such that $V_{o}-V_{o}\subset W$,

$V_{o}.\Big{(}\bigcap_{k=1}^{r}(-g_{k}).B\Big{)}=B_{V_{o}}\cap\Big{(}\bigcap_{k=1}^{r}(-g_{k}).B_{V_{o}}\Big{)},\quad\textrm{for
all $g_{1},\ldots,g_{r}\in\Delta$},$

where $B_{V_{o}}=V_{o}.B$. In particular, for every
$\mu\in\operatorname{Prob}_{G}(X)$,

$\mu_{Y}\Big{(}\bigcap_{k=1}^{r}(-g_{k}).B\Big{)}=\frac{1}{m_{G}(V_{o})}\cdot\mu\Big{(}B_{V_{o}}\cap\Big{(}\bigcap_{k=1}^{r}(-g_{k}).B_{V_{o}}\Big{)}\Big{)}.$

###### Proof.

Fix $V_{o}$ as in the lemma and an $r$-tuple $g_{1},\ldots,g_{r}\in\Delta$. We
first note that the inclusion

$V_{o}.\Big{(}\bigcap_{k=1}^{r}(-g_{k}).B\Big{)}\subseteq
B_{V_{o}}\cap\Big{(}\bigcap_{k=1}^{r}(-g_{k}).B_{V_{o}}\Big{)}$

is trivial. In particular, if the right-hand side is empty, so is the left-
hand side. It thus remains to show that if the right-hand side is non-empty,
then

$V_{o}.\Big{(}\bigcap_{k=1}^{r}(-g_{k}).B\Big{)}\supseteq
B_{V_{o}}\cap\Big{(}\bigcap_{k=1}^{r}(-g_{k}).B_{V_{o}}\Big{)}.$ (2.1)

To do this, fix an element $x$ in the set on the right-hand side of (2.1). By
definition, we can find $v_{o},v_{1},\ldots,v_{r}\in V_{o}$ and
$y_{o},y_{1},\ldots,y_{r}\in B$ such that

$x=v_{o}.y_{o}=(v_{1}-g_{1}).y_{1}=\ldots=(v_{r}-g_{r}).y_{r}.$

In particular, $v_{o}-v_{k}+g_{k}\in\Xi_{Y}$ for all $k=1,\ldots,r$ and thus

$v_{o}-v_{k}\in(\Xi_{Y}-g_{k})\cap(V_{o}-V_{o})\subseteq(\Xi_{Y}-\Delta)\cap
W=\\{0\\},\quad\textrm{for all $k=1,\ldots,r$}.$

We conclude that $v_{o}=v_{1}=\ldots=v_{r}$, and thus
$y_{o}=(-g_{1}).y_{1}=\ldots=(-g_{r}).y_{r}$. In particular,

$x=v_{o}.y_{o}\in v_{o}.\Big{(}\bigcap_{k=1}^{r}(-g_{k}).B\Big{)}\subset
V_{o}.\Big{(}\bigcap_{k=1}^{r}(-g_{k}).B\Big{)}.$

Since $x$ is arbitrary, this shows (2.1), and we are done. ∎

###  2.2. The Chabauty topology and hulls of uniformly discrete sets

We denote by $\mathscr{C}(G)$ the space of closed subsets of $G$, endowed with
the Chabauty topology. We recall from [3, Section E.1] that $\mathscr{C}(G)$
is a compact metrizable space, and a sequence $(P_{n})$ in $\mathscr{C}(G)$
converges to some $P\in\mathscr{C}(G)$ if and only if the following two
conditions hold:

  * (i)

If $p_{n_{k}}\in P_{n_{k}}$ for some sub-sequence $(n_{k})$ and
$p_{n_{k}}\rightarrow p$ in $G$, then $p\in P$

  * (ii)

For every $p\in P$, there exist $p_{n}\in P_{n}$ such that $p_{n}\rightarrow
p$ in $G$.

We equip $\mathscr{C}(G)$ with the jointly continuous $G$-action:

$(g,P)\mapsto g.P=P-g,\quad(g,P)\in G\times\mathscr{C}(G).$

Given $P_{o}\in\mathscr{C}(G)$, we define

$\Omega_{P_{o}}:=\overline{G.P_{o}}\quad\textrm{and}\quad\Omega_{P_{o}}^{\times}:=\Omega_{P_{o}}\setminus\\{\emptyset\\},$

and refer to $\Omega_{P_{o}}$ as the _hull of $P_{o}$_ and to
$\Omega_{P_{o}}^{\times}$ as the _punctured hull of $P_{o}$_. Note that the
space $\Omega_{P_{o}}$ is compact, while $\Omega_{P_{o}}^{\times}$ is only
compact if $\emptyset\notin\Omega_{P_{o}}$. We also define the _canonical
cross-section $\mathcal{T}_{P_{o}}\subset\Omega_{P_{o}}^{\times}$_ by

$\mathcal{T}_{P_{o}}=\\{P\in\Omega_{P_{o}}^{\times}\,:\,0\in P\\}.$

Note that $\mathcal{T}_{P_{o}}$ is a compact subset of
$\Omega_{P_{o}}^{\times}$, and for every open set $V\subset G$, the set

$\mathcal{O}_{V}:=(-V).\mathcal{T}_{P_{o}}=\\{P\in\Omega_{P_{o}}\,:\,P\cap
V\neq\emptyset\\}$

is open in both $\Omega_{P_{o}}$ and $\Omega_{P_{o}}^{\times}$.

We say that $P\subset G$ is _uniformly discrete_ if $(P-P)\cap U=\\{0\\}$ for
some identity neighbourhood $U$ in $G$. If we want to emphasize the dependence
on $U$, we say that the set $P$ is _$U$ -uniformly discrete_. We say that
$P\subset G$ _has finite local complexity_ if $P-P$ is locally finite. Clearly
every set with finite local complexity is uniformly discrete.

In what follows, let $P_{o}\subset G$ be a $U$-uniformly discrete set for some
open and bounded identity neighbourhood $U$ in $G$.

######  Lemma 2.4.

Let $P\in\Omega_{P_{o}}$. Then,

$P-P\subset\overline{P_{o}-P_{o}}\quad\textrm{for all
$P\in\Omega^{\times}_{P_{o}}$},$

and $\mathcal{T}_{P_{o}}$ is a $U$-separated cross-section in
$\Omega_{P_{o}}^{\times}$ with
$\Xi_{\mathcal{T}_{P_{o}}}\subset\overline{P_{o}-P_{o}}$.

###### Proof.

Fix an open identity neighbourhood $W$ in $G$ and a sequence $(g_{n})$ in $G$
such that $g_{n}.P_{o}\rightarrow P$ as $n\rightarrow\infty$. Then, by using
(i) in the characterization of sequential convergence in the Chabauty topology
above, we see that for every pair $p_{1},p_{2}\in P$, we can find
$w_{1,n},w_{2,n}\in W$ and $p^{o}_{1,n},p^{o}_{2,n}\in P_{o}$ such that

$p_{1}=p^{o}_{1,n}-g_{n}+w_{1,n}\quad\textrm{and}\quad
p_{2}=p^{o}_{2,n}-g_{n}+w_{2,n},$

for all sufficiently large $n$. In particular,

$p_{1}-p_{2}=(p_{1,n}^{o}+w_{1,n})-(p_{2,n}^{o}+w_{2,n})\subset
P_{o}-P_{o}+W-W.$

Since $p_{1}$ and $p_{2}$ are arbitrary in $P$, we have $P-P\subset
P_{o}-P_{o}+W-W$. Furthermore, $W$ is arbitrary, and thus
$P-P\subset\overline{P_{o}-P_{o}}$. For the last assertion, pick
$g\in\Xi_{\mathcal{T}_{P_{o}}}$ and an element $P\in\mathcal{T}_{P_{o}}\cap
g.\mathcal{T}_{P_{o}}$. Then $0\in P$ and $-g\in P$, and thus $g\in P-P$. From
the previous inclusion, we conclude that
$\Xi_{\mathcal{T}_{P_{o}}}\subset\overline{P_{o}-P_{o}}$. Since $P_{o}$ is
$U$-uniformly discrete, it follows that $\Xi_{\mathcal{T}_{P_{o}}}\cap
U=\\{0\\}$ and thus $\mathcal{T}_{P_{o}}$ is $U$-separated. ∎

If $M$ is a locally compact metrizable space and $\nu$ is a finite non-
negative Borel measure on $M$, then a Borel set $B\subset M$ is called _$\nu$
-Jordan measurable_ if $\nu(B^{o})=\nu(\overline{B})$. By choosing a
continuous metric $\rho$ on $M$ which induces the topology, it is not hard to
show that for every $m\in M$, there are at most countably many radii $r>0$
such that $\rho$-ball of radius $r$ around $m$ is _not_ $\nu$-Jordan
measurable. In particular, for every $m\in M$ and open neighbourhood $U$
around $m$, there is a $\nu$-Jordan measurable open neighbourhood of $m$
contained in $U$.

######  Lemma 2.5.

Let $P_{o}\subset G$ be a $U$-uniformly discrete set for some open identity
neighbourhood $U$ in $G$ and let $V$ be an open and $m_{G}$-Jordan measurable
subset of $G$ such that $\overline{V}-\overline{V}\subset U$. Then, for every
$\mu\in\operatorname{Prob}_{G}(\Omega_{P_{o}})$, the open set
$\mathcal{O}_{V}\subset\Omega_{P_{o}}$ is $\mu$-Jordan measurable.

###### Proof.

Fix $\mu\in\operatorname{Prob}_{G}(\Omega_{P_{o}})$ and write
$\mu=\mu^{\prime}+\alpha\cdot\delta_{\\{\emptyset\\}}$ for some finite and
non-negative Borel measure $\mu^{\prime}$ on $\Omega_{P_{o}}^{\times}$ and
$\alpha\geq 0$. Recall that Lemma 2.4 tells us that $\mathcal{T}_{P_{o}}$ is a
$U$-separated cross-section in $\Omega_{P_{o}}^{\times}$, and thus the
transverse measure $(\mu^{\prime})_{\mathcal{T}_{P_{o}}}$ is well-defined.
Since $\mathcal{O}_{V}$ is open,
$\overline{\mathcal{O}_{V}}\subset\mathcal{O}_{\overline{V}}$ and
$\delta_{\\{\emptyset\\}}(\mathcal{O}_{\overline{V}})=0$, we have

$\displaystyle\mu(\mathcal{O}_{V})$
$\displaystyle=\mu^{\prime}(\mathcal{O}_{V})=m_{G}(V)\cdot(\mu^{\prime})_{\mathcal{T}_{P_{o}}}(\mathcal{T}_{P_{o}})=m_{G}(\overline{V})\cdot(\mu^{\prime})_{\mathcal{T}_{P_{o}}}(\mathcal{T}_{P_{o}})$
$\displaystyle=\mu^{\prime}(\mathcal{O}_{\overline{V}})=\mu(\mathcal{O}_{\overline{V}})\geq\mu(\overline{\mathcal{O}_{V}}),$

and thus $\mathcal{O}_{V}$ is $\mu$-Jordan measurable. Since $\mu$ is
arbitrary, we are done. ∎

We also record the following lemma for future use.

######  Lemma 2.6.

Let $\Lambda\subset G$ be a locally finite set and suppose that there exists a
uniformly discrete set $\Lambda_{o}$ such that
$\Lambda_{o}-\Lambda_{o}\subset\Lambda$. Then for all identity neighbourhoods
$V_{o}$ and $V$ in $G$ such that $V_{o}-V_{o}\subset V$ and for all
$\nu\in\operatorname{Prob}_{G}(\Omega^{\times}_{\Lambda_{o}})$, we have

$\sum_{\lambda\in\Lambda}\chi_{V}(g+\lambda)\geq\nu(\mathcal{O}_{V_{o}}\cap
g.\mathcal{O}_{V_{o}}),\quad\textrm{for all $g\in G$},$

where $\chi_{V}$ denotes the indicator function of $V$, and
$\mathcal{O}_{V_{o}}=(-V_{o}).\mathcal{T}_{\Lambda_{o}}\subset\Omega_{\Lambda_{o}}^{\times}$.

###### Proof.

It is clearly enough to show that

$\mathcal{O}_{V_{o}}\cap
g.\mathcal{O}_{V_{o}}\neq\emptyset\implies(g+\Lambda)\cap V\neq\emptyset.$

To do this, note that if the open set $\mathcal{O}_{V_{o}}\cap
g.\mathcal{O}_{V_{o}}$ is non-empty, then, since $\Lambda_{o}$ has a dense
$G$-orbit in $\Omega_{\Lambda_{o}}^{\times}$, there is $g_{o}\in G$ such that
$g_{o}.\Lambda_{o}\in\mathcal{O}_{V_{o}}\cap g.\mathcal{O}_{V_{o}}$, or
equivalently,

$(\Lambda_{o}-g_{o})\cap
V_{o}\neq\emptyset\quad\textrm{and}\quad(\Lambda_{o}-g_{o}+g)\cap
V_{o}\neq\emptyset.$

By taking difference, this implies that

$\emptyset\neq(g+\Lambda_{o}-\Lambda_{o})\cap(V_{o}-V_{o})\subset(g+\Lambda)\cap
V,$

and we are done. ∎

###  2.3. Strong Følner sequences

Let $G$ be a lcsc abelian group and let $(F_{n})$ be a sequence of bounded
Borel sets with positive Haar measures. We say that $(F_{n})$ is _a strong
Følner sequence_ if

  * (i)

for every compact subset $C\subset G$, we have
$\lim_{n\rightarrow\infty}\frac{m_{G}(F_{n}+C)}{m_{G}(F_{n})}=1$,

  * (ii)

there exists an open bounded identity neighbourhood $V\subset G$ such that

$\lim_{n\rightarrow\infty}\frac{m_{G}\big{(}\bigcap_{v\in
V}(F_{n}-v)\big{)}}{m_{G}(F_{n})}=1.$

If we want to emphasize the dependence on the identity neighbourhood $V$, we
refer to the sequence $(F_{n})$ as a _$V$ -adapted strong Følner sequence_.

######  Remark 2.7.

Note that if $G$ is a countable discrete group, then every Følner sequence in
$G$ is a strong Følner sequence (we can take $V=\\{e\\}$).

######  Lemma 2.8.

Let $V$ be an open and bounded identity neighbourhood in $G$. Then there
exists a $V$-adapted strong Følner sequence in $G$. Furthermore, if $(L_{n})$
is a $V$-adapted strong Følner sequence in $G$ and $(g_{n})$ is any sequence
in $G$, then

$F_{n}:=L_{n}+g_{n}\quad\textrm{and}\quad\widetilde{F}_{n}:=L_{n}+V$

are both $V$-adapted strong Følner sequences.

###### Proof.

Since $G$ is both amenable and $\sigma$-compact, [13, Theorem 4 and
Proposition 1] guarantee that there exists a sequence $(K_{n})$ of bounded
Borel sets in $G$ with positive Haar measures such that for every compact
subset $B\subset G$,

$\lim_{n\rightarrow\infty}\frac{m_{G}((K_{n}+B)\Delta
K_{n})}{m_{G}(K_{n})}=0,$ (2.2)

where $\Delta$ denotes the set-theoretical difference. In particular, if $0\in
B$, then

$1\leq\frac{m_{G}(K_{n}+B)}{m_{G}(K_{n})}=1+\frac{m_{G}((K_{n}+B)\setminus
K_{n})}{m_{G}(K_{n})}\rightarrow 1,\quad\textrm{as $n\rightarrow\infty$}.$

Let $V$ be an open bounded identity neighbourhood in $G$ and define
$F_{n}=K_{n}+V$. We claim that $(F_{n})$ is a $V$-adapted strong Følner
sequence in $G$. Let us first show that $(F_{n})$ satisfies $(i)$. Let $C$ be
a compact subset of $G$. Since $V$ is pre-compact,
$B:=\overline{C+V}\cup\\{0\\}$ is compact and $F_{n}+C\subset K_{n}+B$, and
thus (2.2) implies that

$1\leq\frac{m_{G}(F_{n}+C)}{m_{G}(F_{n})}\leq\frac{m_{G}(K_{n}+B)}{m_{G}(K_{n})}\rightarrow
1,\quad n\rightarrow\infty,$

proving (i). Let us now show (ii). Since $F_{n}=K_{n}+V$, we have

$K_{n}=\bigcap_{v\in V}(F_{n}-v)\quad\textrm{and}\quad K_{n}+V\subset F_{n},$

and since $V$ is pre-compact, (2.2) tells us that

$1\geq\lim_{n\rightarrow\infty}\frac{m_{G}\big{(}\bigcap_{v\in
V}(F_{n}-v)\big{)}}{m_{G}(F_{n})}\geq\lim_{n\rightarrow\infty}\frac{m_{G}(K_{n})}{m_{G}(K_{n}+V)}=1,$

and thus (ii) holds for $(F_{n})$. For the last assertion, let $(L_{n})$ be a
$V$-adapted strong Følner sequence in $G$, let $(g_{n})$ be any sequence in
$G$ and let $F_{n}=L_{n}+g_{n}$. Then, if $C$ is a compact set in $G$, we have
$F_{n}+C=(L_{n}+C)+g_{n}$, and

$\frac{m_{G}(F_{n}+C)}{m_{G}(F_{n})}=\frac{m_{G}(L_{n}+C)}{m_{G}(L_{n})}\rightarrow
1,\quad n\rightarrow\infty,$

since $(L_{n})$ is a strong Følner sequence in $G$. Furthermore,

$\bigcap_{v\in V}(F_{n}-v)=g_{n}+\bigcap_{v\in V}(L_{n}-v),\quad\textrm{for
all $n$},$

and thus

$\lim_{n\rightarrow\infty}\frac{m_{G}\big{(}\bigcap_{v\in
V}(F_{n}-v)\big{)}}{m_{G}(F_{n})}\lim_{n\rightarrow\infty}\frac{m_{G}\big{(}\bigcap_{v\in
V}(L_{n}-v)\big{)}}{m_{G}(L_{n})}=1,$

since $(L_{n})$ is $V$-adapted. We conclude that $(F_{n})$ is a $V$-adapted
strong Følner sequence in $G$. The proof that $(\widetilde{F}_{n})$ is a
$V$-adapted strong Følner sequence is similar. ∎

###  2.4. Uniformly discrete sets and Banach densities

Let $U$ be an open identity neighbourhood in $G$ and let $P\subset G$ be a
$U$-uniformly discrete set. Given a strong Følner sequence $(F_{n})$ in $G$,
which always exists in view of Lemma 2.8, we define the _upper density
$\overline{d}_{(F_{n})}(P)$ of $P$ along $(F_{n})$_ by

$\overline{d}_{(F_{n})}(P)=\varlimsup_{n\rightarrow\infty}\frac{|P\cap
F_{n}|}{m_{G}(F_{n})},$

and the _upper Banach density $d^{*}(P)$_ by

$d^{*}(P)=\sup\big{\\{}\,\overline{d}_{(F_{n})}(P)\,:\,\textrm{$(F_{n})$ is a
strong F\o lner sequence in $G$}\big{\\}}.$

We will show below (Corollary 2.10) that $d^{*}(P)$ is always finite. To do
this, we first need to prove the following lemma.

######  Lemma 2.9.

Let $U$ be an open identity neighbourhood and let $P\subset G$ be a
$U$-uniformly discrete set. Then, for every symmetric Borel set $V\subset G$
such that $V-V\subset U$ and for every bounded Borel set $Q\subset G$,

$|P\cap Q|\leq\frac{m_{G}(Q+V)}{m_{G}(V)}.$

Furthermore, for every Borel set $V_{o}\subset V$ such that
$V_{o}-V_{o}\subset V$,

$|P\cap(Q+V)|\geq\frac{m_{G}((P+V_{o})\cap(Q+V_{o}))}{m_{G}(V_{o})}.$

###### Proof.

Let $V$ be a symmetric Borel set such that $V-V\subset U$. Then, since $P$ is
$U$-uniformly discrete, all sets of the form $p+V$, for $p\in P$ are disjoint,
and thus

$(P+V)\cap(Q+V)=\bigsqcup_{p\in P}(p+V)\cap(Q+V)\supseteq\bigsqcup_{p\in P\cap
Q}(p+V).$ (2.3)

Hence,

$m_{G}(Q+V)\geq m_{G}((P+V)\cap(Q+V))\geq m_{G}(V)|P\cap Q|.$

For the last inequality in the lemma, let $V_{o}\subset V$ be a Borel set such
that $V_{o}-V_{o}\subset V$. Then (2.3) tells us that

$(P+V_{o})\cap(Q+V_{o})=\bigsqcup_{p\in
P}(p+V_{o})\cap(Q+V_{o})\subseteq\bigsqcup_{p\in P\cap(Q+V)}(p+V_{o}),$

and thus

$m_{G}((P+V_{o})\cap(Q+V_{o}))\leq m_{G}(V_{o})\cdot|P\cap(Q+V)|.$

∎

######  Corollary 2.10.

Let $U$ be an open identity neighbourhood and let $P\subset G$ be a
$U$-uniformly discrete set. Then, $d^{*}(P)\in[0,1/m_{G}(V)]$ for every
identity neighbourhood $V$ in $G$ such that $V-V\subset U$.

###### Proof.

Let $(F_{n})$ be a strong Følner sequence, and let $V$ be an identity
neighbourhood in $G$ such that $V-V\subset U$. By Lemma 2.9,

$\varlimsup_{n\rightarrow\infty}\frac{|P\cap
F_{n}|}{m_{G}(F_{n})}\leq\varlimsup_{n\rightarrow\infty}\frac{m_{G}(F_{n}+V)}{m_{G}(V)\cdot
m_{G}(F_{n})}=\frac{1}{m_{G}(V)}.$

Since $(F_{n})$ is arbitrary, we are done. ∎

###  2.5. Syndetic sets

Let $\Lambda$ be a (not necessarily closed) subset of $G$. We say that
$\Lambda$ is _relatively dense_ in $G$ if there is a compact set $Q$ in $G$
such that $G=\Lambda+Q$. A subset $\Lambda_{o}\subset\Lambda$ is _syndetic_ if
there is a finite set $F$ in $G$ such that $\Lambda\subset\Lambda_{o}+F$. The
following useful lemma will be used several times in the proofs of the main
theorems.

######  Lemma 2.11.

Let $\Lambda\subset G$ be a set with finite local complexity, and suppose that
$f$ is a non-negative and bounded function on $\Lambda$ with the property that
for every strong Følner sequence $(F_{n})$ in $G$,

$\varliminf_{n\rightarrow\infty}\frac{1}{m_{G}(F_{n})}\sum_{\lambda\in\Lambda\cap
F_{n}}f(\lambda)>0.$

Then there exists $c>0$ such that the set
$\\{\lambda\in\Lambda\,:\,f(\lambda)\geq c\\}$ is syndetic in $\Lambda$.

###### Proof.

Given $c>0$, define

$S_{c}=\\{\lambda\in\Lambda\,:\,f(\lambda)\geq c\\}.$

We first show that there exists $c>0$ such that $S_{c}$ is relatively dense in
$G$. To do this, fix a strong Følner sequence $(L_{n})$ in $G$ and assume that
there is no $c>0$ for which $S_{c}$ is relatively dense in $G$. In particular,
for every $n$, we can find an element $g_{n}\in G$ such that $g_{n}\notin
S_{1/n}-L_{n}$, or equivalently $S_{1/n}\cap(L_{n}+g_{n})=\emptyset$. By Lemma
2.8, $F_{n}=L_{n}+g_{n}$ is again a strong Følner sequence in $G$. Now, since
$S_{1/n}\cap F_{n}=\emptyset$, we have

$\displaystyle\sum_{\lambda\in\Lambda\cap F_{n}}f(\lambda)$
$\displaystyle=\sum_{\lambda\in(\Lambda\cap S^{c}_{1/n})\cap
F_{n}}f(\lambda)<\frac{1}{n}|\Lambda\cap F_{n}|,$

for all $n$. Since $\Lambda$ has finite local complexity, it is a
$U$-uniformly discrete set in $G$ for some identity neighbourhood $U$, so by
Lemma 2.9, we have

$|\Lambda\cap F_{n}|\leq m_{G}(F_{n}+V)/m_{G}(V),$

for every identity neighbourhood $V$ in $G$ such that $V-V\subset U$, and thus

$\frac{1}{m_{G}(F_{n})}\cdot\sum_{\lambda\in\Lambda\cap
F_{n}}f(\lambda)\leq\frac{m_{G}(F_{n}+V)}{n\cdot m_{G}(U)\cdot
m_{G}(F_{n})},\quad\textrm{for all $n$}.$

Since $(F_{n})$ is a strong Følner sequence, the right hand side tends to zero
as $n\rightarrow\infty$, contradicting our assumption that the limit inferior
of the left-hand side is strictly positive. We conclude that there exist $c>0$
and a compact set $Q\subset G$ such that $S_{c}+Q=G$. In particular,
$\Lambda\subset S_{c}+Q$. Note that if $q\in Q$ satisfies
$\Lambda\cap(S_{c}+q)\neq\emptyset$, then $q\in\Lambda-
S_{c}\subset\Lambda-\Lambda$, and thus

$\Lambda\subset S_{c}+Q\cap(\Lambda-\Lambda).$

Since $\Lambda$ has finite local complexity, $Q\cap(\Lambda-\Lambda)$ is a
finite set, and thus $S_{c}$ is a syndetic subset of $\Lambda$. ∎

######  Corollary 2.12.

Let $\Gamma$ be a countable abelian group and suppose that $f$ is a non-
negative and bounded function on $\Gamma$ with the property that for every
Følner sequence $(F_{n})$ in $\Gamma$,

$\varliminf_{n\rightarrow\infty}\frac{1}{|F_{n}|}\sum_{\gamma\in
F_{n}}f(\gamma)>0.$

Then there exists $c>0$ such that the set
$\\{\gamma\in\Gamma\,:\,f(\gamma)\geq c\\}$ is syndetic in $\Gamma$.

##  3\. Furstenberg’s Correspondence Principle for uniformly discrete sets in
lcsc groups

Furstenberg’s Correspondence Principle is a fundamental tool in density Ramsey
theory. In this subsection we establish a version of this principle for
uniformly discrete subsets of a lcsc abelian group. The proof follows the
usual route, first outlined by Furstenberg in [14], but we present it here for
completeness as several steps need to be modified due to the (potential) non-
discreteness of $G$. We recall that if $P_{o}\in\mathscr{C}(G)$ is a
$U$-uniformly discrete set in $G$ and $\Omega_{P_{o}}^{\times}$ denotes the
punctured hull of $P_{o}$, then the canonical cross-section
$\mathcal{T}_{P_{o}}\subset\Omega_{P_{o}}^{\times}$ satisfies

$\Xi_{\mathcal{T}_{P_{o}}}\subseteq\overline{P_{o}-P_{o}},$

by Lemma 2.4, and is thus $U$-separated. If
$\mu\in\operatorname{Prob}_{G}(\Omega_{P_{o}}^{\times})$, then
$\mu_{\mathcal{T}_{P_{o}}}$ denotes the transverse measure on
$\mathcal{T}_{P_{o}}$, whose existence is guaranteed by Lemma 2.1.

######  Theorem 3.1 (Transverse Correspondence Principle).

Let $P_{o}\subset G$ be a uniformly discrete set with positive upper Banach
density. Let $\Delta\subset G$ and suppose that $0\in\Delta$ and there is an
open identity neighbourhood $W$ in $G$ such that

$(\overline{P_{o}-P_{o}}-\Delta)\cap W=\\{0\\}.$

Then there exists an ergodic
$\mu\in\operatorname{Prob}_{G}(\Omega_{P_{o}}^{\times})$, such that for all
$r\geq 1$,

$d^{*}(P_{o})=\mu_{\mathcal{T}_{P_{o}}}(\mathcal{T}_{P_{o}})\quad\textrm{and}\quad
d^{*}\Big{(}P_{o}\cap\Big{(}\bigcap_{k=1}^{r}(P_{o}-g_{k})\Big{)}\Big{)}\geq\mu_{\mathcal{T}_{P_{o}}}\Big{(}\bigcap_{k=1}^{r}(-g_{k}).\mathcal{T}_{P_{o}}\Big{)},$

for all $g_{1},\ldots,g_{r}\in\Delta$.

######  Remark 3.2.

Note that the lower bound is non-trivial only if
$\\{g_{1},\ldots,g_{r}\\}\subset\Xi_{\mathcal{T}_{P_{o}}}$, as the right-hand
side otherwise equals zero.

The proof of Theorem 3.1 will be done in several steps. We begin by collecting
a few lemmas. In what follows, let $U$ be an open and bounded identity
neighbourhood in $G$ and suppose $P_{o}\subset G$ is a $U$-uniformly discrete
set with positive upper Banach density, and let $\Delta$ and $W$ be as above.
Note that we may without loss of generality assume that $W\subset U$. Fix two
symmetric and open identity neighbourhoods $V_{o}$ and $V$ in $G$ such that

$V_{o}-V_{o}+V_{o}-V_{o}\subset V\subset W\quad\textrm{and}\quad V-V\subset
U,$

and we require $V$ to be $m_{G}$-Jordan measurable. Recall from Subsection 2.2
that every open set in $G$ contains an open, symmetric and $m_{G}$-Jordan
measurable subset, so the existence of such a identity neighbourhood $V$ is
clear.

######  Lemma 3.3.

Let $V\subset G$ be a symmetric, open and $m_{G}$-Jordan measurable set such
that $V-V\subset U$. Then, for every strong Følner sequence $(F_{n})$ in $G$,
there exists $\mu\in\operatorname{Prob}_{G}(\Omega_{P_{o}})$ such that
$\mu(\mathcal{O}_{V})\geq m_{G}(V)\cdot\overline{d}_{(F_{n})}(P_{o})$.

###### Proof.

Let $(F_{n})$ be a strong Følner sequence in $G$ and define
$\widetilde{F}_{n}:=F_{n}+V$. Then, by Lemma 2.8, $(\widetilde{F}_{n})$ is
again a strong Følner sequence in $G$. Define
$\mu_{n}\in\operatorname{Prob}(\Omega_{P_{o}})$ by

$\mu_{n}=\frac{1}{m_{G}(\widetilde{F}_{n})}\int_{\widetilde{F}_{n}}\delta_{g.P_{o}}\,dm_{G}(g).$

Note that

$\displaystyle\mu_{n}(\mathcal{O}_{V})$
$\displaystyle=\frac{1}{m_{G}(\widetilde{F}_{n})}\int_{\widetilde{F}_{n}}\delta_{g.P_{o}}(\mathcal{O}_{V})\,dm_{G}(g)$
$\displaystyle=\frac{m_{G}(\\{g\in\widetilde{F}_{n}\,:\,(P_{o}-g)\cap
V\neq\emptyset\\}}{m_{G}(\widetilde{F}_{n})}$
$\displaystyle=\frac{m_{G}((P_{o}+V)\cap(F_{n}+V))}{m_{G}(F_{n}+V)},$

where we in the last step have used that $V$ is symmetric. Since $P_{o}$ is
$U$-uniformly discrete and $V-V\subset U$,

$\displaystyle(P_{o}+V)\cap(F_{n}+V)$ $\displaystyle=\bigsqcup_{p\in
P_{o}}\big{(}(p+V)\cap(F_{n}+V)\big{)}$ $\displaystyle\supseteq\bigsqcup_{p\in
P_{o}\cap F_{n}}(p+V),$

and thus

$\mu_{n}(\mathcal{O}_{V})\geq m_{G}(V)\cdot\frac{|P_{o}\cap
F_{n}|}{m_{G}(F_{n})}\cdot\frac{m_{G}(F_{n})}{m_{G}(F_{n}+V)},\quad\textrm{for
all $n$}.$ (3.1)

Pick a weak*-convergent sub-sequence $(\mu_{n_{k}})$ with weak*-limit $\mu$.
Then $\mu$ is $G$-invariant, and since $V$ is $m_{G}$-Jordan measurable,
$\mathcal{O}_{V}$ is $\mu$-Jordan measurable by Lemma 2.5. In particular,
$\mu_{n_{k}}(\mathcal{O}_{V})\rightarrow\mu(\mathcal{O}_{V})$ as
$k\rightarrow\infty$, and since $(F_{n})$ is assumed to be a strong Følner
sequence in $G$, (3.1), implies that

$\mu(\mathcal{O}_{V})\geq m_{G}(V)\cdot\overline{d}_{(F_{n})}(P_{o}).$

∎

######  Lemma 3.4.

There exists an ergodic
$\mu\in\operatorname{Prob}_{G}(\Omega_{P_{o}}^{\times})$ such that
$\mu_{\mathcal{T}_{P_{o}}}(\mathcal{T}_{P_{o}})\geq d^{*}(P_{o})$.

###### Proof.

For every $m$, pick a strong Følner sequence $(F_{n}(m))$ in $G$ such that

$\overline{d}_{(F_{n}(m))}(P_{o})\geq d^{*}(P_{o})-\frac{1}{m}.$

By Lemma 3.3, we can find $\mu_{m}\in\operatorname{Prob}_{G}(\Omega_{P_{o}})$
such that

$\mu_{m}(\mathcal{O}_{V})\geq
m_{G}(V)\cdot\Big{(}d^{*}(P_{o})-\frac{1}{m}\Big{)}.$

We can now extract a weak*-convergent sub-sequence $(\mu_{m_{k}})$ with
weak*-limit $\nu$. Note that $\nu$ is $G$-invariant. Since $V$ is
$m_{G}$-Jordan measurable, $\mathcal{O}_{V}$ is $\mu$-Jordan measurable by
Lemma 2.5 and thus $\nu(\mathcal{O}_{V})\geq m_{G}(V)\cdot d^{*}(P_{o})$. By
[12, Theorem 8.20], we can decompose $\nu$ into ergodic components, i.e. there
is a Borel probability measure $\sigma$ on
$\operatorname{Prob}_{G}^{\textrm{erg}}(\Omega_{P_{o}})$ such that

$\nu=\int_{\operatorname{Prob}_{G}^{\textrm{erg}}(\Omega_{P_{o}})}\mu\,d\sigma(\mu).$

In particular, we can find an ergodic
$\mu\in\operatorname{Prob}_{G}(\Omega_{P_{o}})$ such that

$\mu(\mathcal{O}_{V})\geq\nu(\mathcal{O}_{V})\geq m_{G}(V)\cdot d^{*}(P_{o}).$
(3.2)

Since we assume that $d^{*}(P_{o})>0$, we conclude that
$\mu\neq\delta_{\\{\emptyset\\}}$. In particular, since $\mu$ is ergodic and
$\emptyset$ is a fixed point for the $G$-action, $\mu(\\{\emptyset\\})=0$, and
thus $\mu\in\operatorname{Prob}_{G}(\Omega_{P_{o}}^{\times})$, so the
transverse measure $\mu_{\mathcal{T}_{P_{o}}}$ of $\mu$ is well-defined, and
$\mu(\mathcal{O}_{V})=m_{G}(V)\cdot\mu_{\mathcal{T}_{P_{o}}}(\mathcal{T}_{P_{o}})$.
In particular, (3.2) implies that
$\mu_{\mathcal{T}_{P_{o}}}(\mathcal{T}_{P_{o}})\geq d^{*}(P_{o})$. ∎

######  Lemma 3.5.

Let $(F_{n})$ be a strong Følner sequence in $G$ and suppose
$\mu\in\operatorname{Prob}_{G}(\Omega_{P_{o}}^{\times})$ is ergodic. Let
$g_{1},\ldots,g_{r}\in\Delta$. Then, for $\mu$-almost every
$P\in\Omega_{P_{o}}^{\times}$, there exists a sub-sequence $(F_{n_{j}})$ such
that

$\lim_{j\rightarrow\infty}\frac{m_{G}\big{(}\big{(}P\cap\big{(}\bigcap_{k=1}^{r}(P-g_{k})\big{)}+V_{o}\big{)}\cap
F_{n_{j}}\big{)}}{m_{G}(F_{n_{j}})}=m_{G}(V_{o})\cdot\mu_{\mathcal{T}_{P_{o}}}\big{(}\bigcap_{k=1}^{r}(-g_{k}).\mathcal{T}_{P_{o}}\big{)}.$

###### Proof.

Define the (possibly empty) open set $A\subset\Omega_{P_{o}}^{\times}$ by

$A:=\mathcal{O}_{V_{o}}\cap\Big{(}\bigcap_{k=1}^{r}(-g_{k}).\mathcal{O}_{V_{o}}\Big{)}.$

Lemma 2.3 (applied to $B=\mathcal{T}_{P_{o}}$, and using that $V_{o}$ is
symmetric) implies that

$\mu(A)=m_{G}(V_{o})\cdot\mu_{\mathcal{T}_{P_{o}}}\big{(}\bigcap_{k=1}^{r}(-g_{k}).\mathcal{T}_{P_{o}}\big{)}.$

Since $\mu$ is ergodic and $(F_{n})$ is Følner, the mean ergodic theorem tells
us that

$\lim_{n\rightarrow\infty}\frac{1}{m_{G}(F_{n})}\int_{F_{n}}\chi_{A}(g.\,\cdot)\,dm_{G}(g)=\mu(A),$

in the norm topology on $L^{2}(\Omega_{P_{o}}^{\times},\mu)$. We can thus
extract a sub-sequence $(F_{n_{j}})$ along which the convergence is
$\mu$-almost sure. Hence, for $\mu$-almost every
$P\in\Omega_{P_{o}}^{\times}$,

$\lim_{j\rightarrow\infty}\frac{m_{G}(\\{g\in F_{n_{j}}\,:\,P-g\in
A\\})}{m_{G}(F_{n_{j}})}=\mu(A).$

Note that since $V_{o}$ is symmetric,

$\displaystyle\\{g\in F_{n_{j}}\,:\,P-g\in A\\}$ $\displaystyle=\\{g\in
F_{n_{j}}\,:\,(P-g)\cap V_{o}\neq\emptyset,\enskip(P-g-g_{k})\cap
V_{o}\neq\emptyset,\enskip\textrm{for all $k=1,\ldots,r$}\\}$
$\displaystyle=(P+V_{o})\cap\big{(}\bigcap_{k=1}^{r}(P+V_{o}-g_{k})\big{)}\cap
F_{n_{j}},$

for all $j$. It thus suffices to show that

$(P+V_{o})\cap\big{(}\bigcap_{k=1}^{r}(P+V_{o}-g_{k})\big{)}=\big{(}P\cap\big{(}\bigcap_{k=1}^{r}(P-g_{k})\big{)}+V_{o}\big{)}.$

To do this, first note the $\supseteq$-inclusion is trivial, so it is enough
to prove

$(P+V_{o})\cap\big{(}\bigcap_{k=1}^{r}(P+V_{o}-g_{k})\big{)}\subseteq\big{(}P\cap\big{(}\bigcap_{k=1}^{r}(P-g_{k})\big{)}+V_{o}\big{)}.$
(3.3)

We may without loss of generality assume that the set on the left-hand side is
non-empty, and pick $g$ in this set, allowing us to write

$g=p_{o}+v_{o}=p_{1}+v_{1}-g_{1}=\ldots=p_{r}+v_{r}-g_{r},$

for some $p_{o},\ldots,p_{r}\in P$ and $v_{o},v_{1},\ldots,v_{r}\in V_{o}$.
Then, by Lemma 2.4,

$p_{k}-p_{o}-g_{k}=v_{o}-v_{k}\in(P-P-\Delta)\cap(V_{o}-V_{o})\subset(\overline{P_{o}-P_{o}}-\Delta)\cap
W=\\{0\\},$

for all $k$, and thus $v_{k}=v_{o}$ and $p_{o}=p_{k}-g_{k}$ for all $k$. In
particular,

$g=p_{o}+v_{o}\in P\cap\big{(}\bigcap_{k=1}^{r}(P-g_{k})\big{)}+V_{o}.$

Since $g$ is arbitrary, this proves (3.3), and we are done. ∎

######  Lemma 3.6.

Let $P\in\Omega_{P_{o}}^{\times}$ and let $V_{1}\subset V_{o}$ be an open
identity neighbourhood. Then, for all $g_{1},\ldots,g_{r}\in\Delta$ and for
every pre-compact set $Q\subset G$, there exists $h\in G$ such that

$\big{(}P\cap\big{(}\bigcap_{k=1}^{r}(P-g_{k})\big{)}+V_{o}\big{)}\cap
Q\subseteq
P_{o}\cap\big{(}\bigcap_{k=1}^{r}(P_{o}-g_{k})\big{)}-h+V_{o}+V_{1}.$

###### Proof.

Set $g_{o}=0$, and define

$P^{\prime}:=\bigcap_{k=0}^{r}(P-g_{k})\quad\textrm{and}\quad
P_{o}^{\prime}:=\bigcap_{k=0}^{r}(P_{o}-g_{k}).$

Fix a pre-compact set $Q\subset G$. If $(P^{\prime}+V_{o})\cap Q$ is empty,
the inclusion in the lemma is trivial, so we assume that this set is non-
empty. Since $P$ is $U$-uniformly discrete, we can write

$(P^{\prime}+V_{o})\cap Q=\bigsqcup_{j=1}^{q}(p_{j}+V_{o})$

for some $q\geq 1$ and $p_{1},\ldots,p_{q}\in P^{\prime}$. Furthermore, for
all $k=0,\ldots,r$, we can find elements $p_{j,k}\in P$ such that
$p_{j}=p_{j,k}-g_{k}$ for all $j=1,\ldots,q$. Let us now fix a sequence
$(h_{n})$ in $G$ such that $h_{n}.P_{o}=P_{o}-h_{n}\rightarrow P$ as
$n\rightarrow\infty$. Then, by (i) in the characterization of sequential
convergence in the Chabauty topology in Subsection 2.2, we can find, for all
$j$ and $k$, an integer $n_{j,k}$ such that for all $n\geq n_{j,k}$, there
exist elements $p^{o}_{j,k}(n)\in P_{o}$ and $v_{j,k}(n)\in V_{1}$ such that

$p_{j,k}=p^{o}_{j,k}(n)-h_{n}+v_{j,k}(n).$

Let $n_{o}:=\max_{j,k}n_{j,k}$. Then, for all $n\geq n_{o}$ and
$v_{o},\ldots,v_{r}\in V_{o}$,

$p_{j}+v_{j}=p_{j,k}-g_{k}+v_{j}\in
P_{o}-(g_{k}+h_{n})+V_{o}+V_{1},\quad\textrm{for all $k=0,1,\ldots,r$},$

and thus (since $v_{o},\ldots,v_{r}\in V_{o}$ are arbitrary),

$(P^{\prime}+V_{o})\cap
Q\subset\bigcap_{k=0}^{r}\big{(}P_{o}-g_{k}+V_{o}+V_{1}\big{)}-h_{n},\quad\textrm{for
all $n\geq n_{o}$}.$

Let $h=h_{n}$ for some $n\geq n_{o}$. Then to prove the lemma, it suffices to
show that

$\bigcap_{k=0}^{r}\big{(}P_{o}-g_{k}+V_{o}+V_{1}\big{)}=\bigcap_{k=0}^{r}\big{(}P_{o}-g_{k}\big{)}+V_{o}+V_{1}.$

The $\supseteq$-inclusion is trivial, so we only need to establish the other
inclusion. To do this, pick $g$ in the set on the left-hand side, and write

$g=p_{k}-g_{k}+v_{k},\quad\textrm{for $k=0,\ldots,r$},$

and $p_{o},\ldots,p_{r}\in P_{o}$ and $v_{0},\ldots,v_{r}\in V_{o}+V_{1}$.
Hence, since $V_{1}\subset V_{o}$ and $V_{o}+V_{o}-V_{o}-V_{o}\subset W$, it
follows from Lemma 2.4,

$p_{k}-p_{o}-g_{k}=v_{o}-v_{k}\in(P-P-\Delta)\cap(V_{o}+V_{1}-V_{o}-V_{1})\subset(\overline{P_{o}-P_{o}}-\Delta)\cap
W=\\{0\\},$

for all $k=1,\ldots,r$. We conclude that $v_{k}=v_{o}$ and $p_{o}=p_{k}-g_{k}$
for all $k$, and thus $p_{o}\in P_{o}^{\prime}$ and $g=p_{o}+v_{o}\in
P_{o}^{\prime}+V_{o}+V_{1}$, which finishes the proof. ∎

#### _Proof of Theorem 3.1_

Let the sets $P_{o},\Delta,W,V_{o},V$ and $U$ be as above. By Lemma 3.4, there
is at least one ergodic
$\mu\in\operatorname{Prob}_{G}(\Omega_{P_{o}}^{\times})$ such that
$\mu_{\mathcal{T}_{P_{o}}}(\mathcal{T}_{P_{o}})\geq d^{*}(P_{o})$. We want to
show that

$d^{*}\Big{(}P_{o}\cap\Big{(}\bigcap_{k=1}^{r}(P_{o}-g_{k})\Big{)}\Big{)}\geq\mu_{\mathcal{T}_{P_{o}}}\Big{(}\bigcap_{k=1}^{r}(-g_{k}).\mathcal{T}_{P_{o}}\Big{)},$
(3.4)

for all $g_{1},\ldots,g_{r}\in\Delta$. In particular, since $0\in\Delta$, we
conclude from (3.4) that
$d^{*}(P_{o})\geq\mu_{\mathcal{T}_{P_{o}}}(\mathcal{T}_{P_{o}})$, and thus
$d^{*}(P_{o})=\mu_{\mathcal{T}_{P_{o}}}(\mathcal{T}_{P_{o}})$.

To prove (3.4), fix $g_{1},\ldots,g_{r}\in\Delta$. Moreover, fix a strong
Følner sequence $(F_{n})$ in $G$ and an open identity neighbourhood
$V_{1}\subset V_{o}$. Define $\widetilde{F}_{n}=F_{n}+V_{o}+V_{1}$, and note
that $(\widetilde{F}_{n})$ is a strong Følner sequence by Lemma 2.8. Hence, by
Lemma 3.5, we can find a point $P\in\Omega_{P_{o}}^{\times}$ and a sub-
sequence $(\widetilde{F}_{n_{k}})$ such that

$\lim_{j\rightarrow\infty}\frac{m_{G}\big{(}\big{(}P\cap\big{(}\bigcap_{k=1}^{r}(P-g_{k})\big{)}+V_{o}\big{)}\cap\widetilde{F}_{n_{j}}\big{)}}{m_{G}(\widetilde{F}_{n_{j}})}=m_{G}(V_{o})\cdot\mu_{\mathcal{T}_{P_{o}}}\big{(}\bigcap_{k=1}^{r}(-g_{k}).\mathcal{T}_{P_{o}}\big{)}.$

For every $j$, we can apply Lemma 3.6 to find $h_{j}\in G$ such that

$\displaystyle\frac{m_{G}\big{(}\big{(}P\cap\big{(}\bigcap_{k=1}^{r}(P-g_{k})\big{)}+V_{o}\big{)}\cap\widetilde{F}_{n_{j}}\big{)}}{m_{G}(\widetilde{F}_{n_{j}})}$
$\displaystyle\leq\frac{m_{G}\big{(}\big{(}P_{o}\cap\big{(}\bigcap_{k=1}^{r}(P_{o}-g_{k})\big{)}-h_{j}+V_{o}+V_{1}\big{)}\cap\widetilde{F}_{n_{j}}\big{)}}{m_{G}(\widetilde{F}_{n_{j}})}$
$\displaystyle=\frac{m_{G}\big{(}\big{(}P_{o}\cap\big{(}\bigcap_{k=1}^{r}(P_{o}-g_{k})\big{)}+V_{o}+V_{1}\big{)}\cap(F_{n_{j}}+V_{o}+V_{1}+h_{j})\big{)}}{m_{G}(F_{n_{j}}+V_{o}+V_{1}+h_{j})}.$

Since

$(V_{o}+V_{1})-(V_{o}+V_{1})\subset V_{o}+V_{o}-V_{o}-V_{o}\subset V,$

Lemma 2.9 (applied to the $U$-uniformly discrete set
$P_{o}\cap\big{(}\bigcap_{k=1}^{r}(P_{o}-g_{k})\big{)}$ and the bounded set
$Q=F_{n_{j}}+h_{j}$) yields,

$\displaystyle
m_{G}\big{(}\big{(}P_{o}\cap\big{(}\bigcap_{k=1}^{r}(P_{o}-g_{k})\big{)}+V_{o}+V_{1}\big{)}\cap(F_{n_{j}}+V_{o}+V_{1}+h_{j})\big{)}$
$\displaystyle\leq
m_{G}(V_{o}+V_{1})\cdot|P_{o}\cap\big{(}\bigcap_{k=1}^{r}(P_{o}-g_{k})\big{)}\cap(F_{n_{j}}+h_{j}+V)|.$

Set $L_{j}=F_{n_{j}}+h_{j}+V$, and note that Lemma 2.8 again guarantees that
$(L_{j})$ is a strong Følner sequence in $G$. Then,

$\displaystyle\mu_{\mathcal{T}_{P_{o}}}\big{(}\bigcap_{k=1}^{r}(-g_{k}).\mathcal{T}_{P_{o}}\big{)}$
$\displaystyle=\lim_{j\rightarrow\infty}\frac{m_{G}(V_{o}+V_{1})}{m_{G}(V_{o})}\cdot\frac{|P_{o}\cap\big{(}\bigcap_{k=1}^{r}(P_{o}-g_{k})\big{)}\cap
L_{j}|}{m_{G}(L_{j})}\cdot\frac{m_{G}(L_{j})}{m_{G}(F_{n_{j}}+V_{o}+V_{1}+h_{j})}$
$\displaystyle\leq\frac{m_{G}(V_{o}+V_{1})}{m_{G}(V_{o})}\cdot\overline{d}_{(L_{j})}(P_{o}\cap\big{(}\bigcap_{k=1}^{r}(P_{o}-g_{k})\big{)})\leq\frac{m_{G}(V_{o}+V_{1})}{m_{G}(V_{o})}\cdot
d^{*}(P_{o}\cap\big{(}\bigcap_{k=1}^{r}(P_{o}-g_{k})\big{)}),$

where we in the second to last step have used that $(L_{j})$ is a strong
Følner sequence (and thus the last factor on the middle line tends to $1$ as
$j\rightarrow\infty$) Note that this inequality holds for any open set
$V_{1}\subset V_{o}$. In particular, the quotient
$\frac{m_{G}(V_{o}+V_{1})}{m_{G}(V_{o})}$ can be made arbitrarily close to $1$
by taking $V_{1}$ small enough. We conclude that

$\mu_{\mathcal{T}_{P_{o}}}\big{(}\bigcap_{k=1}^{r}(-g_{k}).\mathcal{T}_{P_{o}}\big{)}\leq
d^{*}(P_{o}\cap\big{(}\bigcap_{k=1}^{r}(P_{o}-g_{k})\big{)}),$

and thus (3.4) is proved.

##  4\. Furstenberg-Katznelson Theorem for lcsc abelian groups

Let $G$ be a locally compact and second countable (lcsc) abelian group. If
$(Z,\theta)$ is a standard Borel probability measure space and $a,b:G\times
Z\rightarrow Z$ are two Borel $G$-actions on $Z$, we say that they _commute_
if

$a(g_{1},b(g_{2},z))=b(g_{2},a(g_{1},z)),\quad\textrm{for all $g_{1},g_{2}\in
G$ and $z\in Z$}.$

For countable abelian groups, the following theorem is due to Furstenberg and
Katznelson ([15, Theorem B] and [16, Secion 3]) and Austin [2, Theorem B]. We
show below that the general case follows from a simple approximation argument.

######  Theorem 4.1.

Let $(Z,\theta)$ be a standard Borel probability measure space and let
$a_{1},\ldots,a_{r}$ be commuting $\theta$-preserving Borel $G$-actions on
$Z$. Then, for every $\theta$-measurable set $A\subset Z$ with positive
$\theta$-measure, there exists $c>0$ such that the set

$S:=\Big{\\{}g\in
G\,:\,\theta\Big{(}A\cap\Big{(}\bigcap_{k=1}^{r}a_{k}(g,A)\Big{)}\Big{)}\geq
c\Big{\\}}$

is syndetic in $G$. In particular, for every strong Følner sequence $(F_{n})$
in $G$,

$\varliminf_{n\rightarrow\infty}\frac{1}{m_{G}(F_{n})}\int_{F_{n}}\theta\Big{(}A\cap\Big{(}\bigcap_{k=1}^{r}a_{k}(g,A)\Big{)}\Big{)}\,dm_{G}(g)>0.$

###  4.1. Proof of Theorem 4.1

Let $a_{1},\ldots,a_{r}$ be commuting $\theta$-preserving Borel $G$-actions on
$Z$, and let $A\subset Z$ be a $\theta$-measurable set with positive
$\theta$-measure. Since $G$ is locally compact and second countable, it is
separable, and thus we can find a dense countable subgroup $\Gamma<G$. The
following theorem is due to Austin [2, Theorem B].

######  Theorem 4.2.

Let $(Z,\theta)$ be a standard Borel probability measure space and let
$b_{1},\ldots,b_{r}$ be commuting $\theta$-preserving Borel $\Gamma$-actions
on $Z$. Then, for every $\theta$-measurable set $A\subset Z$ with positive
$\theta$-measure and for every Følner sequence $(F_{n})$ in $\Gamma$,

$\varliminf_{n\rightarrow\infty}\frac{1}{|F_{n}|}\sum_{\gamma\in
F_{n}}\theta\Big{(}A\cap\Big{(}\bigcap_{k=1}^{r}a_{k}(\gamma,A)\Big{)}\Big{)}>0.$

Define $\varphi:G\rightarrow[0,1]$ by

$\varphi(g)=\theta\Big{(}A\cap\Big{(}\bigcap_{k=1}^{r}a_{k}(g,A)\Big{)}\Big{)},\quad
g\in G.$

By Theorem 4.2 and Corollary 2.12 (applied to the function
$f=\varphi|_{\Gamma}$), we conclude that there exists $c_{o}>0$ such that

$S_{\Gamma}=\Big{\\{}\gamma\in\Gamma\,:\,\varphi(\gamma)\geq c_{o}\Big{\\}}$

is syndetic in $\Gamma$. The following simple lemma will be proved below.

######  Lemma 4.3.

$\varphi$ is uniformly continuous on $G$.

Taking the lemma for granted for now, we can thus find an open set $V\subset
G$ such that

$|\varphi(g_{1})-\varphi(g_{2})|<c_{o}/2,\quad\textrm{for all $g_{1},g_{2}\in
G$ such that $g_{1}-g_{2}\in V$}.$ (4.1)

Define $S_{o}=S_{\Gamma}+V\subset G$. Since $S_{\Gamma}$ is syndetic in
$\Gamma$, there is a finite set $F\subset\Gamma$ such that
$S_{\Gamma}+F=\Gamma$, and thus

$S_{o}+F=(S_{\Gamma}+F)+V=\Gamma+V=G,$

so $S_{o}$ is syndetic in $G$. Now if $g\in S_{o}$, we can clearly write
$g=\gamma+v$ for some $\gamma\in S_{\Gamma}$ and $v\in V$, and thus by (4.1)
(with $g_{1}=\gamma+v$ and $g_{2}=\gamma$),

$\varphi(g)=\varphi(\gamma+v)-\varphi(\gamma)+\varphi(\gamma)\geq\varphi(\gamma)-c_{o}/2\geq
c_{o}/2.$

Let $c=c_{o}/2$ and define

$S=\\{g\in G\,:\,\varphi(g)\geq c\\}.$

The argument above shows that $S_{o}\subset S$ and thus $S$ is syndetic as
well, finishing the proof of Theorem 4.1.

###  4.2. Proof of Lemma 4.3

By [1, Lemme A.1.1], the induced $L^{1}$-representations associated with the
actions $a_{1},\ldots,a_{r}$ are strongly continuous, so for every
$\varepsilon>0$, we can find an open identity neighbourhood $U\subset G$ such
that

$\int_{Z}|\chi_{A}(z)-\chi_{A}(a_{k}(g,z))|\,d\theta(z)<\varepsilon/r,\quad\textrm{for
all $g\in U$ and $k=1,\ldots,r$}.$ (4.2)

Note that

$\varphi(g)=\int_{Z}\chi_{A}(z)\,\prod_{k=1}^{r}\chi_{A}(a_{k}(-g,z))\,d\theta(z),\quad\textrm{for
all $g\in G$}.$

A simple induction argument shows that for all $g_{1},g_{2}\in G$,

$\varphi(g_{1})-\varphi(g_{2})=\sum_{k=1}^{r}\int_{Z}\beta_{k}(z)\cdot(\chi_{A}(a_{k}(-g_{1},z)-\chi_{A}(a_{k}(-g_{2},z))\,d\eta(z),$

where

$\beta_{k}(z)=\big{(}\prod_{j=k+1}^{r}\chi_{A}(a_{j}(-g_{1},z))\big{)}\cdot\big{(}\prod_{j=1}^{k-1}\chi_{A}(a_{j}(-g_{2},z))\big{)}.$

Here we use the convention that products over empty sets are always equal to
$1$. Note that $\|\beta_{k}\|_{\infty}\leq 1$ for all $k$.

Fix $\varepsilon>0$ and let $U$ be as above. Note that if $g_{1}-g_{2}\in U$,
then by (4.2),

$\displaystyle|\varphi(g_{1})-\varphi(g_{2})|$
$\displaystyle\leq\sum_{k=1}^{r}\int_{Z}|\chi_{A}(a_{k}(-g_{1},z))-\chi_{A}(a_{k}(-g_{2},z))|\,d\theta(z)$
$\displaystyle=\sum_{k=1}^{r}\int_{Z}|\chi_{A}(z)-\chi_{A}(a_{k}(g_{1}-g_{2},z))|\,d\theta(z)<\varepsilon.$

Since $\varepsilon>0$ and $g_{1},g_{2}\in G$ such that $g_{1}-g_{2}\in U$ are
arbitrary, $\varphi$ is uniformly continuous.

##  5\. Proof of Theorem 1.9

Let $(X,\mathscr{B}_{X})$ be a Borel $G$-space, equipped with a $U$-separated
cross-section $Y\subset X$ for some identity neighbourhood $U$ in $G$.
Furthermore, let $\Lambda$ and $\Delta$ be uniformly discrete subsets of $G$
and suppose that there exist
$\alpha_{1},\ldots,\alpha_{r}\in\operatorname{End}(G)$ such that
$\alpha_{k}(\Lambda)\subseteq\Delta$ for all $k=1,\ldots,r$.

We further assume that $\Lambda$ has finite local complexity and that there is
a set $\Lambda_{o}$ with positive upper Banach density such that
$\Lambda_{o}-\Lambda_{o}\subseteq\Lambda$. We also assume there is an identity
neighbourhood $V\subset U$ such that $(\Xi_{Y}-\Delta)\cap V=\\{0\\}$.

In this section, we show the following slightly stronger version of Theorem
1.9 (which corresponds to the case when $\Delta=\Lambda^{q}$, where $q$ is
chosen as above).

######  Theorem 5.1.

For every $\mu\in\operatorname{Prob}_{G}(X)$ and Borel set $B\subset Y$ with
positive $\mu_{Y}$-measure, there exists $c>0$ such that the set

$S=\Big{\\{}\lambda\in\Lambda\,:\,\mu_{Y}\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).B\Big{)}\geq
c\Big{\\}}$

is syndetic in $\Lambda$.

In view of Lemma 2.11, Theorem 5.1 is an immediate consequence of the
following theorem, whose proof will be presented below.

######  Theorem 5.2.

Let $\Delta,\Lambda$ and $\alpha_{1},\ldots,\alpha_{r}$ be as in Theorem 5.1.
Let $\mu\in\operatorname{Prob}_{G}(X)$ and let $B\subset Y$ be a Borel set
with positive $\mu_{Y}$-measure. Then, for every strong Følner sequence
$(F_{n})$ in $G$,

$\varliminf_{n\rightarrow\infty}\frac{1}{m_{G}(F_{n})}\sum_{\lambda\in\Lambda\cap
F_{n}}\mu_{Y}\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).B\Big{)}>0.$

###  5.1. Proof of Theorem 5.2

In what follows, let us fix $\mu\in\operatorname{Prob}_{G}(X)$ and a Borel set
$B\subset Y$ with positive $\mu_{Y}$-measure. We first show how Theorem 5.2
can be deduced from the following proposition.

######  Proposition 5.3.

Let $\Delta,\Lambda,\Lambda_{o}$ and $\alpha_{1},\ldots,\alpha_{r}$ be as in
Theorem 5.1. Let
$\nu\in\operatorname{Prob}_{G}(\Omega_{\Lambda_{o}}^{\times})$ and let
$(F_{n})$ be a strong Følner sequence in $G$. Then there exist $\beta_{o}>0$,
a non-negative null sequence $(\delta_{n})$, $r$ commuting
$\mu\otimes\nu$-preserving Borel $G$-actions $a_{1},\ldots,a_{r}$ on
$X\times\Omega_{\Lambda_{o}}^{\times}$ and a Borel set $A_{o}\subset
X\times\Omega_{\Lambda_{o}}^{\times}$ with positive $\mu\otimes\nu$-measure
such that

$\displaystyle\sum_{\lambda\in\Lambda\cap
F_{n}}\mu_{Y}\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).B\Big{)}$
$\displaystyle\geq\beta_{o}\cdot\int_{F_{n}}(\mu\otimes\nu)\Big{(}A_{o}\cap\Big{(}\bigcap_{k=1}^{r}a_{k}(g,A_{o})\Big{)}\Big{)}\,dm_{G}(g)-\delta_{n}\cdot
m_{G}(F_{n}),\quad\textrm{for all $n$}.$

###### Proof of Theorem 5.2 assuming Proposition 5.3.

Let $(F_{n})$ be a strong Følner sequence in $G$. Then Proposition 5.3 tells
us that there exist $\beta_{o}>0$, $r$ commuting $\mu\otimes\nu$-preserving
Borel $G$-actions $a_{1},\ldots,a_{r}$ on
$X\times\Omega_{\Lambda_{o}}^{\times}$ and a Borel set $A_{o}\subset
X\times\Omega_{\Lambda_{o}}^{\times}$ with positive $\mu\otimes\nu$-measure
such that

$\displaystyle\varliminf_{n\rightarrow\infty}\frac{1}{m_{G}(F_{n})}\cdot\sum_{\lambda\in\Lambda\cap
F_{n}}\mu_{Y}\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).B\Big{)}\geq$
$\displaystyle\geq\beta_{o}\cdot\varliminf_{n\rightarrow\infty}\frac{1}{m_{G}(F_{n})}\cdot\int_{F_{n}}(\mu\otimes\nu)\Big{(}A_{o}\cap\Big{(}\bigcap_{k=1}^{r}a_{k}(g,A_{o})\Big{)}\Big{)}\,dm_{G}(g).$

The positivity of the last expression now follows from Theorem 4.1, applied to
the Borel space $(Z,\theta)=(X\times\Omega_{\Lambda_{o}},\mu\otimes\nu)$. ∎

###  5.2. Proof of Proposition 5.3

Before we begin the proof, let us briefly recall the main players in this
proposition.

  * (i)

$(X,\mathscr{B}_{X})$ is a Borel $G$-space, equipped with a $U$-separated
cross-section $Y$ for some identity neighbourhood $U$ in $G$. Furthermore, we
fix $\mu\in\operatorname{Prob}_{G}(X)$ and a Borel set $B\subset Y$ with
positive $\mu_{Y}$-measure.

  * (ii)

$\Lambda_{o}\subset G$ is a uniformly discrete set with positive upper Banach
density, and we fix
$\nu\in\operatorname{Prob}_{G}(\Omega_{\Lambda_{o}}^{\times})$. Note that
$\nu(\mathcal{O}_{W})>0$ for all identity neighbourhoods $W$ in $G$, where
$\mathcal{O}_{W}=(-W).\mathcal{T}_{\Lambda_{o}}\subset\Omega_{\Lambda_{o}}^{\times}$.

  * (iii)

$\Lambda$ and $\Delta$ are uniformly discrete sets in $G$ such that

$\Lambda_{o}-\Lambda_{o}\subset\Lambda\quad\textrm{and}\quad(\Xi_{Y}-\Delta)\cap
V=\\{0\\}$

for some identity neighbourhood $V\subset U$. Furthermore, we assume that
$\Lambda$ has finite local complexity in $G$.

  * (iv)

Fix $\alpha_{1},\ldots,\alpha_{r}\in\operatorname{End}(G)$ such that
$\alpha_{k}(\Lambda)\subset\Delta$ for all $k=1,\ldots,r$.

  * (v)

Fix a strong Følner sequence $(F_{n})$ in $G$.

In what follows, we fix an identity neighbourhood $V_{o}$ such that
$V_{o}-V_{o}\subset V$. By (iv), we have $\alpha_{k}(\Lambda)\subseteq\Delta$
for all $k$, so Lemma 2.3 (applied to $g_{k}=\alpha_{k}(\lambda)$ for
$\lambda\in\Lambda)$ tells us that

$\mu_{Y}\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).B\Big{)}=\frac{1}{m_{G}(V_{o})}\cdot\mu\Big{(}B_{V_{o}}\cap\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).B_{V_{o}}\Big{)}\Big{)},\quad\textrm{for
all $\lambda\in\Lambda$}.$

In particular,

$\sum_{\lambda\in\Lambda\cap
F_{n}}\mu_{Y}\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).B\Big{)}=\frac{1}{m_{G}(V_{o})}\cdot\sum_{\lambda\in\Lambda\cap
F_{n}}\mu\Big{(}B_{V_{o}}\cap\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).B_{V_{o}}\Big{)}\Big{)}$
(5.1)

for all $n$. Since $\alpha_{1},\ldots,\alpha_{r}$ are continuous endomorphisms
of $G$, we can find a _symmetric_ identity neighbourhood $V_{1}\subset V_{o}$
such that $V_{1}+\alpha_{k}(V_{1})\subset V_{o}$ for all $k$. In particular,

$B_{V_{1}}\subset
B_{V_{o}}\quad\textrm{and}\quad\alpha_{k}(h).B_{V_{1}}\subset
B_{V_{o}},\quad\textrm{for all $h\in V_{1}$ and $k$},$

and thus

$\mu\Big{(}B_{V_{o}}\cap\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(g)).B_{V_{o}}\Big{)}\Big{)}\geq\mu\Big{(}B_{V_{1}}\cap\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(g-h)).B_{V_{1}}\Big{)}\Big{)},$
(5.2)

for all $g\in G$ and $h\in V_{1}$. Let us now define

$F_{n}^{\prime}:=\bigcap_{h\in V_{1}}(F_{n}-h)\enskip\textrm{and note that
$F_{n}^{\prime}+V_{1}\subset F_{n}$}.$

Hence,

$\chi_{F_{n}}(g)\geq\frac{1}{m_{G}(V_{1})}\int_{G}\chi_{V_{1}}(h)\chi_{F_{n}^{\prime}}(g-h)\,dm_{G}(h)\quad\textrm{for
all $g\in G$},$

and it follows from (5.2) that,

$\displaystyle\int_{G}\chi_{V_{1}}(h)\chi_{F_{n}^{\prime}}(g-h)\,dm_{G}(h)\cdot\mu\Big{(}B_{V_{o}}\cap\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(g)).B_{V_{o}}\Big{)}\Big{)}$
$\displaystyle\geq\int_{G}\chi_{V_{1}}(h)\chi_{F_{n}^{\prime}}(g-h)\,\mu\Big{(}B_{V_{1}}\cap\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(g-h)).B_{V_{1}}\Big{)}\Big{)}\,dm_{G}(h),\quad\textrm{for
all $g\in G$}.$

Let us now put of all this together. It follows from the last inequality above
that

$\displaystyle\sum_{\lambda\in\Lambda\cap
F_{n}}\mu\Big{(}B_{V_{o}}\cap\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).B_{V_{o}}\Big{)}\Big{)}=\sum_{\lambda\in\Lambda}\chi_{F_{n}}(\lambda)\cdot\mu\Big{(}B_{V_{o}}\cap\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).B_{V_{o}}\Big{)}\Big{)}$
$\displaystyle\geq\frac{1}{m_{G}(V_{1})}\cdot\sum_{\lambda\in\Lambda}\int_{G}\chi_{V_{1}}(h)\chi_{F_{n}^{\prime}}(\lambda-h)\,dm_{G}(h)\cdot\mu\Big{(}B_{V_{o}}\cap\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).B_{V_{o}}\Big{)}\Big{)}$
$\displaystyle\geq\frac{1}{m_{G}(V_{1})}\cdot\sum_{\lambda\in\Lambda}\int_{G}\chi_{V_{1}}(h)\chi_{F_{n}^{\prime}}(\lambda-h)\mu\Big{(}B_{V_{1}}\cap\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda-h)).B_{V_{1}}\Big{)}\Big{)}\,dm_{G}(h)$
$\displaystyle=\frac{1}{m_{G}(V_{1})}\cdot\sum_{\lambda\in\Lambda}\int_{G}\chi_{V_{1}}(\lambda-h)\chi_{F_{n}^{\prime}}(h)\mu\Big{(}B_{V_{1}}\cap\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(h)).B_{V_{1}}\Big{)}\Big{)}\,dm_{G}(h)$
$\displaystyle=\frac{1}{m_{G}(V_{1})}\cdot\int_{F_{n}^{\prime}}\mu\Big{(}B_{V_{1}}\cap\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(h)).B_{V_{1}}\Big{)}\Big{)}\cdot\Big{(}\sum_{\lambda\in\Lambda}\chi_{V_{1}}(\lambda-h)\Big{)}\,dm_{G}(h),$

where we in the last step used Fubini’s Theorem to interchange summation and
integration. Let us now fix an identity neigbourhood $V_{2}$ in $G$ such that
$V_{2}-V_{2}\subset V_{1}$. Since both $V_{1}$ and $\Lambda$ are symmetric,
Lemma 2.6 tells us that

$\sum_{\lambda\in\Lambda}\chi_{V_{1}}(\lambda-h)=\sum_{\lambda\in\Lambda}\chi_{V_{1}}(h+\lambda)\geq\nu(\mathcal{O}_{V_{2}}\cap
h.\mathcal{O}_{V_{2}}),\quad\textrm{for all $h\in G$},$

where $\nu$ is defined in (ii) above. Note that $\nu(\mathcal{O}_{V_{2}})>0$.
We have now shown that

$\displaystyle\sum_{\lambda\in\Lambda\cap
F_{n}}\mu\Big{(}B_{V_{o}}\cap\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).B_{V_{o}}\Big{)}\Big{)}$
$\displaystyle\geq\frac{1}{m_{G}(V_{1})}\cdot\int_{F_{n}^{\prime}}\mu\Big{(}B_{V_{1}}\cap\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(h)).B_{V_{1}}\Big{)}\Big{)}\cdot\nu(\mathcal{O}_{V_{2}}\cap
h.\mathcal{O}_{V_{2}})\,dm_{G}(h).$ (5.3)

Let us now rewrite the integrand. First, let
$Z=X\times\Omega_{\Lambda_{o}}^{\times}$ and

$A_{o}=B_{V_{1}}\times\mathcal{O}_{V_{2}}\subset
Z\quad\textrm{and}\quad\theta=\mu\otimes\nu\in\operatorname{Prob}_{G}(Z).$

Then,
$\theta(A_{o})=\mu(B_{V_{1}})\cdot\nu(\mathcal{O}_{V_{2}})=m_{G}(V_{1})\mu_{Y}(B)\nu(\mathcal{O}_{V_{2}})>0$.

Furthermore, we define the Borel $G$-actions $a_{k}:G\times Z\rightarrow Z$ by

$a_{1}(g,(x,\Lambda_{o}^{\prime}))=(-\alpha_{1}(g).x,g.\Lambda_{o}^{\prime})$

and

$a_{k}(g,(x,\Lambda_{o}^{\prime}))=(-\alpha_{k}(g).x,\Lambda_{o}^{\prime}),\quad
k=2,\ldots,r,$

for $(x,\Lambda_{o}^{\prime})\in Z$. Clearly these actions preserve $\theta$,
they all commute with each other, and

$\mu\Big{(}B_{V_{1}}\cap\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(g)).B_{V_{1}}\Big{)}\Big{)}\cdot\nu(\mathcal{O}_{V_{2}}\cap
h.\mathcal{O}_{V_{2}})=(\mu\otimes\nu)\Big{(}A_{o}\cap\Big{(}\bigcap_{k=1}^{r}a_{k}(g,A_{o})\Big{)}\Big{)},$

for all $g\in G$. We now conclude from (5.1) and (5.3) that

$\displaystyle\sum_{\lambda\in\Lambda\cap
F_{n}}\mu_{Y}\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).B\Big{)}\geq$
$\displaystyle\frac{1}{m_{G}(V_{o})m_{G}(V_{1})}\cdot\int_{F_{n}^{\prime}}(\mu\otimes\nu)\Big{(}A_{o}\cap\Big{(}\bigcap_{k=1}^{r}a_{k}(g,A_{o})\Big{)}\Big{)}\,dm_{G}(g)$

for all $n$. Since both $\mu$ and $\nu$ are probability measures,

$\displaystyle\int_{F_{n}^{\prime}}(\mu\otimes\nu)\Big{(}A_{o}\cap\Big{(}\bigcap_{k=1}^{r}a_{k}(g,A_{o})\Big{)}\Big{)}\,dm_{G}(g)\geq$
$\displaystyle\int_{F_{n}}(\mu\otimes\nu)\Big{(}A_{o}\cap\Big{(}\bigcap_{k=1}^{r}a_{k}(g,A_{o})\Big{)}\Big{)}\,dm_{G}(g)-m_{G}(F_{n}\setminus
F_{n}^{\prime}),$

for all $n$. Hence, if we set

$\beta_{o}=\frac{1}{m_{G}(V_{o})m_{G}(V_{1})}\quad\textrm{and}\quad\delta_{n}=\frac{\beta_{o}\cdot
m_{G}(F_{n}\setminus F_{n}^{\prime})}{m_{G}(F_{n})},$

then

$\displaystyle\sum_{\lambda\in\Lambda\cap
F_{n}}\mu_{Y}\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).B\Big{)}\geq$
$\displaystyle\beta_{o}\cdot\int_{F_{n}}(\mu\otimes\nu)\Big{(}A_{o}\cap\Big{(}\bigcap_{k=1}^{r}a_{k}(g,A_{o})\Big{)}\Big{)}\,dm_{G}(g)-\delta_{n}\cdot
m_{G}(F_{n}),$

for all $n$. Since $(F_{n})$ is a strong Følner sequence, $(\delta_{n})$ is a
null sequence, and we are done.

##  6\. Proof of Theorem 1.5

Let $\Lambda,\Lambda,P_{o},q$ and $\alpha_{1},\ldots,\alpha_{r}$ be as in
Theorem 1.5. Set $\Delta=\Lambda^{q}$, so that
$\alpha_{k}(\Lambda)\subseteq\Delta$ for all $k$. Furthermore, by Lemma 2.4,

$\Xi_{\mathcal{T}_{P_{o}}}-\Delta\subset\overline{P_{o}-P_{o}}-\Lambda^{q}\subset\Lambda-\Lambda-\Lambda^{q}\subset\Lambda^{q+2},$

which is uniformly discrete in $G$ since $\Lambda$ is an approximate lattice.
We conclude that the conditions on $\Lambda$ and $\Delta$ in Theorem 1.9 are
satisfied.

Since $d^{*}(P_{o})>0$, Theorem 3.1 further tells us that there exists
$\mu\in\operatorname{Prob}_{G}(\Omega^{\times}_{P_{o}})$ such that
$\mu_{\mathcal{T}_{P_{o}}}(\mathcal{T}_{P_{o}})>0$ and

$d^{*}\Big{(}P_{o}\cap\Big{(}\bigcap_{k=1}^{r}(P_{o}-\alpha_{k}(\lambda))\Big{)}\Big{)}\geq\mu_{\mathcal{T}_{P_{o}}}\Big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).\mathcal{T}_{P_{o}}\Big{)},$
(6.1)

for all $\lambda\in\Lambda$. Hence, by Theorem 1.9, applied to
$X=\Omega_{P_{o}}^{\times},Y=\mathcal{T}_{P_{o}}$ and $B=\mathcal{T}_{P_{o}}$,
there exists $c>0$ such that

$S_{o}=\big{\\{}\lambda\in\Lambda\,:\,\mu_{\mathcal{T}_{P_{o}}}\big{(}\bigcap_{k=1}^{r}(-\alpha_{k}(\lambda)).\mathcal{T}_{P_{o}}\big{)}\geq
c\big{\\}}$

is syndetic in $\Lambda$. By (6.1), we have $S_{o}\subset S$, where

$S=\Big{\\{}\lambda\in\Lambda\,:\,d^{*}\Big{(}P_{o}\cap\Big{(}\bigcap_{k=1}^{r}(P_{o}-\alpha_{k}(\lambda))\Big{)}\Big{)}\geq
c\Big{\\}},$

and thus $S$ is syndetic in $\Lambda$ as well.

##  Appendix A Szemerédi’s Theorem for cut-and-project sets in Euclidean
spaces via the IP-Szemerédi Theorem

The aim of this section is to prove the following strengthening of Theorem 1.2
using the deep ergodic $\operatorname{IP}$-theory developed by Furstenberg and
Katznelson in [16], and further extended by Bergelson and McCutcheon in [7].

######  Theorem A.1.

Let $\Lambda\subset\mathbb{R}^{d}$ be an approximate lattice and suppose that

$\Delta=\Delta(\mathbb{R},\mathbb{R}^{m},\Gamma,W)\subset\mathbb{R}$

is a cut-and-project set for some lattice
$\Gamma<\mathbb{R}\times\mathbb{R}^{m}$ which projects injectively to
$\mathbb{R}$ and densely to $\mathbb{R}^{m}$, and for some bounded Borel set
$W\subset\mathbb{R}^{m}$ whose interior is a convex and symmetric set
containing $0$. Assume that $\delta\cdot\Lambda\subset\Lambda$ for every
$\delta\in\Delta$. Then, for every $P_{o}\subset\Lambda$ with positive upper
Banach density and finite subset $F\subset\Lambda^{\infty}$, there exist an
integer $n$ and a syndetic and symmetric subset $\Delta_{n}\subset\Delta$ such
that $\Delta_{n}^{n}\subset\Delta$ with the property that for every
$\delta\in\Delta_{n}$, there exist an integer $j=1,\ldots,n$ and $p_{o}\in
P_{o}$, such that

$p_{o}+j\delta\cdot F\subset P_{o}.$

######  Remark A.2.

Theorem 1.2 corresponds to

$\Lambda=\Delta=\Delta(\mathbb{R},\mathbb{Z}[\sqrt{D}],[-1,1]).$

Furthermore, since $\Delta_{n}^{n}\subset\Delta$, we have $j\delta\in\Delta$
for all $j=1,\ldots,n$. In particular, this shows that the set $S_{F}$ in
Theorem 1.2 is non-empty.

### Ergodic $\operatorname{IP}$-theory

Let $H$ be an abelian semi-group and $n$ a positive integer. We denote by
$2^{[n]}$ the set of all subsets of the set $[n]=\\{1,\ldots,n\\}$. A function
$\varphi:2^{[n]}\rightarrow H$ such that

$\varphi(\alpha\cup\beta)=\varphi(\alpha)+\varphi(\beta),\quad\textrm{for all
$\alpha,\beta\in 2^{[n]}$ such that $\alpha\cap\beta=\emptyset$}$

is called an _$\operatorname{IP}_{n}$ -system with values in $H$_.

######  Example A.3.

Let $L\subset H$ be a set with at most $n$ elements, and fix
$h_{1},\ldots,h_{n}\in L$ (allowing repetitions). Then

$\varphi:2^{[n]}\rightarrow H,\quad\varphi(\alpha)=\sum_{k\in\alpha}h_{k}$

is an $\operatorname{IP}_{n}$-system with values in $H$. In particular, if we
fix $h\in H$, and set $h_{k}=h$ for all $k=1,\ldots,n$, then

$\varphi(\alpha)=|\alpha|\cdot h,\quad\textrm{for all $\alpha\in 2^{[n]}$}.$

We denote this special $\operatorname{IP}_{n}$-system by $\varphi_{n,h}$.

We will deduce Theorem A.1 from the following result ([16, Theorem 9.5]),
which is only stated for subsets in $\mathbb{R}^{d}$ in [16]. We denote by
$\mathbb{R}^{+}$ the additive semigroup of positive real numbers.

######  Theorem A.4 (Furstenberg-Katznelson).

Let $A\subset\mathbb{R}^{d}$ be a Borel set and suppose that

$\varepsilon:=\varlimsup_{n\rightarrow\infty}\frac{m_{\mathbb{R}^{d}}(A\cap[-n,n]^{d})}{n^{d}}>0,$

Then, for every finite set $F\subset\mathbb{R}^{d}$, there exists an integer
$n=n(\varepsilon,|F|)$ with the property that for every
$\operatorname{IP}_{n}$-system $\varphi$ with values in $\mathbb{R}^{+}$,
there exist $\alpha\in 2^{[n]}$ and $x\in\mathbb{R}^{d}$ such that

$x+\varphi(\alpha)\cdot F\subset A.$

### Proof of Theorem A.1

Let $\Lambda,\Delta,\Gamma$ and $W$ be as in Theorem A.1. Let
$F_{o}=\\{\lambda_{1},\ldots,\lambda_{r}\\}\subset\Lambda^{\infty}$ be a
finite set, and define $F=\\{0,\lambda_{1},\ldots,\lambda_{r}\\}$. Let $q\geq
1$ be the smallest integer such that $F\subset\Lambda^{q}$. Since $\Lambda$ is
an approximate lattice, there is an open identity neighbourhood $U$ in
$\mathbb{R}^{d}$ such that $\Lambda^{q+2}\cap U=\\{0\\}$.

Suppose $P_{o}\subset\Lambda$ has positive upper asymptotic density along
$([-n,n]^{d})$ in $\mathbb{R}^{d}$. Fix an open identity neighbourhood $V$ in
$\mathbb{R}^{d}$ such that $V-V\subset U$, and define the open set
$A=P_{o}+V\subset\mathbb{R}^{d}$. Since $P_{o}$ has positive upper asymptotic
density along $([-n,n]^{d})$, we have

$\varepsilon:=\varlimsup_{n\rightarrow\infty}\frac{m_{\mathbb{R}^{d}}(A\cap[-n,n]^{d})}{n^{d}}>0.$

By Theorem A.4, there exists an integer $n$, only depending on $\varepsilon$
and $r$ with the property that for every $\operatorname{IP}_{n}$-system
$\varphi$ with values in $\mathbb{R}^{+}$, there exist $\alpha\in 2^{[n]}$ and
$x\in\mathbb{R}^{d}$ such that

$x+\varphi(\alpha)\cdot F\subset A=P_{o}+V.$ (A.1)

Define $\Delta_{n}=\Delta(\mathbb{R}^{m},\Gamma,\frac{1}{n}W^{o})$, where
$W^{o}\subset W$ denotes the interior of $W$, which is assumed to be convex
and symmetric. In particular, $\Delta_{n}$ is again a cut-and-project set, and
is thus both relatively dense and uniformly discrete in $\mathbb{R}^{d}$.
Furthermore,

$\Delta_{n}^{n}\subset\Delta\Big{(}\mathbb{R}^{m},\Gamma,\frac{1}{n}W^{o}+\ldots+\frac{1}{n}W^{o}\Big{)}\subset\Delta(\mathbb{R}^{m},\Gamma,W^{o})\subset\Delta.$

Hence, if we pick any $\delta\in\Delta_{n}\cap\mathbb{R}^{+}$, then
$\varphi_{n,\delta}:2^{[n]}\rightarrow\mathbb{R}^{+}$, as defined in the
example above, is an $\operatorname{IP}_{n}$-system such that

$\varphi_{n,\delta}(\alpha)=|\alpha|\cdot\delta\in\Delta,\quad\textrm{for all
$\alpha\in 2^{[n]}$}.$

Fix $\delta\in\Delta_{n}$. By (A.1), there exist $j=1,\ldots,n$ and
$x\in\mathbb{R}^{d}$ such that

$x+j\delta\cdot\\{0,\lambda_{1},\ldots,\lambda_{r}\\}\subset P_{o}+V.$

Set $\lambda_{o}=0$. The inclusion above implies that there are
$p_{o},p_{1},\ldots,p_{r}\in P_{o}$ and $v_{o},v_{1},\ldots,v_{r}\in V$ such
that

$x+j\delta\cdot\lambda_{k}=p_{k}+v_{k},\quad\textrm{for all
$k=0,1,\ldots,r$}.$

Note that since $j\delta\in\Delta$ for all $j=1,\ldots,n$ and
$j\delta\cdot\Lambda\subset\Lambda$, we have

$j\delta\cdot\lambda_{k}\in
j\delta\cdot\Lambda^{q}\subset\Lambda^{q},\quad\textrm{for all
$k=0,1,\ldots,r$}.$

By taking differences,

$(p_{k}+v_{k})-(p_{o}+v_{o})=j\delta\cdot\lambda_{k}\in\Lambda^{q},\quad\textrm{for
all $k$},$

and thus
$v_{k}-v_{o}\in(\Lambda^{q}-P_{o}+P_{o})\cap(V-V)\subset\Lambda^{q+2}\cap
U=\\{0\\}$ for all $k$. We conclude that $v_{o}=v_{1}=\ldots,v_{r}$, so

$p_{k}=p_{o}+j\delta\cdot\lambda_{k},\quad\textrm{for all $k=1,\ldots,r$}.$

Hence, $p_{o}+j\delta\cdot\\{\lambda_{1},\ldots,\lambda_{r}\\}\subset P_{o}$
and $p_{o}\in P_{o}$, which finishes the proof.

### A few comments about the proof

This proof can be extended to other lcsc abelian groups, thus providing _some_
(potentially weaker) version of our main combinatorial Theorem 1.5 for _cut-
and-project sets_. To also establish syndeticity of the sets $S$ in Theorem
1.5 along these lines (at least in the case when the ambient group $G$
contains a dense finitely generated group), a theory of
$\operatorname{IP}^{*}$-sets in approximate lattices needs to be developed.
However, the proof above uses that the approximate lattice
$\Delta\subset\mathbb{R}$ is a cut-and-project set, and thus contains, for
every $n\geq 1$, a syndetic set $\Delta_{n}\subset\Delta$ such that
$\Delta_{n}^{n}\subset\Delta$. As we have already pointed out in Remark 1.6,
this property does not hold for arbitrary approximate lattices, so if one
wants to prove Theorem 1.5 in its full generality using this approach, then
some other way of producing $\operatorname{IP}$-systems inside an approximate
lattice is needed.

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