1. Introduction
1.1. Szemerédi’s theorem and the aim of the paper
The upper asymptotic density 
of a subset
$P_o \subset \mathbb {Z}$
is defined by
The celebrated theorem of Szemerédi [Reference Szemerédi20], with the ergodic-theoretical proof of Furstenberg [Reference Furstenberg14, Theorem 11.13], asserts that, for every
$P_o \subset \mathbb {Z}$
with positive upper asymptotic density and for every finite set
$F \subset \mathbb {Z}$
, the set
$$ \begin{align*} S_F = \{ \unicode{x3bb} \in \mathbb{Z} :\text{ there exists } \, \unicode{x3bb}_o \in P_o\text{ such that } \unicode{x3bb}_o + \unicode{x3bb} \cdot F \subset P_o \} \end{align*} $$
is syndetic in
$\mathbb {Z}$
: that is, there is a finite set
$K \subset \mathbb {Z}$
such that
$S_F + K = \mathbb {Z}$
. In this article, we show that 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 that
$\Lambda \subset R$
is a symmetric, uniformly discrete and multiplicatively closed subset of R, which is an approximate subgroup under addition: that is, 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
$$ \begin{align*} S_F = \{ \unicode{x3bb} \in \Lambda :\text{ there exists } \unicode{x3bb}_o \in P_o\text{ such that } \unicode{x3bb}_o + \unicode{x3bb} \cdot F \subset P_o \}. \end{align*} $$
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 [Reference Klick, Strungaru and Tcaciuc17].
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
$$ \begin{align*} \Lambda = \{ m + n \sqrt{D} :|m-n\sqrt{D}| \leq 1 \}. \end{align*} $$
It is not hard to check that
$\Lambda $
is symmetric, uniformly discrete in
$\mathbb {R}$
and multiplicatively closed. Furthermore,
$$ \begin{align*} \Lambda + \Lambda \subset \Lambda + \{-1,0,1\}, \end{align*} $$
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
$$ \begin{align*} \varlimsup_{n \rightarrow \infty} \frac{|P_o \cap [-n,n]|}{2n}> 0, \end{align*} $$
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 that
$P_o \subset \Lambda $
has positive upper asymptotic density. Then, for every finite set
$F \subset \mathbb {Z}[\sqrt {D}]$
, the set
$$ \begin{align*} S_F = \{ \unicode{x3bb} \in \Lambda :\text{ there exists } \unicode{x3bb}_o \in P_o\text{ such that } \unicode{x3bb}_o + \unicode{x3bb} \cdot F \subset P_o \} \end{align*} $$
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 [Reference Furstenberg and Katznelson16]. 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 locally compact and second countable (lcsc) abelian groups like
$\mathbb {Q}_p$
, without dense finitely generated subgroups, it seems difficult (but certainly not impossible) to establish syndeticity of sets such as
$S_F$
using this approach, and we believe that such a proof might provide a way to prove Conjecture 1.2 below. Our proof of syndeticity of
$S_F$
, instead, employs some recent results of Austin [Reference Austin2].
1.2.2. p-adic examples
Let p be a prime number and define the set
$\Lambda _p \subset \mathbb {Q}_p$
by
$$ \begin{align*} \Lambda_p = \{ \gamma \in \mathbb{Z}[1/p] :|\gamma|_\infty \leq 1 \}, \end{align*} $$
where
$|\cdot |_\infty $
denotes the (real) absolute value when
$\mathbb {Z}[1/p]$
is viewed as a subring of
$\mathbb {R}$
. Even though
$\mathbb {Q}_p$
does not admit a lattice, the set
$\Lambda _p$
is, nevertheless, an approximate lattice: that is, it is a 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
$$ \begin{align*} \varlimsup_{n \rightarrow \infty} \frac{|P_o \cap F_n|}{p^n}> 0. \end{align*} $$
The following result is a special case of Corollary 1.7 below.
Theorem 1.4. Suppose that
$P_o \subset \Lambda _p$
has positive upper asymptotic density. Then, for every finite set
$F \subset \mathbb {Z}[1/p]$
, the set
$$ \begin{align*} S_F = \{ \unicode{x3bb} \in \Lambda :\text{ there exists } \unicode{x3bb}_o \in P_o\text{ such that}\ \unicode{x3bb}_o + \unicode{x3bb} \cdot F \subset P_o \} \end{align*} $$
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 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
where 
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 they do simplify their exposition.
The examples in the previous subsection are of this form with
$$ \begin{align*} G = H = \mathbb{R}, \Gamma = \{ (m+n\sqrt{D},m-n\sqrt{D}) :m,n \in \mathbb{Z}\}, W = [-1,1] \end{align*} $$
and
$$ \begin{align*} G = \mathbb{Q}_p, H = \mathbb{R}, \Gamma = \{ (\gamma,\gamma) :\gamma \in \mathbb{Z}[1/p] \}, W = [-1,1], \end{align*} $$
respectively. It is not difficult to see that every
$\Lambda $
of this form is uniformly discrete and relatively dense (that is, there is a compact set
$Q \subset G$
such that
$Q + \Lambda = G$
). Furthermore, there is a finite
$F \subset G$
such that
$$ \begin{align*} \Lambda - \Lambda \subset \Lambda + F. \end{align*} $$
The study of cut-and-project sets was initiated by Meyer [Reference Meyer18].
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
$$ \begin{align*} \Lambda^q = \underbrace{\Lambda + \cdots + \Lambda} \end{align*} $$
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
$W \subset H$
whose interior contains
$0$
, is an approximate lattice.
Fix a Haar measure
$m_G$
on G. Given a strong Følner sequence
$(F_n)$
(see §2.3 for definitions), we define the upper asymptotic density 
of a uniformly discrete subset
$P_o \subset G$
by
and we define the upper Banach density
$d^*(P_o)$
by
We show below (§2.4) that
$d^*(P_o)$
is always finite.
1.4. Main combinatorial results
We can now state our first main result, the proof of which will be given in §6. Let G be an lcsc abelian group and denote by 
the space of continuous endomorphisms on G.
Theorem 1.5. Let
$\Lambda \subset G$
be an approximate lattice and suppose that 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 a subset with positive Banach upper density. Suppose that there exist 
and
$q \geq 1$
such that
$\alpha _k(\Lambda ) \subset \Lambda ^q$
for all k. Then, there exists
$c> 0$
such that the set
$$ \begin{align*} S = \bigg\{ \unicode{x3bb} \in \Lambda :d^*\bigg(P_o \cap \bigg( \bigcap_{k=1}^r (P_o - \alpha_k(\unicode{x3bb})) \bigg)\bigg) \geq c \bigg\} \end{align*} $$
is syndetic in
$\Lambda $
.
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
$G = \mathbb {R}$
) that do not contain the difference set of any relatively dense set [Reference Björklund and Hartnick9, Example, §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: that is, 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, then we can deduce the following corollary from Theorem 1.5.
Corollary 1.7. Let R be an lcsc 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 finite set
$F \subset \Lambda ^\infty $
, the set
$$ \begin{align*} S_o = \{ \unicode{x3bb} \in \Lambda \, : \unicode{x3bb}_o + \unicode{x3bb} \cdot F \subset P_o \text{ for some }\unicode{x3bb}_o \in P_o \} \end{align*} $$
is syndetic in
$\Lambda $
.
Proof. Let
$\Lambda = \Lambda (R,H,\Gamma ,W) \subset R$
for some
$H,\Gamma $
and W, and suppose that
$\Lambda $
is multiplicatively closed, that is,
$\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
$W_o \subset W$
that 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 an approximate lattice in R (and thus has positive upper Banach density) and
$$ \begin{align*} \Lambda_o - \Lambda_o \subset \Lambda. \end{align*} $$
Let
$F = \{\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _r\} \subset \Lambda ^\infty $
be a finite set, and let
$q \geq 1$
denote the smallest integer such that
$F \subset \Lambda ^q$
. Define 
by
$\alpha _k(g) = g \unicode{x3bb} _k$
for
$k=1,\ldots ,r$
. Then, since
$\Lambda \cdot \Lambda \subset \Lambda $
,
$$ \begin{align*} \alpha_k(\Lambda) \subset \Lambda \cdot \Lambda^q \subset \Lambda^q \quad \text{for all }k=1,\ldots,r. \end{align*} $$
Hence, all of the conditions of Theorem 1.5 are satisfied and there is a constant
$c> 0$
such that
$$ \begin{align*} S = \bigg\{ \unicode{x3bb} \in \Lambda :d^*\bigg( P_o \cap \bigg( \bigcap_{k=1}^r (P_o - \unicode{x3bb} \unicode{x3bb}_k ) \bigg) \bigg) \geq c \bigg\} \end{align*} $$
is syndetic in
$\Lambda $
. Clearly,
$S \subset S_o$
, so the proof is complete.
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 [Reference Klick, Strungaru and Tcaciuc17, Conjecture 6.7]. Recall that a subset
$\{\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _r\}$
in an lcsc group G is an arithmetic progression of length r if
$\unicode{x3bb} = \unicode{x3bb} _{k+1} - \unicode{x3bb} _k \neq 0$
for all
$k=1,\ldots ,r-1$
: that is, if
$\unicode{x3bb} _k = \unicode{x3bb} _o + k \cdot \unicode{x3bb} $
for all
$k=1,\ldots ,r$
, for some
$\unicode{x3bb} _o$
and for
$\unicode{x3bb} \in G$
. We refer to
$\unicode{x3bb} $
as the gap of the arithmetic progression.
Conjecture 1.1. Let G be an 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.
We now briefly show how Theorem 1.5 establishes the following strengthening of this conjecture.
Theorem 1.8. Let G be an 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 exists a syndetic set
$S_r \subset \Lambda $
with the property that, for every
$\unicode{x3bb} \in S_r$
, there is a set
$P_\unicode{x3bb} \subset P_o$
with positive upper Banach density such that
$$ \begin{align*} \unicode{x3bb}_o + \{\unicode{x3bb},2 \unicode{x3bb},\ldots,r\unicode{x3bb}\} \subset P_o \quad \text{for all }\unicode{x3bb}_o \in P_\unicode{x3bb}. \end{align*} $$
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 
by
$$ \begin{align*} \alpha_k(g) = k \cdot g \quad \text{for }k=1,\ldots,r. \end{align*} $$
Then, since
$\Lambda \subset \Lambda ^2 \subset \cdots \subset \Lambda ^r$
, we have
$\alpha _k(\Lambda ) \subset \Lambda ^{r}$
for all
$k=1,\ldots ,r$
, so the conditions of Theorem 1.5 are satisfied with
$q=r$
, and we conclude that there exists a syndetic set
$S_r \subset \Lambda $
with the property that, for every
$\unicode{x3bb} \in S_r$
, the set
$$ \begin{align*} P_\unicode{x3bb} = P_o \cap \bigg( \bigcap_{k=1}^r (P_o - k \unicode{x3bb}) \bigg) \subset P_o \end{align*} $$
has positive upper Banach density.
1.6. Main dynamical result
In §3, we establish a version of Furstenberg’s famous correspondence principle for uniformly discrete subsets of an lcsc abelian group G. Using this correspondence principle, it is not hard to deduce Theorem 1.5 from the following dynamical statement, the proof of which will be given in §5.
Let
$(X,\mathscr {B}_X)$
be a standard Borel space, equipped with a Borel measurable G-action
$G \times X \rightarrow X, (g,x) \mapsto g.x$
. 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
$$ \begin{align*} \Xi_Y = \{ g \in G :g.Y \cap Y \neq \emptyset \} \subset G. \end{align*} $$
For every 
, 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 the reader to §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 
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 
and for every Borel set
$B \subset Y$
with positive
$\mu _Y$
-measure, there exists
$c> 0$
such that the set
$$ \begin{align*} S = \bigg\{ \unicode{x3bb} \in \Lambda :\mu_Y\bigg( \bigcap_{k=1}^r (-\alpha_k(\unicode{x3bb})).B\bigg) \geq c \bigg\} \end{align*} $$
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 [Reference Bergelson and Leibman4–Reference Bergelson and McCutcheon6]. In view of these works, specifically [Reference Bergelson and McCutcheon6, 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
$$ \begin{align*} p(x) = \sum_{k=0}^r a_k x^k \quad \text{for some }a_o, a_1,\ldots,a_r \in R. \end{align*} $$
Conjecture 1.2. Let R be an lcsc 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
$$ \begin{align*} S_F = \{ \unicode{x3bb} \in \Lambda :\text{ there exists } \unicode{x3bb}_o \in P_o\text{ such that } \unicode{x3bb}_o + \{p_1(\unicode{x3bb}),\ldots,p_r(\unicode{x3bb})\} \subset P_o \} \end{align*} $$
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 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 [Reference Björklund and Hartnick8, Reference Björklund and Hartnick9].
1.8. Organization of the paper
In §2, we introduce notation and some fundamental concepts such as cross-sections, transverse measures, Banach densities and strong Følner sequences. The most important result in §3 is a transverse version of Furstenberg’s correspondence principle (Theorem 3.1). In §4, we extend Furstenberg and Katznelson’s multiple recurrence theorem to general lcsc abelian groups, using recent works of Austin. In §5, we prove our main dynamical theorem (Theorem 1.9) and in §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 [Reference Furstenberg and Katznelson16].
2. Preliminaries
Let G be an lcsc abelian group. In particular, as a topological space, G is
$\sigma $
-compact [Reference Cornulier and de la Harpe11, Theorem 2.B.4]. We 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 
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
$$ \begin{align*} \Xi_Y = \{ g \in G :g.Y \cap Y \neq \emptyset \}. \end{align*} $$
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., [Reference Slutsky19, 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., [Reference Björklund, Hartnick and Karasik10, 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 
, there exists a unique non-negative finite Borel measure
$\mu _Y$
on Y such that
$$ \begin{align*} \mu(V.B) = m_G(V) \cdot \mu_Y(B) \end{align*} $$
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 
, 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, for any Borel set
$B \subset Y$
,
$$ \begin{align*} \mu_Y(B)> 0\! \iff \mu(V.B) > 0 \end{align*} $$
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
$W \subset U$
such that
$(\Xi _Y - \Delta ) \cap W = \{0\}$
. For every Borel set
$V_o$
in G such that
$V_o-V_o \subset W$
, and for every Borel set
$B \subset Y$
,
$$ \begin{align*} V_o.\bigg(B \cap \bigcap_{k=1}^r (-g_k).B\bigg) = B_{V_o} \cap \bigg( \bigcap_{k=1}^r (-g_k).B_{V_o}\bigg) \quad \text{for all }g_1,\ldots,g_r \in \Delta, \end{align*} $$
where
$B_{V_o} = V_o.B$
. In particular, for every 
,
$$ \begin{align*} \mu_Y\bigg( B \cap \bigcap_{k=1}^r (-g_k).B\bigg) = \frac{1}{m_G(V_o)} \cdot \mu\bigg(B_{V_o} \cap \bigg( \bigcap_{k=1}^r (-g_k).B_{V_o}\bigg)\bigg). \end{align*} $$
Proof. Fix
$V_o$
and B, as in the lemma, and an r-tuple
$g_1,\ldots ,g_r \in \Delta $
. We first note that the inclusion
$$ \begin{align*} V_o.\bigg(B \cap \bigcap_{k=1}^r (-g_k).B\bigg) \subseteq B_{V_o} \cap \bigg( \bigcap_{k=1}^r (-g_k).B_{V_o}\bigg) \end{align*} $$
is trivial. In particular, if the right-hand side is empty, then so is the left-hand side. It thus remains to show that if the right-hand side is non-empty, then
$$ \begin{align} V_o.\bigg(B \cap \bigcap_{k=1}^r (-g_k).B\bigg) \supseteq B_{V_o} \cap \bigg( \bigcap_{k=1}^r (-g_k).B_{V_o}\bigg). \end{align} $$
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
$$ \begin{align*} x = v_o.y_o = (v_1-g_1).y_1 = \cdots = (v_r - g_r).y_r. \end{align*} $$
In particular,
$v_o - v_k + g_k \in \Xi _Y$
for all
$k=1,\ldots ,r$
and thus
$$ \begin{align*} v_o - v_k \in (\Xi_Y - g_k) \cap (V_o - V_o) \subseteq (\Xi_Y - \Delta) \cap W = \{0\} \quad \text{for all }k=1,\ldots, r. \end{align*} $$
We conclude that
$v_o = v_1 = \cdots = v_r$
, and thus
$y_o = (-g_1).y_1 = \cdots = (-g_r).y_r$
. In particular,
$$ \begin{align*} x = v_o.y_o \in v_o.\bigg(B \cap \bigcap_{k=1}^r (-g_k).B\bigg) \subset V_o.\bigg(B \cap \bigcap_{k=1}^r (-g_k).B\bigg). \end{align*} $$
Since x is arbitrary, this shows (2.1), and the proof is complete.
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 [Reference Benedetti and Petronio3, §E.1] that
$\mathscr {C}(G)$
is a compact metrizable space and that 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 subsequence
$(n_k)$
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
$$ \begin{align*} (g,P) \mapsto g.P = P - g, \quad (g,P) \in G \times \mathscr{C}(G). \end{align*} $$
Given
$P_o \in \mathscr {C}(G)$
, we define
and we refer to 
as the hull of
$P_o$
and to 
as the punctured hull of
$P_o$
. Note that the space 
is compact, whereas 
is only compact if 
. We also define the canonical cross-section 
by
Note that
$\mathcal {T}_{P_o}$
is a compact subset of 
and, for every open set
$V \subset G$
, the set
is open in both 
and 
.
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 
. Then,
and
$\mathcal {T}_{P_o}$
is a U-separated cross-section in 
with 
.
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
$$ \begin{align*} p_1 = p^{o}_{1,n} - g_n + w_{1,n} \quad \text{and} \quad p_2 = p^{o}_{2,n} - g_n + w_{2,n} \end{align*} $$
for all sufficiently large n. In particular,
$$ \begin{align*} p_1 - p_2 = (p_{1,n}^o + w_{1,n}) - (p_{2,n}^o + w_{2,n}) \subset P_o - P_o + W-W. \end{align*} $$
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 
. 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 
. 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 
. By choosing a continuous metric
$\rho $
on M that 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 a
$\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 a non-empty, open and
$m_G$
-Jordan measurable subset of G such that 
. Then, for every 
, the open set 
is
$\mu $
-Jordan measurable.
Proof. Fix 
and write
$\mu = \mu ' + \alpha \cdot \delta _{\{\emptyset \}}$
for some finite and non-negative Borel measure
$\mu '$
on 
and
$\alpha \geq 0$
. Recall that Lemma 2.4 tells us that
$\mathcal {T}_{P_o}$
is a U-separated cross-section in 
, and thus the transverse measure
$(\mu ')_{\mathcal {T}_{P_o}}$
is well defined. Since
$\mathcal {O}_V$
is open, 
and 
,
and thus
$\mathcal {O}_V$
is
$\mu $
-Jordan measurable. Since
$\mu $
is arbitrary, the proof is complete.
We also record the following lemma for future use.
Lemma 2.6. Let
$\Lambda \subset G$
be a countable 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 
,
$$ \begin{align*} \sum_{\unicode{x3bb} \in \Lambda} \chi_V(g+\unicode{x3bb}) \geq \nu(\mathcal{O}_{V_o} \cap g.\mathcal{O}_{V_o}) \quad \text{for all }g \in G, \end{align*} $$
where
$\chi _V$
denotes the indicator function of V, and 
.
Proof. It is clearly enough to show that
$$ \begin{align*} \mathcal{O}_{V_o} \cap g.\mathcal{O}_{V_o} \neq \emptyset \implies\! (g + \Lambda) \cap V \neq \emptyset. \end{align*} $$
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 
, 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,
$$ \begin{align*} (\Lambda_o - g_o) \cap V_o \neq \emptyset \quad \text{and} \quad (\Lambda_o - g_o+g) \cap V_o \neq \emptyset. \end{align*} $$
By taking difference, this implies that
$$ \begin{align*} \emptyset \neq (g + \Lambda_o - \Lambda_o) \cap (V_o - V_o) \subset (g + \Lambda) \cap V, \end{align*} $$
and the proof is complete.
2.3. Strong Følner sequences
Let G be an 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
$\lim _{n \rightarrow \infty } ({m_G(F_n+C)}/{m_G(F_n)}) = 1$
; and -
(ii) there exists an open bounded identity neighbourhood
$V \subset G$
such that
$$ \begin{align*} \lim_{n \rightarrow \infty} \frac{m_G( \bigcap_{v \in V} (F_n-v) )}{m_G(F_n)} = 1. \end{align*} $$
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
$$ \begin{align*} F_n := L_n + g_n \quad \text{and} \quad \widetilde{F}_n := L_n + V \end{align*} $$
are both V-adapted strong Følner sequences.
Proof. Since G is both amenable and
$\sigma $
-compact, [Reference Emerson13, 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
$B \subset G$
,
$$ \begin{align} \lim_{n \rightarrow \infty} \frac{m_G((K_n+B) \Delta K_n)}{m_G(K_n)} = 0, \end{align} $$
where
$\Delta $
denotes the set-theoretical difference. In particular, if
$0 \in B$
, then
$$ \begin{align} 1 \leq \frac{m_G(K_n + B)}{m_G(K_n)} = 1 + \frac{m_G((K_n + B) \setminus K_n)}{m_G(K_n)} \rightarrow 1 \quad \text{as }n \rightarrow \infty. \end{align} $$
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. First, we show that
$(F_n)$
satisfies
$(i)$
. Let C be a compact subset of G. Since V is pre-compact, 
is compact and
$F_n + C \subset K_n + B$
, and thus (2.2) implies that
$$ \begin{align*} 1 \leq \frac{m_G(F_n + C)}{m_G(F_n)} \leq \frac{m_G(K_n + B)}{m_G(K_n)} \rightarrow 1, \quad n \rightarrow \infty, \end{align*} $$
which proves (i). We now show (ii). Since
$F_n = K_n + V$
and V is a pre-compact identity neighbourhood, (2.3) tells us that
$$ \begin{align*} 1 \geq \lim_{n \rightarrow \infty} \frac{m_G( \bigcap_{v \in V} (F_n-v) )}{m_G(F_n)} \geq \lim_{n \rightarrow \infty} \frac{m_G(K_n)}{m_G(K_n + V)} = 1, \end{align*} $$
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
$$ \begin{align*} \frac{m_G(F_n + C)}{m_G(F_n)} = \frac{m_G(L_n + C)}{m_G(L_n)} \rightarrow 1, \quad n \rightarrow \infty, \end{align*} $$
since
$(L_n)$
is a strong Følner sequence in G. Furthermore,
$$ \begin{align*} \bigcap_{v \in V} (F_n - v) = g_n + \bigcap_{v \in V} (L_n - v) \quad \text{for all }n, \end{align*} $$
and thus
$$ \begin{align*} \lim_{n \rightarrow \infty} \frac{m_G( \bigcap_{v \in V} (F_n-v) )}{m_G(F_n)} = \lim_{n \rightarrow \infty} \frac{m_G( \bigcap_{v \in V} (L_n-v) )}{m_G(L_n)} = 1, \end{align*} $$
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 
of P along
$(F_n)$
by
and the upper Banach density
$d^*(P)$
by
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 Borel set
$V \subset G$
such that
$V-V \subset U$
and for every bounded Borel set
$Q \subset G$
,
$$ \begin{align*} |P \cap Q| \leq \frac{m_G(Q+V)}{m_G(V)}. \end{align*} $$
Furthermore, for every Borel set
$V_o \subset V$
such that
$V_o - V_o \subset V$
,
$$ \begin{align*} |P \cap (Q+V)| \geq \frac{m_G((P+V_o) \cap (Q+V_o))}{m_G(V_o)}. \end{align*} $$
Proof. Let V be a Borel set that contains
$0$
and such that
$V - V \subset U$
, and let
$Q \subset G$
be a bounded Borel set. Then, since P is U-uniformly discrete, all sets of the form
$p+V$
for
$p \in P$
are disjoint, and thus
$$ \begin{align} (P + V) \cap (Q+V) = \bigsqcup_{p \in P} (p + V) \cap (Q+V) \supseteq \bigsqcup_{p \in P \cap Q} (p+V). \end{align} $$
Hence,
$$ \begin{align*} m_G(Q + V) \geq m_G((P + V) \cap (Q+V)) \geq m_G(V) |P \cap Q|. \end{align*} $$
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,
$$ \begin{align*} (P + V_o) \cap (Q+V_o) = \bigsqcup_{p \in P} (p + V_o) \cap (Q+V_o) \subseteq \bigsqcup_{p \in P \cap (Q+V)} (p+V_o), \end{align*} $$
and thus
$$ \begin{align*} m_G((P + V_o) \cap (Q+V_o)) \leq m_G(V_o) \cdot |P \cap (Q+V)|.\\[-37pt] \end{align*} $$
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,
$$ \begin{align*} \varlimsup_{n \rightarrow \infty} \frac{|P \cap F_n|}{m_G(F_n)} \leq \varlimsup_{n \rightarrow \infty} \frac{m_G(F_n+V)}{m_G(V) \cdot m_G(F_n)} = \frac{1}{m_G(V)}. \end{align*} $$
Since
$(F_n)$
is arbitrary, the proof is complete.
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,
$$ \begin{align*} \varliminf_{n \rightarrow \infty} \frac{1}{m_G(F_n)} \sum_{\unicode{x3bb} \in \Lambda \cap F_n} f(\unicode{x3bb})> 0. \end{align*} $$
Then, there exists
$c> 0$
such that the set
$\{\unicode{x3bb} \in \Lambda :f(\unicode{x3bb} ) \geq c\}$
is syndetic in
$\Lambda $
.
Proof. Given
$c> 0$
, define
$$ \begin{align*} S_c = \{ \unicode{x3bb} \in \Lambda :f(\unicode{x3bb}) \geq c \}. \end{align*} $$
First, we 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 $
,
$$ \begin{align*} \sum_{\unicode{x3bb} \in \Lambda \cap F_n} f(\unicode{x3bb}) &= \sum_{\unicode{x3bb} \in (\Lambda \cap S^c_{1/n}) \cap F_n} f(\unicode{x3bb}) < \frac{1}{n} |\Lambda \cap F_n| \end{align*} $$
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,
$$ \begin{align*} |\Lambda \cap F_n| \leq m_G(F_n+V)/m_G(V) \end{align*} $$
for every identity neighbourhood V in G such that
$V-V \subset U$
, and thus
$$ \begin{align*} \frac{1}{m_G(F_n)} \cdot \sum_{\unicode{x3bb} \in \Lambda \cap F_n} f(\unicode{x3bb}) \leq \frac{m_G(F_n +V)}{n \cdot m_G(V) \cdot m_G(F_n)} \quad \text{for all }n. \end{align*} $$
Since
$(F_n)$
is a strong Følner sequence, the right-hand side tends to zero as
$n \rightarrow \infty $
, which contradicts 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
$$ \begin{align*} \Lambda \subset S_c + Q \cap (\Lambda - \Lambda). \end{align*} $$
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 $
,
$$ \begin{align*} \varliminf_{n \rightarrow \infty} \frac{1}{|F_n|} \sum_{\gamma \in F_n} f(\gamma)> 0. \end{align*} $$
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 an lcsc abelian group. The proof follows the usual route, first outlined by Furstenberg in [Reference Furstenberg14], 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 
denotes the punctured hull of
$P_o$
, then the canonical cross-section 
satisfies
by Lemma 2.4, and is thus U-separated. If 
, 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
Then, there exists an ergodic 
such that, for all
$r \geq 1$
,
$$ \begin{align*} d^*(P_o) = \mu_{\mathcal{T}_{P_o}}(\mathcal{T}_{P_o}) \quad \text{and} \quad d^*\bigg(P_o \cap \bigg( \bigcap_{k=1}^r (P_o - g_k) \bigg)\bigg) \geq \mu_{\mathcal{T}_{P_o}}\bigg(\bigcap_{k=1}^r (-g_k).\mathcal{T}_{P_o}\bigg) \end{align*} $$
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 that
$P_o \subset G$
is a U-uniformly discrete set with positive upper Banach density. Let
$\Delta $
and W be as above. Note that we may, without loss of generality, assume that
$W \subset U$
. We fix two symmetric and open identity neighbourhoods
$V_o$
and V in G such that
$$ \begin{align*} V_o - V_o + V_o - V_o \subset V \subset W \quad \text{and} \quad V-V \subset U, \end{align*} $$
and we require V to be
$m_G$
-Jordan measurable. Recall from §2.2 that every open set in G contains an open, symmetric and
$m_G$
-Jordan measurable subset, so the existence of such an 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 
such that 
.
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 
by
$$ \begin{align*} \mu_n = \frac{1}{m_G(\widetilde{F}_n)} \int_{\widetilde{F}_n} \delta_{g.P_o} \, dm_G(g). \end{align*} $$
Note that
$$ \begin{align*} \mu_n(\mathcal{O}_V) &= \frac{1}{m_G(\widetilde{F}_n)} \int_{\widetilde{F}_n} \delta_{g.P_o}(\mathcal{O}_V) \, dm_G(g) \\[0.2cm] &= \frac{m_G(\{ g \in \widetilde{F}_n :(P_o - g) \cap V \neq \emptyset\}}{m_G(\widetilde{F}_n)} \\[0.2cm] &= \frac{m_G((P_o + V) \cap (F_n+V))}{m_G(F_n +V)}, \end{align*} $$
where, in the last step, we have used that V is symmetric. Since
$P_o$
is U-uniformly discrete and
$V-V \subset U$
,
$$ \begin{align*} (P_o + V) \cap (F_n+V) &= \bigsqcup_{p \in P_o} ((p + V) \cap (F_{n} + V)) \\[0.2cm] &\supseteq \bigsqcup_{p \in P_o \cap F_{n}} (p+V), \end{align*} $$
and thus
$$ \begin{align} \mu_n(\mathcal{O}_V) \geq m_G(V) \cdot \frac{|P_o \cap F_n|}{m_G(F_n)} \cdot \frac{m_G(F_n)}{m_G(F_n+V)} \quad \text{for all }n. \end{align} $$
Pick a weak*-convergent subsequence
$(\mu _{n_k})$
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
Lemma 3.4. There exists an ergodic 
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
By Lemma 3.3, we can find 
such that
$$ \begin{align*} \mu_m(\mathcal{O}_V) \geq m_G(V) \cdot \bigg(d^*(P_o) - \frac{1}{m}\bigg). \end{align*} $$
We can now extract a weak*-convergent subsequence
$(\mu _{m_k})$
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 [Reference Einsiedler and Ward12, Theorem 8.20], we can decompose
$\nu $
into ergodic components: that is, there is a Borel probability measure
$\sigma $
on 
such that
In particular, we can find an ergodic 
such that
$$ \begin{align} \mu(\mathcal{O}_V) \geq \nu(\mathcal{O}_V) \geq m_G(V) \cdot d^*(P_o). \end{align} $$
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 
, 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 that 
is ergodic. Let
$g_1,\ldots ,g_r \in \Delta $
. Then, for
$\mu $
-almost every 
, there exists a subsequence
$(F_{n_j})$
such that
$$ \begin{align*} \lim_{j \rightarrow \infty} \frac{m_G((P \cap (\bigcap_{k=1}^r (P-g_k))+V_o) \cap F_{n_j})}{m_G(F_{n_j})} = m_G(V_o) \cdot \mu_{\mathcal{T}_{P_o}}\bigg(\bigcap_{k=1}^r (-g_k).\mathcal{T}_{P_o}\bigg). \end{align*} $$
Proof. Define the (possibly empty) open set 
by
$$ \begin{align*} A := \mathcal{O}_{V_o} \cap \bigg( \bigcap_{k=1}^r (-g_k).\mathcal{O}_{V_o} \bigg). \end{align*} $$
Lemma 2.3 (applied to
$B = \mathcal {T}_{P_o}$
, and using that
$V_o$
is symmetric) implies that
$$ \begin{align*} \mu(A) = m_G(V_o) \cdot \mu_{\mathcal{T}_{P_o}}\bigg(\mathcal{T}_{P_o} \cap \bigg(\bigcap_{k=1}^r (-g_k).\mathcal{T}_{P_o}\bigg)\bigg). \end{align*} $$
Since
$\mu $
is ergodic and
$(F_n)$
is Følner, the mean ergodic theorem tells us that
$$ \begin{align*} \lim_{n \rightarrow \infty} \frac{1}{m_G(F_n)} \int_{F_n} \chi_{A}(g. \, \cdot) \, dm_G(g) = \mu(A) \end{align*} $$
in the norm topology on 
. We can thus extract a subsequence
$(F_{n_j})$
along which the convergence is
$\mu $
-almost sure. Hence, for
$\mu $
-almost every 
,
$$ \begin{align*} \lim_{j \rightarrow \infty} \frac{m_G(\{ g \in F_{n_j} :P - g \in A\})}{m_G(F_{n_j})} = \mu(A). \end{align*} $$
Note that, since
$V_o$
is symmetric,
$$ \begin{align*} &\{ g \in F_{n_j} :P - g \in A\} \\[0.2cm] &\quad= \{g \in F_{n_j} :(P - g) \cap V_o \neq \emptyset, (P-g-g_k) \cap V_o \neq \emptyset, \text{ for all }k=1,\ldots,r\} \\ &\quad= (P+V_o) \cap \bigg( \bigcap_{k=1}^r (P + V_o - g_k) \bigg) \cap F_{n_j} \end{align*} $$
for all j. It thus suffices to show that
$$ \begin{align*} (P+V_o) \cap \bigg( \bigcap_{k=1}^r (P + V_o - g_k) \bigg) = \bigg(P \cap \bigg( \bigcap_{k=1}^r (P-g_k)\bigg) + V_o \bigg). \end{align*} $$
To do this, first note that the
$\supseteq $
-inclusion is trivial, so it is enough to prove that
$$ \begin{align} (P+V_o) \cap \bigg( \bigcap_{k=1}^r (P + V_o - g_k) \bigg) \subseteq \bigg(P \cap \bigg( \bigcap_{k=1}^r (P-g_k)\bigg) + V_o \bigg). \end{align} $$
We may, without loss of generality, assume that the set on the left-hand side is non-empty and pick g in this set, which allows us to write
$$ \begin{align*} g = p_o + v_o = p_1 + v_1 - g_1 = \cdots = p_r + v_r - g_r \end{align*} $$
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,
for all k, and thus
$v_k = v_o$
and
$p_o = p_k - g_k$
for all k. In particular,
$$ \begin{align*} g = p_o + v_o \in P \cap \bigg( \bigcap_{k=1}^r (P-g_k)\bigg) + V_o. \end{align*} $$
Since g is arbitrary, this proves (3.3), and the proof is complete.
Lemma 3.6. Let 
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
$$ \begin{align*} \bigg(P \cap \bigg( \bigcap_{k=1}^r (P-g_k)\bigg) + V_o\bigg) \cap Q \subseteq P_o \cap \bigg( \bigcap_{k=1}^r (P_o-g_k)\bigg) - h + V_o + V_1. \end{align*} $$
Proof. Set
$g_o = 0$
and define
$$ \begin{align*} P' := \bigcap_{k=0}^r (P-g_k) \quad \text{and} \quad P_o' := \bigcap_{k=0}^r (P_o - g_k). \end{align*} $$
Fix a pre-compact set
$Q \subset G$
. If
$(P' + V_o) \cap Q$
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
$$ \begin{align*} (P' + V_o) \cap Q = \bigsqcup_{j=1}^q ((p_j + V_o) \cap Q) \end{align*} $$
for some
$q \geq 1$
and
$p_1,\ldots ,p_q \in P'$
. 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
$j=1,\ldots ,q$
. We 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 (ii) in the characterization of sequential convergence in the Chabauty topology in §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
$$ \begin{align*} p_{j,k} = p^o_{j,k}(n) - h_n + v_{j,k}(n). \end{align*} $$
Let
$n_o := \max _{j,k} n_{j,k}$
. Then, for all
$n \geq n_o$
and
$v_o,\ldots ,v_r \in V_o$
,
$$ \begin{align*} p_j + v_j = p_{j,k} - g_k + v_j \in P_o - (g_k+h_n) + V_o + V_1 \quad \text{for all }k=0,1,\ldots,r, \end{align*} $$
and thus (since
$v_o,\ldots ,v_r \in V_o$
are arbitrary),
$$ \begin{align*} (P' + V_o) \cap Q \subset \bigcap_{k=0}^r (P_o - g_k + V_o + V_1) - h_n \quad \text{for all }n \geq n_o. \end{align*} $$
Let
$h = h_n$
for some
$n \geq n_o$
. Then, to prove the lemma, it suffices to show that
$$ \begin{align*} \bigcap_{k=0}^r (P_o - g_k + V_o + V_1) = \bigcap_{k=0}^r (P_o - g_k) + V_o + V_1. \end{align*} $$
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
$$ \begin{align*} g = p_k - g_k + v_k \quad \text{for }k=0,\ldots,r, \end{align*} $$
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 that
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
$p_o \in P_o'$
and
$g = p_o + v_o \in P_o' + V_o + V_1$
, 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 
such that
$\mu _{\mathcal {T}_{P_o}}(\mathcal {T}_{P_o}) \geq d^*(P_o)$
. We want to show that
$$ \begin{align} d^*\bigg(P_o \cap \bigg( \bigcap_{k=1}^r (P_o - g_k) \bigg)\bigg) \geq \mu_{\mathcal{T}_{P_o}}\bigg(\bigcap_{k=1}^r (-g_k).\mathcal{T}_{P_o}\bigg) \end{align} $$
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 
and a subsequence
$(\widetilde {F}_{n_k})$
such that
$$ \begin{align*} \lim_{j \rightarrow \infty} \frac{m_G((P \cap (\bigcap_{k=1}^r (P-g_k))+V_o) \cap \widetilde{F}_{n_j})}{m_G(\widetilde{F}_{n_j})} = m_G(V_o) \cdot \mu_{\mathcal{T}_{P_o}}\bigg(\bigcap_{k=1}^r (-g_k).\mathcal{T}_{P_o}\bigg). \end{align*} $$
For every j, we can apply Lemma 3.6 to find
$h_j \in G$
such that
$$ \begin{align*} &\frac{m_G((P \cap (\bigcap_{k=1}^r (P-g_k))+V_o) \cap \widetilde{F}_{n_j})}{m_G(\widetilde{F}_{n_j})} \\& \quad \leq \frac{m_G\bigg(\bigg(P_o \cap \bigg(\bigcap_{k=1}^r (P_o-g_k)\bigg) - h_j + V_o + V_1 \bigg) \cap \widetilde{F}_{n_j}\bigg)}{m_G(\widetilde{F}_{n_j})} \\& \quad = \frac{m_G((P_o \cap (\bigcap_{k=1}^r (P_o-g_k))+V_o + V_1 ) \cap (F_{n_j} + V_o + V_1 + h_j) )}{m_G(F_{n_j} + V_o + V_1 + h_j)}. \end{align*} $$
Since
$$ \begin{align*} (V_o + V_1) - (V_o + V_1) \subset V_o + V_o - V_o - V_o \subset V, \end{align*} $$
Lemma 2.9 (applied to the U-uniformly discrete set
$P_o \cap (\bigcap _{k=1}^r (P_o-g_k))$
and the bounded set
$Q = F_{n_j} + h_j$
) yields,
$$ \begin{align*} &m_G\bigg(\bigg(P_o \cap \bigg(\bigcap_{k=1}^r (P_o-g_k)\bigg)+V_o + V_1 \bigg) \cap (F_{n_j} + V_o + V_1 + h_j) \bigg) \\ &\quad\leq m_G(V_o + V_1) \cdot |P_o \cap \bigg(\bigcap_{k=1}^r (P_o-g_k)\bigg) \cap (F_{n_j} + h_j + V)|. \end{align*} $$
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,
where, in the second to last step, we 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
${m_G(V_o + V_1)}/{m_G(V_o)}$
can be made arbitrarily close to
$1$
by taking
$V_1$
small enough. We conclude that
$$ \begin{align*} \mu_{\mathcal{T}_{P_o}}\bigg(\bigcap_{k=1}^r (-g_k).\mathcal{T}_{P_o}\bigg) \leq d^*\bigg(P_o \cap \bigg(\bigcap_{k=1}^r (P_o-g_k)\bigg)\bigg), \end{align*} $$
and thus (3.4) is proved.
4. Furstenberg–Katznelson theorem for lcsc abelian groups
Let G be an 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
$$ \begin{align*} a(g_1,b(g_2,z)) = b(g_2,a(g_1,z)) \quad \text{for all }g_1,g_2 \in G\text{ and }z \in Z. \end{align*} $$
For countable abelian groups, the following theorem is due to Furstenberg and Katznelson ([Reference Furstenberg and Katznelson15, Theorem B] and [Reference Furstenberg and Katznelson16, §3]) and Austin [Reference Austin2, 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
$$ \begin{align*} S := \bigg\{ g \in G :\theta\bigg(A \cap \bigg( \bigcap_{k=1}^r a_k(g,A) \bigg)\bigg) \geq c \bigg\} \end{align*} $$
is syndetic in G. In particular, for every strong Følner sequence
$(F_n)$
in G,
$$ \begin{align*} \varliminf_{n \rightarrow \infty} \frac{1}{m_G(F_n)} \int_{F_n} \theta\bigg(A \cap \bigg( \bigcap_{k=1}^r a_k(g,A) \bigg)\bigg) \, dm_G(g)> 0. \end{align*} $$
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 lcsc, it is separable, and thus we can find a dense countable subgroup
$\Gamma < G$
. The following theorem is due to Austin [Reference Austin2, Theorem B].
Theorem 4.2. Let
$(Z,\theta )$
be a standard Borel probability measure space and let
$a_1,\ldots ,a_r$
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 $
,
$$ \begin{align*} \varliminf_{n \rightarrow \infty} \frac{1}{|F_n|} \sum_{\gamma \in F_n} \theta\bigg(A \cap \bigg( \bigcap_{k=1}^r a_k(\gamma,A) \bigg)\bigg)> 0. \end{align*} $$
Define
$\varphi : G \rightarrow [0,1]$
by
$$ \begin{align*} \varphi(g) = \theta\bigg(A \cap \bigg( \bigcap_{k=1}^r a_k(g,A) \bigg)\bigg), \quad g \in G. \end{align*} $$
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
$$ \begin{align*} S_\Gamma = \{ \gamma \in \Gamma :\varphi(\gamma) \geq c_o \} \end{align*} $$
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
$$ \begin{align} |\varphi(g_1) - \varphi(g_2)| < c_o/2 \quad \text{for all }g_1,g_2 \in G\text{ such that }g_1 - g_2 \in V. \end{align} $$
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
$$ \begin{align*} S_o + F = (S_\Gamma + F) + V = \Gamma + V = G, \end{align*} $$
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 $
),
$$ \begin{align*} \varphi(g) = \varphi(\gamma + v) - \varphi(\gamma) + \varphi(\gamma) \geq \varphi(\gamma) - c_o/2 \geq c_o/2. \end{align*} $$
Let
$c = c_o/2$
and define
$$ \begin{align*} S = \{ g \in G :\varphi(g) \geq c \}. \end{align*} $$
The argument above shows that
$S_o \subset S$
and thus S is syndetic as well, which finishes the proof of Theorem 4.1.
4.2. Proof of Lemma 4.3
By [Reference Anantharaman, Anker, Babillot, Bonami, Demange, Grellier, Havard, Jaming, Lesigne, Maheux, Otal, Schapira and Schreiber1, 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
$$ \begin{align} \int_Z |\chi_A(z) - \chi_{A}(a_k(g,z)) | \, d\theta(z) < \varepsilon/r \quad \text{for all }g \in U\text{ and }k=1,\ldots,r. \end{align} $$
Note that
$$ \begin{align*} \varphi(g) = \int_Z \chi_A(z) \, \prod_{k=1}^r \chi_A(a_k(-g,z)) \, d\theta(z) \quad \text{for all }g \in G. \end{align*} $$
A simple induction argument shows that, for all
$g_1, g_2 \in G$
,
$$ \begin{align*} \varphi(g_1) - \varphi(g_2) = \sum_{k=1}^r \int_Z \beta_k(z) \cdot (\chi_{A}(a_k(-g_1,z) - \chi_{A}(a_k(-g_2,z)) \, d\eta(z), \end{align*} $$
where
$$ \begin{align*} \beta_k(z) = \bigg(\prod_{j=k+1}^{r} \chi_{A}(a_j(-g_1,z))\bigg) \cdot \bigg(\prod_{j=1}^{k-1} \chi_{A}(a_j(-g_2,z))\bigg). \end{align*} $$
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),
$$ \begin{align*} |\varphi(g_1) - \varphi(g_2)| &\leq \sum_{k=1}^r \int_Z |\chi_A(a_k(-g_1,z)) - \chi_{A}(a_k(-g_2,z)) | \, d\theta(z) \\ &= \sum_{k=1}^r \int_Z |\chi_A(z) - \chi_{A}(a_k(g_1-g_2,z)) | \, d\theta(z) < \varepsilon. \end{align*} $$
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 
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 that 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
$\Delta = \Lambda ^q$
).
Theorem 5.1. For every 
and Borel set
$B \subset Y$
with positive
$\mu _Y$
-measure, there exists
$c> 0$
such that the set
$$ \begin{align*} S = \bigg\{ \unicode{x3bb} \in \Lambda :\mu_Y\bigg( \bigcap_{k=1}^r (-\alpha_k(\unicode{x3bb})).B\bigg) \geq c \bigg\} \end{align*} $$
is syndetic in
$\Lambda $
.
Since
$\Lambda $
is assumed to have finite local complexity in G, Lemma 2.11 tells us that 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 
and let
$B \subset Y$
be a Borel set with positive
$\mu _Y$
-measure. Then, for every strong Følner sequence
$(F_n)$
in G,
$$ \begin{align*} \varliminf_{n \rightarrow \infty} \frac{1}{m_G(F_n)} \sum_{\unicode{x3bb} \in \Lambda \cap F_n} \mu_Y\bigg( \bigcap_{k=1}^r (-\alpha_k(\unicode{x3bb})).B\bigg)> 0. \end{align*} $$
5.1. Proof of Theorem 5.2
In what follows, we fix 
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 
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 
-preserving Borel G-actions
$a_1,\ldots ,a_r$
on 
and a Borel set 
with positive 
-measure such that
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 
-preserving Borel G-actions
$a_1,\ldots ,a_r$
on 
and a Borel set 
with positive 
-measure such that
The positivity of the last expression now follows from Theorem 4.1, applied to the Borel space 
.
5.2. Proof of Proposition 5.3
Before we begin the proof, we briefly recall the main features of 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 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 . Note that
$\nu (\mathcal {O}_{W})> 0$
for all identity neighbourhoods W in G, where . -
(iii)
$\Lambda $
and
$\Delta $
are uniformly discrete sets in G such that for some identity neighbourhood
$$ \begin{align*} \Lambda_o - \Lambda_o \subset \Lambda \quad \text{and} \quad (\Xi_Y - \Delta) \cap V = \{0\} \end{align*} $$
$V \subset U$
.
-
(iv) Fix
such that
$\alpha _k(\Lambda ) \subset \Delta $
for all
$k=1,\ldots ,r$
. -
(v) Fix a strong Følner sequence
$(F_n)$
in G.
and a Borel set 
. Note that 
.
such that 
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
$g_k = \alpha _k(\unicode{x3bb} )$
for
$\unicode{x3bb} \in \Lambda ) $
tells us that
$$ \begin{align*} \mu_Y\bigg( B \cap \bigcap_{k=1}^r (-\alpha_k(\unicode{x3bb})).B\bigg) = \frac{1}{m_G(V_o)} \cdot \mu\bigg(B_{V_o} \cap \bigg( \bigcap_{k=1}^r (-\alpha_k(\unicode{x3bb})).B_{V_o}\bigg)\!\bigg) \quad \text{for all }\unicode{x3bb} \in \Lambda. \end{align*} $$
In particular,
$$ \begin{align} \sum_{\unicode{x3bb} \in \Lambda \cap F_n} \mu_Y\bigg( B \cap \bigcap_{k=1}^r (-\alpha_k(\unicode{x3bb})).B\bigg) = \frac{1}{m_G(V_o)} \cdot \sum_{\unicode{x3bb} \in \Lambda \cap F_n} \mu\bigg(B_{V_o} \cap \bigg( \bigcap_{k=1}^r (-\alpha_k(\unicode{x3bb})).B_{V_o}\bigg)\!\bigg) \end{align} $$
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,
$$ \begin{align*} B_{V_1} \subset B_{V_o} \quad \text{and} \quad \alpha_k(h).B_{V_1} \subset B_{V_o} \quad \text{for all }h \in V_1\text{ and }k, \end{align*} $$
and thus
$$ \begin{align} \mu\bigg(B_{V_o} \cap \bigg( \bigcap_{k=1}^r (-\alpha_k(g)).B_{V_o}\bigg)\!\bigg) \geq \mu\bigg(B_{V_1} \cap \bigg( \bigcap_{k=1}^r (-\alpha_k(g-h)).B_{V_1}\bigg)\bigg) \end{align} $$
for all
$g \in G$
and
$h \in V_1$
. We now define
$$ \begin{align*} F_n' := \bigcap_{h \in V_1} (F_n - h) \quad\text{and note that }F_n' + V_1 \subset F_n. \end{align*} $$
Hence,
$$ \begin{align*} \chi_{F_n}(g) \geq \frac{1}{m_G(V_1)} \int_G \chi_{V_1}(h) \chi_{F_n'}(g-h) \, dm_G(h) \quad \text{for all }g \in G, \end{align*} $$
and it follows from (5.2) that,
$$ \begin{align*} &\int_G \chi_{V_1}(h) \chi_{F_n'}(g-h) \, dm_G(h) \cdot \mu\bigg(B_{V_o} \cap \bigg( \bigcap_{k=1}^r (-\alpha_k(g)).B_{V_o}\bigg)\!\bigg) \\&\quad\geq \int_G \chi_{V_1}(h) \chi_{F_n'}(g-h) \, \mu\bigg(B_{V_1} \cap \bigg( \bigcap_{k=1}^r (-\alpha_k(g-h)).B_{V_1}\bigg)\!\bigg) \, dm_G(h) \!\quad \text{for all }g \in G. \end{align*} $$
We now put of all this together. It follows from the last inequality above that
$$ \begin{align*} &\sum_{\unicode{x3bb} \in \Lambda \cap F_n} \mu\bigg(B_{V_o} \cap \bigg( \bigcap_{k=1}^r (-\alpha_k(\unicode{x3bb})).B_{V_o}\bigg)\!\bigg) = \sum_{\unicode{x3bb} \in \Lambda} \chi_{F_n}(\unicode{x3bb}) \cdot \mu\bigg(B_{V_o} \cap \bigg( \bigcap_{k=1}^r (-\alpha_k(\unicode{x3bb})).B_{V_o}\bigg)\!\bigg) \\&\quad\geq \frac{1}{m_G(V_1)} \cdot \sum_{\unicode{x3bb} \in \Lambda} \int_G \chi_{V_1}(h) \chi_{F_n'}(\unicode{x3bb}-h) \, dm_G(h) \cdot \mu\bigg(B_{V_o} \cap \bigg( \bigcap_{k=1}^r (-\alpha_k(\unicode{x3bb})).B_{V_o}\bigg)\!\bigg) \\&\quad\geq \frac{1}{m_G(V_1)} \cdot \sum_{\unicode{x3bb} \in \Lambda} \int_G \chi_{V_1}(h) \chi_{F_n'}(\unicode{x3bb}-h) \mu\bigg(B_{V_1} \cap \bigg( \bigcap_{k=1}^r (-\alpha_k(\unicode{x3bb}-h)).B_{V_1}\bigg)\!\bigg)\, dm_G(h) \\&\quad= \frac{1}{m_G(V_1)} \cdot \sum_{\unicode{x3bb} \in \Lambda} \int_G \chi_{V_1}(\unicode{x3bb}-h) \chi_{F_n'}(h) \mu\bigg(B_{V_1} \cap \bigg( \bigcap_{k=1}^r (-\alpha_k(h)).B_{V_1}\bigg)\!\bigg)\, dm_G(h) \\&\quad= \frac{1}{m_G(V_1)} \cdot \int_{F_n'} \mu\bigg(B_{V_1} \cap \bigg( \bigcap_{k=1}^r (-\alpha_k(h)).B_{V_1}\bigg)\!\bigg) \cdot \bigg( \sum_{\unicode{x3bb} \in \Lambda} \chi_{V_1}(\unicode{x3bb}-h) \bigg) \, dm_G(h), \end{align*} $$
where, in the last step, we used Fubini’s theorem to interchange summation and integration. We now fix an identity neighbourhood
$V_2$
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
$$ \begin{align*} \sum_{\unicode{x3bb} \in \Lambda} \chi_{V_1}(\unicode{x3bb}-h) = \sum_{\unicode{x3bb} \in \Lambda} \chi_{V_1}(h+\unicode{x3bb}) \geq \nu(\mathcal{O}_{V_2} \cap h.\mathcal{O}_{V_2}) \quad \text{for all }h \in G, \end{align*} $$
where
$\nu $
is defined in (ii) above. Note that
$\nu (\mathcal {O}_{V_2})> 0$
. We have now shown that
$$ \begin{align} &\sum_{\unicode{x3bb} \in \Lambda \cap F_n} \mu\bigg(B_{V_o} \cap \bigg( \bigcap_{k=1}^r (-\alpha_k(\unicode{x3bb})).B_{V_o}\bigg)\!\bigg) \nonumber \\&\quad\geq \frac{1}{m_G(V_1)} \cdot \int_{F_n'} \mu\bigg(B_{V_1} \cap \bigg( \bigcap_{k=1}^r (-\alpha_k(h)).B_{V_1}\bigg)\!\bigg) \cdot \nu(\mathcal{O}_{V_2} \cap h.\mathcal{O}_{V_2}) \, dm_G(h). \end{align} $$
We now rewrite the integrand. First, let 
and
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
$$ \begin{align*} a_1(g,(x,\Lambda_o')) = (-\alpha_1(g).x,g.\Lambda_o') \end{align*} $$
and
$$ \begin{align*} a_k(g,(x,\Lambda_o')) = (-\alpha_k(g).x,\Lambda_o'), \quad k = 2,\ldots,r, \end{align*} $$
for
$(x,\Lambda _o') \in Z$
. Clearly, these actions preserve
$\theta $
, they all commute with each other and
for all
$g \in G$
. We now conclude from (5.1) and (5.3) that
for all n. Since both
$\mu $
and
$\nu $
are probability measures,
for all n. Hence, if we set
$$ \begin{align*} \beta_o = \frac{1}{m_G(V_o)m_G(V_1)} \quad \text{and} \quad \delta_n = \frac{\beta_o \cdot m_G(F_n \setminus F_n')}{m_G(F_n)}, \end{align*} $$
then
for all n. Since
$(F_n)$
is a strong Følner sequence,
$(\delta _n)$
is a null sequence, and the proof is complete.
6. Proof of Theorem 1.5
Let
$\Lambda , \Lambda _o, P_o,q$
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,
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 
such that
$\mu _{\mathcal {T}_{P_o}}(\mathcal {T}_{P_o})> 0$
and
$$ \begin{align} d^*\bigg( P_o \cap \bigg( \bigcap_{k=1}^r (P_o - \alpha_k(\unicode{x3bb})) \bigg)\! \bigg) \geq \mu_{\mathcal{T}_{P_o}}\bigg(\bigcap_{k=1}^r (-\alpha_k(\unicode{x3bb})).\mathcal{T}_{P_o}\bigg) \end{align} $$
for all
$\unicode{x3bb} \in \Lambda $
. Hence, by Theorem 1.9, applied to 
and
$B = \mathcal {T}_{P_o}$
, there exists
$c> 0$
such that
$$ \begin{align*} S_o = \bigg\{ \unicode{x3bb} \in \Lambda :\mu_{\mathcal{T}_{P_o}}\bigg(\bigcap_{k=1}^r (-\alpha_k(\unicode{x3bb})).\mathcal{T}_{P_o}\bigg) \geq c\bigg\} \end{align*} $$
is syndetic in
$\Lambda $
. By (6.1), we have
$S_o \subset S$
, where
$$ \begin{align*} S= \bigg\{ \unicode{x3bb} \in \Lambda :d^*\bigg( P_o \cap \bigg( \bigcap_{k=1}^r (P_o - \alpha_k(\unicode{x3bb})) \bigg) \bigg) \geq c\bigg\}, \end{align*} $$
and thus S is syndetic in
$\Lambda $
as well.
Acknowledgements
This work was completed during M.B’s research visit to 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 grants 11253320 and 2023-03803 from the Swedish Research Council. A.F. was supported by the ARC via grants DP210100162 and DP240100472. Finally, we are grateful to the referee who provided a very careful report.
A. Appendix. 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 variation of Theorem 1.2 using the deep ergodic 
-theory developed by Furstenberg and Katznelson in [Reference Furstenberg and Katznelson16] and further extended by Bergelson and McCutcheon in [Reference Bergelson and McCutcheon7].
Theorem A.1. Let
$\Lambda \subset \mathbb {R}^d$
be an approximate lattice and suppose that
$$ \begin{align*} \Delta = \Delta(\mathbb{R},\mathbb{R}^m,\Gamma,W) \subset \mathbb{R} \end{align*} $$
is a cut-and-project set for some lattice
$\Gamma < \mathbb {R} \times \mathbb {R}^m$
that 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
$$ \begin{align*} p_o + j\delta \cdot F \subset P_o. \end{align*} $$
Remark A.2. Theorem 1.2 corresponds to
$$ \begin{align*} d = m = 1 \quad \text{and} \quad \Lambda = \Delta = \Delta(\mathbb{R},\mathbb{Z}[\sqrt{D}],[-1,1]). \end{align*} $$
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 -theory
-theory Let H be an abelian semigroup and let n be 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
$$ \begin{align*} \varphi(\alpha \cup \beta) = \varphi(\alpha) + \varphi(\beta) \quad \text{for all }\alpha, \beta \in 2^{[n]}\text{ such that }\alpha \cap \beta = \emptyset \end{align*} $$
is called an 
-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,
$$ \begin{align*} \varphi : 2^{[n]} \rightarrow H, \quad \varphi(\alpha) = \sum_{k \in \alpha} h_{k} \end{align*} $$
is an 
-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
$$ \begin{align*} \varphi(\alpha) = |\alpha| \cdot h \quad \text{for all }\alpha \in 2^{[n]}. \end{align*} $$
We denote this special 
-system by
$\varphi _{n,h}$
.
We will deduce Theorem A.1 from the following result [Reference Furstenberg and Katznelson16, Theorem 9.5], which is only stated for subsets in
$\mathbb {R}^d$
in [Reference Furstenberg and Katznelson16]. 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
$$ \begin{align*} \varepsilon := \varlimsup_{n \rightarrow \infty} \frac{m_{\mathbb{R}^d}(A \cap [-n,n]^d)}{n^d}> 0. \end{align*} $$
Then, for every finite set
$F \subset \mathbb {R}^d$
, there exists an integer
$n = n(\varepsilon ,|F|)$
with the property that, for every 
-system
$\varphi $
with values in
$\mathbb {R}^{+}$
, there exist
$\alpha \in 2^{[n]}$
and
$x \in \mathbb {R}^d$
such that
$$ \begin{align*} x + \varphi(\alpha) \cdot F \subset A. \end{align*} $$
Proof of Theorem A.1
Let
$\Lambda , \Delta , \Gamma $
and W be as in Theorem A.1. Let
$F_o = \{\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _r\} \subset \Lambda ^\infty $
be a finite set and define
$F = \{0,\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _r\}$
. 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 that
$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
$$ \begin{align*} \varepsilon := \varlimsup_{n \rightarrow \infty} \frac{m_{\mathbb{R}^d}(A \cap [-n,n]^d)}{n^d}> 0. \end{align*} $$
By Theorem A.4, there exists an integer n, depending only on
$\varepsilon $
and r and with the property that, for every 
-system
$\varphi $
with values in
$\mathbb {R}^{+}$
, there exist
$\alpha \in 2^{[n]}$
and
$x \in \mathbb {R}^d$
such that
$$ \begin{align} x + \varphi(\alpha) \cdot F \subset A = P_o + V. \end{align} $$
Define
$\Delta _n = \Delta (\mathbb {R},\mathbb {R}^m,\Gamma ,\frac {1}{n}W^o)$
, 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,
$$ \begin{align*} \Delta_n^n \subset \Delta\bigg(\mathbb{R},\mathbb{R}^m,\Gamma,\frac{1}{n}W^o + \cdots + \frac{1}{n}W^o\bigg) \subset \Delta(\mathbb{R}^m,\Gamma,W^o) \subset \Delta. \end{align*} $$
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 
-system such that
$$ \begin{align*} \varphi_{n,\delta}(\alpha) = |\alpha| \cdot \delta \in \Delta \quad \text{for all }\alpha \in 2^{[n]}. \end{align*} $$
Fix
$\delta \in \Delta _n$
. By (A.1), there exist
$j=1,\ldots ,n$
and
$x \in \mathbb {R}^d$
such that
$$ \begin{align*} x + j \delta \cdot \{0,\unicode{x3bb}_1,\ldots,\unicode{x3bb}_r\} \subset P_o + V. \end{align*} $$
Set
$\unicode{x3bb} _o = 0$
. 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
$$ \begin{align*} x + j\delta \cdot \unicode{x3bb}_k = p_k + v_k \quad \text{for all }k=0,1,\ldots,r. \end{align*} $$
Note that, since
$j\delta \in \Delta $
for all
$j = 1,\ldots ,n$
and
$j\delta \cdot \Lambda \subset \Lambda $
,
$$ \begin{align*} j\delta \cdot \unicode{x3bb}_k \in j\delta \cdot \Lambda^q \subset \Lambda^q \quad \text{for all }k=0,1,\ldots,r. \end{align*} $$
By taking differences,
$$ \begin{align*} (p_k + v_k) - (p_o + v_o) = j\delta \cdot \unicode{x3bb}_k \in \Lambda^q \quad \text{for all }k, \end{align*} $$
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
$v_o = v_1 = \cdots = v_r$
, so
$$ \begin{align*} p_k = p_o + j\delta \cdot \unicode{x3bb}_k \quad \text{for all }k=1,\ldots,r. \end{align*} $$
Hence,
$p_o + j\delta \cdot \{\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _r\} \subset P_o$
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 and thus provides some (potentially weaker) versions 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 
-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 
-systems inside an approximate lattice is needed.
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