An inverse of Furstenberg’s correspondence principle and applications to van der Corput sets

Saúl Rodríguez Martín The Ohio State University
[email protected]

Abstract

In this article we give characterizations of the notions of van der Corput (vdC) set, nice vdC set and set of nice recurrence (defined below) in countable amenable groups. This allows us to prove that nice vdC sets are sets of nice recurrence and that vdC sets are independent of the Følner sequence used to define them, answering questions from [BL] in the context of countable amenable groups. We also give a spectral characterization of vdC sets in abelian groups. The methods developed in this paper allow us to establish a converse to the Furstenberg correspondence principle. In addition, we introduce vdC sets in general non amenable groups and establish some basic properties of them, such as partition regularity.

Several results in this paper, including the converse to Furstenberg’s correspondence principle, have also been proved independently by Robin Tucker-Drob and Sohail Farhangi in the article [FT], which is being uploaded to arXiv simultaneously to this one.

1 Introduction

In this paper we obtain new results about van der Corput (vdC) sets. Among other things, we will provide answers to some questions posed in [BL] and [BF]. The notion of van der Corput set was introduced in [KM] in connection with the study of uniform distribution of sequences in $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ . Recall that a sequence $(x_{n})_{n\in\mathbb{N}}$ in $\mathbb{T}$ is uniformly distributed mod $1$ (u.d. mod $1$ ) if for any continuous function $f:\mathbb{T}\to\mathbb{C}$ we have

\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(x_{n})=\int_{\mathbb{T}}fdm,

where $m$ is Lebesgue measure.

Definition 1.1 ([KM, Page 1]).

A set $H\subseteq\mathbb{N}:=\{1,2,\dots\}$ is a van der Corput set (vdC set) if, for any sequence $(x_{n})_{n\in\mathbb{N}}$ in $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ such that $(x_{n+h}-x_{n})_{n\in\mathbb{N}}$ is u.d. mod $1$ for all $h\in H$ , the sequence $(x_{n})_{n}$ is itself u.d. mod $1$ .

In [Ru] the following characterization of vdC sets in terms of Cesaro averages of bounded sequences of complex numbers was established:

Theorem 1.2 (cf. [Ru, Theorem 1]).

A set $H\subseteq\mathbb{N}$ is a van der Corput set iff for any sequence $(z_{n})_{n\in\mathbb{N}}$ of complex numbers with $|z_{n}|\leq 1$ for all $n$ such that

\forall h\in H,\quad\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}z_{h+n}\overline{z_{n}}=0,

we have

\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}z_{n}=0.

Kamae and Mendès France proved in [KM, Theorem 2] that any vdC set $H\subseteq\mathbb{N}$ is a set of recurrence, meaning that for any m.p.s.¹¹1From now, by ‘m.p.s.’ we mean probability measure preserving system. $(X,\mathcal{B},\mu,T)$ and any $A\in\mathcal{B}$ with $\mu(A)>0$ we have $\mu(A\cap T^{h}A)>0$ for some $h\in H$ . It was observed in [KM] that many classical examples of sets of recurrence, such as sets of differences, images of polynomial functions and shifted primes, are vdC sets. This led to the question of whether the notions of set of recurrence and vdC set coincide. It was Bourgain who answered this question in the negative by giving in [Bo] a rather complicated example of a set of recurrence which is not vdC (also see [Mou] for a more refined example).

In [BL] the notion of vdC set was extended to subsets of $\mathbb{Z}^{d}$ ( $d\geq 2$ ), and a natural notion of ‘ $F$ -vdC set’ was introduced for any Følner sequence²²2From now, by ‘Følner sequence’ we mean left-Følner sequence. $F=(F_{N})_{N}$ in $\mathbb{Z}$ . The definition of $F$ -vdC given in [BL, Section 4.2] naturally extends to Følner sequences in any countable amenable group.

Definition 1.3.

Let $F=(F_{N})_{N\in\mathbb{N}}$ be a Følner sequence in a countable amenable group $G$ . We say that a sequence $(x_{g})_{g\in G}$ in $\mathbb{T}$ is $F$ -u.d. mod $1$ if for any continuous function $f:\mathbb{T}\to\mathbb{\mathbb{C}}$ we have

\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}f(x_{g})=\int_{\mathbb{T}}fdm.

Definition 1.4 (cf. [BL, Page 44]).

Let $F=(F_{N})_{N\in\mathbb{N}}$ be a Følner sequence in a countable amenable group $G$ . We say a subset $H$ of $G$ is $F$ -vdC if any sequence $(x_{g})_{g\in G}$ in $\mathbb{T}$ such that $(x_{hg}-x_{g})_{g\in G}$ is $F$ -u.d. mod $1$ for all $h\in H$ , is itself $F$ -u.d. mod $1$ .

In particular, any set $H$ containing $1_{G}$ is $F$ -vdC. Note that a subset $H$ of $\mathbb{N}$ is vdC in the sense of Definition 1.1 iff it is $(F_{N})_{N}$ -vdC in $\mathbb{Z}$ , where $F_{N}=\{1,\dots,N\}$ . It was asked in [BL, Section 4.2] whether, for any Følner sequence $F$ in $\mathbb{Z}$ , a subset $H\subseteq\mathbb{N}$ is $F$ -vdC if and only if it is vdC. We show that the answer is yes by giving the following characterization of $F$ -vdC sets in a countable amenable group $G$ , which does not depend on the Følner sequence:

Theorem 1.5 (See Theorem 3.1).

Let $G$ be a countably infinite amenable group with a Følner sequence $F=(F_{N})_{N}$ . A set $H\subseteq G$ is $F$ -vdC in $G$ if and only if for any m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and for any function $f\in L^{\infty}(\mu)$ ,

\int_{X}f(T_{h}(x))\cdot\overline{f(x)}d\mu(x)=0\text{ for all }h\in H\textup{ implies }\int_{X}fd\mu=0.

The condition given in Theorem 1.5 makes sense for any countable group³³3The definition makes sense for any (discrete) group, even if it is not countable. We will not be interested in uncountable discrete groups in this article, but many of the properties of vdC sets generalize to this setting., not only for amenable ones. We introduce it below as a definition of vdC subset of a countable group. This makes the notation ‘vdC set’ independent of the Følner sequence and allows us to prove properties of vdC sets in a more general setting.

Definition 1.6.

Let $G$ be a (discrete) countable group. We will say that $H\subseteq G$ is vdC in $G$ if, for every m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and every $f\in L^{\infty}(\mu)$ ,

\int_{X}f(T_{h}(x))\cdot\overline{f(x)}d\mu(x)=0\text{ for all }h\in H\text{ implies }\int_{X}fd\mu=0.

From Definition 1.6 it easily follows that any vdC set in a countable group $G$ is of recurrence, in the sense that for every m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and for every $B\in\mathcal{B}$ such that $\mu(B)>0$ , we have $\mu(B\cap T_{h}B)>0$ for some $h\in H$ . Indeed, if instead of allowing any $f\in L^{\infty}(\mu)$ we restrict our attention to characteristic functions (or positive functions), Definition 1.6 becomes the definition of set of recurrence. This supports the idea implied by [BL, Section 3.2] that sets of recurrence are a ‘positive version’ of vdC sets.

In Theorem 1.7 below we give a spectral characterization of vdC subsets of any countable abelian group $G$ . This generalizes similar characterizations which were given in [KM, BL, Ru] for vdC subsets of $\mathbb{Z}$ or $\mathbb{Z}^{d}$ for $d\geq 2$ (see e.g. [BL, Theorem 1.8.]). These characterizations are useful both for proving properties of vdC sets and for finding (non-) examples of them. For example, Bourgain used a version of Theorem 1.7 for $G=\mathbb{Z}$ to construct his set of recurrence which is not vdC.

For a countable abelian group $G$ we denote by $0$ its identity element and by $\widehat{G}$ its Pontryagin dual (which is compact and metrizable, see [RuFA, Theorems 1.2.5, 2.2.6]). Also, for any Borel probability measure $\mu$ in $\widehat{G}$ we denote the Fourier coefficients of $\mu$ by $\widehat{\mu}(h)=\int_{\widehat{G}}\gamma(h)d\mu(\gamma)$ (for $h\in G$ ).

Theorem 1.7 below is proved in this article; another very elegant proof of it can be found in [FT].

Theorem 1.7 (cf. [BL, Theorem 1.8]).

Let $G$ be a countable abelian group. A set $H\subseteq G$ is vdC in $G$ iff any Borel probability measure $\mu$ in $\widehat{G}$ with $\widehat{\mu}(h)=0\;\forall h\in H$ satisfies $\mu(\{0\})=0$ .

Taking definition Definition 1.6 as the starting point, we will prove in Section 5 several properties of vdC sets. Most of them were proved in [Ru] or [BL] for vdC sets in $\mathbb{Z}$ and $\mathbb{Z}^{d}$ respectively. [Ru, BL] used the spectral criterion to prove many of these properties (which we avoid), so their proofs would only work for countable abelian groups. We now state some of the properties we prove in Section 5.

We show that the family of vdC sets in a countable group satisfies the Ramsey property⁴⁴4Also known as partition regularity., that is, if $H\subseteq G$ is vdC and $H=H_{1}\cup H_{2}$ , then either $H_{1}$ or $H_{2}$ is vdC. We also prove that, if a vdC set in a countable group does not contain the identity, then it contains two disjoint vdC sets. Proving that requires a finitistic criterion for the notion of vdC sets; letting $\mathbb{D}:=\{z\in\mathbb{C};|z|\leq 1\}$ , we have

Proposition 1.8.

Let $G$ be a countably infinite group, let $H\subseteq G$ . Then $H$ is vdC in $G$ if and only if for any $\varepsilon>0$ there exists $\delta>0$ and a finite subset $H_{0}$ of $H$ such that, for any m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and any measurable $f:X\to\mathbb{D}$ we have

\left|\int_{X}f(T_{h}x)\overline{f(x)}d\mu(x)\right|<\delta\;\forall h\in H_{0}\text{ implies }\left|\int_{X}fd\mu\right|<\varepsilon.

We give a proof of 1.8 at the end of Section 5. We also study the behaviour of vdC sets in subgroups and under group homomorphisms and give some easy examples and non-examples of vdC sets.

We now discuss another question in [BL]. The following definition was introduced in [Be2] for subsets of $\mathbb{Z}$ .

Definition 1.9 (cf. [Be2, Definition 2.2]).

Let $G$ be a group. A subset $H$ of $G$ is a set of nice recurrence if for any m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and for any $B\in\mathcal{B}$ ,

\mu(B)^{2}\leq\limsup_{h\in H}\mu(B\cap T_{h}B).

In [BL, Definition 10], a notion of ‘nice vdC set’ was introduced:

Definition 1.10 (cf. [BL, Definition 10]).

Let $G$ be a countable amenable group with a Følner sequence $(F_{N})_{N}$ . A subset $H$ of $G$ is nice $F$ -vdC if for any sequence $(z_{g})_{g\in G}$ in $\mathbb{D}$ ,

\limsup_{N}\left|\frac{1}{|F_{N}|}\sum_{g\in F_{N}}z_{g}\right|^{2}\leq\limsup_{h\in H}\limsup_{N}\left|\frac{1}{|F_{N}|}\sum_{g\in F_{N}}z_{hg}\overline{z_{g}}\right|

Note that nice vdC sets are vdC by Theorem 1.2. Also note that Definition 1.10 and Definition 1.9 are expressed in different settings: Definition 1.10 is about Cesaro averages of sequences of complex numbers, while Definition 1.9 is about integrals. We will translate each of these definitions to the setting of the other one in Propositions 1.12 and 1.13 below. First, we need to recall some notions of density.

Definition 1.11 ([BF, Definitions 2.1, 2.2]).

Let $E$ be a subset of a countable amenable group $G$ . The upper density of $E$ along a Følner sequence $F=(F_{N})_{N}$ is defined by

\overline{d_{F}}(E):=\limsup_{N\to\infty}\frac{|E\cap F_{N}|}{|F_{N}|}.

If the $\limsup$ is actually a limit, we call it $d_{F}(E)$ , the density of $E$ along $(F_{N})_{N}$ . The upper Banach density of $E$ , $d^{*}(E)$ , is defined by

d^{*}(E)=\sup\left\{\overline{d_{F}}(E);F\text{ F{\o}lner sequence in }G\right\}.

Proposition 1.12.

Let $G$ be a countable amenable group with a Følner sequence $F=(F_{N})_{N}$ . Then a subset $H\subseteq G$ is a set of nice recurrence iff for any $E\subseteq G$ we have

\overline{d_{F}}(E)^{2}\leq\limsup_{h\in H}\overline{d_{F}}(E\cap hE).

Proposition 1.13.

Let $G$ be a countable amenable group with a Følner sequence $(F_{N})_{N}$ . Then a subset $H$ of $G$ is nice $F$ -vdC iff for any m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and any $f\in L^{\infty}(X,\mu)$ we have

\left|\int_{X}fd\mu\right|^{2}\leq\limsup_{h\in H}\left|\int_{X}f(T_{h}x)\overline{f(x)}d\mu(x)\right|.

In [BL, Question 8] it is asked whether there is any implication between the notions ‘nice vdC set’ and ‘set of nice recurrence’. It was also proved in [BL] that if $H$ is a nice vdC set, then $H$ satisfies the following, weak version of nice recurrence: for any m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and for any $B\in\mathcal{B}$ , we have $\mu(B)^{4}\leq\limsup_{h\in H}\mu(B\cap T_{h}B)$ . For subsets of $\mathbb{N}$ , it was proved by S. Farhangi that nice vdC implies nice recurrence in [Fa, Theorem 5.2.4]; both 1.13 and 1.12 generalize this fact to all amenable groups.

Note also that 1.13 also implies that the notion of nice $F$ -vdC set is independent of the Følner sequence.

As we mentioned above, both 1.12 and 1.13 are just translations of Definitions 1.9 and 1.10 between the language of Cesaro averages and the language of integrals. Both of them, as well as Theorem 1.5, are easy corollaries of the following theorem, which is the machinery behind most results in this article:

Theorem 1.14.

Let $G$ be a countably infinite amenable group with a Følner sequence $(F_{N})_{N}$ and let $D\subseteq\mathbb{C}$ be compact. Then for any m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and any measurable function $f:X\to D$ there is a sequence $(z_{g})_{g\in G}$ of complex numbers in $D$ such that, for all $j\in\mathbb{N}$ , $h_{1},\dots,h_{j}\in G$ and all continuous functions $p:D^{j}\to\mathbb{C}$ ,

\lim_{N}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}p(z_{h_{1}g},\dots,z_{h_{j}g})=\int_{X}p(f(T_{h_{1}}x),\dots,f(T_{h_{j}}x))d\mu.

(1)

Conversely, given a sequence $(z_{g})_{g\in G}$ in $D$ there is a m.p.s. $(X,\mathcal{B},\mu,T)$ and a measurable function $f:X\to D$ such that, for all $j\in\mathbb{N},h_{1},\dots,h_{j}\in G$ and $p:D^{j}\to\mathbb{C}$ continuous, Equation 1 holds if the limit in the left hand side exists.

Note that Theorem 1.14 is about bounded sequences in $\mathbb{C}$ ; in some cases one may adapt it to sequences $(z_{g})_{g\in G}$ such that $\limsup_{N}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}\|z_{g}\|^{2}<\infty$ , see [FT, Theorem 3.3].

We remark that in our study of vdC sets we only need cases $j=1,2$ of Theorem 1.14. However, the proof is essentially the same for all $j$ , and allowing $j$ to take arbitrary values allows us to obtain applications to the theory of multiple recurrence. In particular, we can use Theorem 1.14 to prove Theorem 1.16 below, a converse to the Furstenberg correspondence principle which we can use to answer a question from [BF]. We state the Furstenberg correspondence principle and Theorem 1.16 below.

Theorem 1.15 (Furstenberg correspondence principle, cf. [Be5, Theorem 1.8]).

Let $G$ be a countable amenable group, let $A\subseteq G$ . Then there exists a m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and $B\in\mathcal{B}$ with $\mu(B)=d^{*}(A)$ such that, for all $k\in\mathbb{N}$ and $h_{1},\dots,h_{k}\in G$ one has:

d^{*}(h_{1}A\cap\dots\cap h_{k}A)\geq\mu(T_{h_{1}}(B)\cap\dots\cap T_{h_{k}}(B))

Theorem 1.16.

Let $G$ be a countably infinite amenable group with a Følner sequence $(F_{N})_{N}$ . For every m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and every $B\in\mathcal{B}$ there exists a subset $A\subseteq G$ such that, for all $k\in\mathbb{N}$ and $h_{1},\dots,h_{k}\in G$ we have

d_{F}\left(h_{1}A\cap\dots\cap h_{k}A\right)=\mu\left(T_{h_{1}}B\cap\dots\cap T_{h_{k}}B\right).

(2)

Reciprocally, for any subset $A\subseteq G$ there is a m.p.s. $(X,\mathcal{B},\mu,T)$ and $B\in\mathcal{B}$ satisfying Equation 2 for all $k,h_{1},\dots,h_{k}$ as above whenever $d_{F}\left(h_{1}A\cap\dots\cap h_{k}A\right)$ exists.

Theorem 1.16 can be viewed as a strengthening of [BF2, Theorem 5.1]: in our case, we do not need the m.p.s. to be ergodic, and we do not need to take a subsequence of $(F_{N})_{N}$ . In [Av, Theorem 3.1] they also give a (weaker) inverse of the Furstenberg correspondence in the case $G=\mathbb{Z}$ .

As an application of Theorem 1.16 we prove Theorem 1.17 below, which answers a question asked in [BF] after Remark 3.6. The question is whether for all countable amenable groups $G$ and all Følner sequences $F$ in $G$ there is a set $E\subseteq G$ such that $\overline{d}_{F}(E)>0$ but for all finite $A\subseteq G$ , $\overline{d}_{F}\left(\cup_{g\in A}g^{-1}E\right)<\frac{3}{4}$ .

Theorem 1.17.

Let $G$ be a countably infinite amenable group with a Følner sequence $F=(F_{N})_{N}$ . Then there is $E\subseteq G$ such that, for all finite $\varnothing\neq A\subseteq G$ ,

d_{F}(E)=d_{F}\left(\cup_{g\in A}gE\right)=\frac{1}{2}.

The structure of the article is as follows. In Section 2 we prove Theorem 1.14, which is the machinery behind Theorem 1.5. At the end of the section we prove several corollaries of Theorem 1.14: Theorem 2.14 (a generalization of Theorem 1.16), Propositions 1.12 and 1.13, and Theorem 1.17. In Section 3 we give a characterization theorem for $F$ -vdC sets in countable amenable groups, Theorem 3.1, which expands upon Theorem 1.5. We also prove in 3.5 that there is a close relationship between Cesaro averages of sequences taking values in a compact set $D\subseteq\mathbb{C}$ , and Cesaro averages of sequences in the convex hull of $D$ . In Section 4 we prove the spectral characterization of vdC sets in countable abelian groups, Theorem 1.7. In Section 5 we prove the properties of vdC sets mentioned above and 1.8.

Acknowledgements.

Special thanks to Vitaly Bergelson for his guidance while writing this article, and for some interesting discussions and suggestions. The author gratefully acknowledges support from the grants BSF 2020124 and NSF CCF AF 2310412.

Several months before uploading this article, it came to our attention that Sohail Farhangi and Robin Tucker-Drob had been independently studying the topic of vdC sets. Their paper [FT] contains a long list of characterizations of vdC sets, including the ones from Theorem 1.5 and Theorem 1.7. Thanks to Sohail Farhangi for his input and suggestions, and especially for bringing [DHZ, Theorem 5.2] to my attention; this result allowed the author to simplify the proof of Theorem 1.7 and to state Theorem 1.14 in full generality (for all amenable groups instead of only monotileable ones).

Some of the results contained in this article (in particular Theorem 1.14 and Theorem 1.16) and in [FT] were announced by Sohail Farhangi in the Conference on Ergodic group actions and unitary representations at IMPAN, June 2024.

After this article was completed, the author was made aware of the existence of the article [FS], which also touches upon an inverse Furstenberg correspondence principle. In this article, among other things, Fish and Skinner obtain the particular case of Theorem 1.16 which corresponds to the Følner sequence $F_{N}=\{1,\dots,N\}$ in $\mathbb{Z}$ (see [FS, Theorem 1.4]).

2 Correspondence between Cesaro and integral averages

The main objective of this section is proving Theorem 1.14. It will be deduced from 2.1, a more technical version of Theorem 1.14 which also includes a finitistic criterion for the existence of sequences with given Cesaro averages.

At the end of the section we prove Theorem 2.14, the converse of a general converse of the Furstenberg correspondence principle. We also give proofs of Propositions 1.12 and 1.13, answering a question of [BL] by proving that nice vdC sets are sets of nice recurrence, and of Theorem 1.17, answering a question of [BF].

Proposition 2.1.

Let $G$ be a countably infinite amenable group with a Følner sequence $(F_{N})_{N}$ , and let $D\subseteq\mathbb{C}$ be compact. For each $l\in\mathbb{N}$ let $j_{l}\in\mathbb{N}$ , $h_{l,1},\dots,h_{l,j_{l}}\in G$ and let $p_{l}:D^{j_{l}}\to\mathbb{C}$ be continuous. Finally, let $\gamma:\mathbb{N}\to\mathbb{C}$ be a sequence of complex numbers. The following are equivalent:

1.

There exists a sequence $(z_{g})_{g\in G}$ of elements of $D$ such that, for all $l\in\mathbb{N}$ ,

$\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}p_{l}(z_{h_{l,1}g},\dots,z_{h_{l,j_{l}}g})=\gamma(l).$ (3)
2.

There exists a m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and a measurable function $f:X\to D$ such that, for all $l\in\mathbb{N}$ ,

$\int_{X}p_{l}(f(T_{h_{l,1}}(x)),\dots,f(T_{h_{l,j_{l}}}(x))d\mu(x)=\gamma(l).$

(Finitistic criterion) For all $A\subseteq G$ finite and for all $L\in\mathbb{N},\delta>0$ there exists some $K\in\mathbb{N}$ and sequences $(z_{g,k})_{g\in G}$ in $D$ , for $k=1,\dots,K$ , such that for all $l=1,\dots,L$ we have

\left|\gamma(l)-\frac{1}{K|A|}\sum_{k=1}^{K}\sum_{g\in A}p_{l}(z_{h_{l,1}g,k},\dots,z_{h_{l,j_{l}}g,k})\right|<\delta.

(4)

Remark 2.2.

For the equivalence 1 $\iff$ 3 to hold $D$ need not be compact, and $p_{l}:D^{j_{l}}\to\mathbb{C}$ can be any bounded function (not necessarily continuous). Indeed, all we will use in the proof of 3 $\implies$ 1 is that the functions $p_{l}$ are bounded, and it is not hard to show 1 $\implies$ 3 if the functions $p_{l}$ are bounded; to do it, one can let $N$ be big enough, let $K=|F_{N}|$ and for all $k\in F_{N}$ let $(z_{g,k})_{g}$ be given by $z_{g,k}=z_{gk}$ , where $(z_{g})_{g}$ satisfies 1.

Proof of Theorem 1.14 from 2.1.

Let $G,F=(F_{N})_{N},D,(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and $f:X\to D$ be as in Theorem 1.14.

Now, for each $k\in\mathbb{N}$ , the set $C(D^{k}):=\{p:D^{k}\to\mathbb{C};p\text{ continuous}\}$ is separable in the supremum norm. So we can consider for each $l\in\mathbb{N}$ elements $h_{l,1},\dots,h_{l,j_{l}}$ and functions $p_{l}\in C(D^{j_{l}})$ such that for any $h_{1},\dots,h_{j}\in G$ , for any $p\in C(D^{j})$ continuous and for any $\varepsilon>0$ there exists $l\in\mathbb{N}$ such that $j_{l}=j$ , $(h_{l,1},\dots,h_{l,j_{l}})=(h_{1},\dots,h_{j})$ and $\|p-p_{l}\|_{\infty}<\varepsilon$ .

If we now apply 2.1 to the sequence

\gamma(l)=\int_{X}p_{l}(f(T_{h_{l,1}}(x)),\dots,f(T_{h_{l,j_{l}}}(x))d\mu(x),

we obtain a sequence $(z_{g})_{g\in G}$ of elements of $D$ such that, for all $l\in\mathbb{N}$ ,

\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}p_{l}(z_{h_{l,1}g},\dots,z_{h_{l,j_{l}}g})=\int_{X}p_{l}(f(T_{h_{l,1}}(x)),\dots,f(T_{h_{l,j_{l}}}(x))d\mu(x).

(5)

This implies that Equation 1 holds for any $h_{1},\dots,h_{j}\in G$ and $p\in C(D^{k})$ , so we are done proving the first implication.

For the converse we can use the same argument with the dense sequence $(p_{l})_{l\in\mathbb{N}}$ from above, except that we first need to take a Følner subsequence $(F_{N}^{\prime})_{N}$ such that the following limit exists for all $l$ .

\gamma(l)=\lim_{N\to\infty}\frac{1}{|F_{N}^{\prime}|}\sum_{g\in F_{N}^{\prime}}p_{l}(z_{h_{l,1}g},\dots,z_{h_{l,j_{l}}g}).\qed

We will now prove 2.1. We will show 1 $\implies$ 2 $\implies$ 3 $\implies$ 1. The complicated implication is 3 $\implies$ 1. During the rest of the section we fix an amenable group $G$ , a Følner sequence $(F_{N})_{N}$ and a compact set $D\subseteq\mathbb{C}$ .

Proof of 1 $\implies$ 2.

Let $\mathbf{z}=(z_{a})_{a\in G}$ be as in 1. Consider the (compact, metrizable) product space $D^{G}$ of sequences $(z_{g})_{g\in G}$ in $D$ , with the Borel $\sigma$ -algebra. We have an action $(R_{g})_{g\in G}$ of $G$ on $D^{G}$ by $R_{g}((z_{a})_{a\in G})=(z_{ag})_{a\in G}$ .

Now, for each $N$ consider the average of Dirac measures $\nu_{N}=\frac{1}{|F_{N}|}\sum_{g\in F_{N}}\delta_{R_{g}\mathbf{z}}$ . Let $\nu$ be the weak limit of some subsequence $(\nu_{N_{m}})_{m}$ ; then $\nu$ is invariant by $R_{h}$ for all $h\in G$ , because $\lim_{N}\frac{|F_{N}\Delta hF_{N}|}{|F_{N}|}=0$ and for all $N\in\mathbb{N}$ ,

\nu_{N}-(R_{h})_{*}\nu_{N}=\frac{1}{|F_{N}|}\left(\sum_{g\in F_{N}\setminus hF_{N}}\delta_{R_{g}\mathbf{z}}-\sum_{g\in hF_{N}\setminus F_{N}}\delta_{R_{g}\mathbf{z}}\right).

Finally, we define the function $f:D^{G}\to\mathbb{C}$ by $f((z_{a})_{a\in G})=z_{1}$ . Then for all $l\in\mathbb{N}$ we have

		$\displaystyle\int_{D^{G}}p_{l}(f(R_{h_{l,1}}(x)),\dots,f(R_{h_{l,j_{l}}}(x))d\nu(x)$
	$\displaystyle=$	$\displaystyle\lim_{m}\int_{D^{G}}p_{l}(f(R_{h_{l,1}}(x)),\dots,f(R_{h_{l,j_{l}}}(x))d\nu_{N_{m}}(x)$
	$\displaystyle=$	$\displaystyle\lim_{m}\frac{1}{\|F_{N_{m}}\|}\sum_{g\in F_{N_{m}}}p_{l}(f(R_{h_{l,1}}(R_{g}\mathbf{z})),\dots,f(R_{h_{l,j_{l}}}(R_{g}(\mathbf{z})))$
	$\displaystyle=$	$\displaystyle\lim_{m}\frac{1}{\|F_{N_{m}}\|}\sum_{g\in F_{N_{m}}}p_{l}(z_{h_{l,1}g},\dots,z_{h_{l,j_{l}}g})=\gamma(l).\qed$

Proof of 2 $\implies$ 3.

Let $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and $f$ be as in 2, and let $(D^{G},\mathcal{B}(D^{G}),(R_{g})_{g\in G})$ be as in the proof of 1 $\implies$ 2.

Consider the function $\Phi:X\to D^{G}$ given by $\Phi(x)=(f(T_{a}x))_{a\in G}$ and let $\nu$ be the measure $\Phi_{*}\mu$ in $D^{G}$ . As a Borel probability measure in a compact metric space, $\nu$ can be weakly approximated by a sequence of finitely supported measures $(\nu_{N})_{N}$ , which we may assume are of the form

\nu_{N}=\frac{1}{|K_{N}|}\sum_{k\in K_{N}}\delta_{(z_{a,k})_{a}}.

For some finite index sets $K_{N}$ and some sequences $(z_{a,k})_{a\in G}$ , for $k\in K_{N}$ . Now, the fact that $\nu_{N}\to\nu$ weakly means that for all $l\in\mathbb{N}$ and $A\subseteq G$ finite we have

		$\displaystyle\lim_{N}\frac{1}{\|K_{N}\|\cdot\|A\|}\sum_{k\in K_{N}}\sum_{g\in A}p_{l}(z_{h_{l,1}g,k},\dots,z_{h_{l,j_{l}}g,k})$
	$\displaystyle=$	$\displaystyle\lim_{N}\frac{1}{\|A\|}\sum_{g\in A}\int_{D^{G}}p_{l}(z_{h_{l,1}g},\dots,z_{h_{l,j_{l}}g})d\nu_{N}((z_{a})_{a})$
	$\displaystyle=$	$\displaystyle\frac{1}{\|A\|}\sum_{g\in A}\int_{D^{G}}p_{l}(z_{h_{l,1}g},\dots,z_{h_{l,j_{l}}g})d\nu((z_{a})_{a})$
	$\displaystyle=$	$\displaystyle\frac{1}{\|A\|}\sum_{g\in A}\int_{X}p_{l}(f(T_{h_{l,1}g}x),\dots,f(T_{h_{l,j_{l}}g}x))d\mu(x)$
	$\displaystyle=$	$\displaystyle\frac{1}{\|A\|}\sum_{g\in A}\int_{X}p_{l}(f(T_{h_{l,1}}x),\dots,f(T_{h_{l,j_{l}}}x))d\mu=\frac{1}{\|A\|}\sum_{g\in A}\gamma(l)=\gamma(l).$

So for any $L\in\mathbb{N}$ and $\delta>0$ , Equation 4 will be satisfied for big enough $N$ . ∎

In order to prove 3 $\implies$ 1, we will first need some lemmas about averages of sequences of complex numbers.

Proposition 2.3.

Let $M>0$ and $\delta\in(0,1)$ . For any finite sets $E^{\prime}\subseteq E\subseteq G$ such that $\frac{|E\setminus E^{\prime}|}{|E|}<\delta$ and any complex numbers $(z_{g})_{g\in G}$ with $|z_{g}|\leq M\;\forall g\in G$ , we have

\left|\frac{1}{|E|}\sum_{g\in E}z_{g}-\frac{1}{|E^{\prime}|}\sum_{g\in E^{\prime}}z_{g}\right|<2M\delta.

Proof.

Indeed, the LHS above is equal to

\left|\frac{|E^{\prime}|-|E|}{|E|\cdot|E^{\prime}|}\sum_{g\in E^{\prime}}z_{g}+\frac{1}{|E|}\sum_{g\in E\setminus E^{\prime}}z_{g}\right|\\ \leq\frac{|E|-|E^{\prime}|}{|E|\cdot|E^{\prime}|}\cdot M|E^{\prime}|+\frac{1}{|E|}\cdot M|E\setminus E^{\prime}|<M\delta+M\delta.\qed

Definition 2.4.

Let $S,T$ be subsets of a group $G$ . We denote

\partial_{S}T:=\{g\in G;Sg\cap T\neq\varnothing\text{ and }Sg\cap(G\setminus T)\neq\varnothing\}=\left(\cup_{s\in S}s^{-1}T\right)\setminus\left(\cap_{s\in S}s^{-1}T\right).

Remark 2.5.

Note that if $(F_{N})_{N}$ is a Følner sequence in $G$ and $S$ is finite, then $\lim_{N}\frac{|\partial_{S}F_{N}|}{|F_{N}|}=0$ , because $\lim_{N}\frac{|s^{-1}F_{N}\Delta F_{N}|}{|F_{N}|}=0$ for all $s\in G$ . Also note that if $c\in G$ , then $\partial_{S}(Tc)=(\partial_{S}T)c$ . Finally, if $e\in S$ , then for all $g\in T\setminus\partial_{S}T$ we have $Sg\subseteq T$ .

The following lemma is a crucial part of the proof of 2.1. Intuitively, it says that if you have a finite set of bounded sequences $f_{i}:T_{i}\to\mathbb{C}$ defined in finite subsets $T_{1},\dots,T_{k}$ of a group $G$ , you can reassemble them into a sequence $f:A\to\mathbb{C}$ defined in a bigger finite set $A\subseteq G$ which is a union of right translates of the sets $T_{i}$ , and the average value of $f$ will be a weighted average of the average values of the $f_{i}$ .

Lemma 2.6.

Let $G$ be a group, $\delta,M>0,K\in\mathbb{N}$ and $c_{1},\dots,c_{K}\in G$ . Let $T_{1},\dots,T_{K},A$ be finite subsets of $G$ such that $T_{1}c_{1},\dots,T_{K}c_{K}$ are pairwise disjoint, contained in $A$ and $\frac{\sum_{k=1}^{K}|T_{k}|}{|A|}>1-\frac{\delta}{M}$ . For $k=1,\dots,K$ let $(z_{g,k})_{g\in G}$ be sequences of complex numbers in $D$ , and consider a sequence $(z_{g})_{g\in G}$ such that

z_{g}=z_{gc_{k}^{-1},k}\text{ for all }g\in T_{k}c_{k},

Finally, let $S=\{h_{1},\dots,h_{j}\}\subseteq G$ be finite with $e\in S$ and let $p:D^{j}\to\mathbb{C}$ satisfy $\|p\|_{\infty}\leq M$ . If we have $\frac{|\partial_{S}T_{k}|}{|T_{k}|}\leq\frac{1}{M\delta}$ for all $k$ , then

\left|\frac{1}{|A|}\sum_{g\in A}p(z_{h_{1}g},\dots,z_{h_{j}g})-\sum_{k=1}^{K}\frac{|T_{k}|}{\sum_{i=1}^{K}|T_{i}|}\cdot\left(\frac{1}{|T_{k}|}\sum_{g\in T_{k}}p(z_{h_{1}g,k},\dots,z_{h_{j}g,k})\right)\right|\leq 4\delta.

Proof.

To abbreviate, for any sequence $(z_{a})_{a\in G}$ we will denote $p((z_{a})_{a})=p(z_{h_{1}},\dots,z_{h_{j}})$ .

Note that, given $g\in G$ , we will have $p((z_{agc_{k}^{-1},k})_{a})=p((z_{ag})_{a})$ whenever $z_{agc_{k}^{-1},k}=z_{ag}$ for all $a\in S$ , which happens when $ag\in T_{k}c_{k}$ for all $a\in S$ , that is, when $g\in T_{k}c_{k}\setminus\partial_{S}T_{k}c_{k}$ . Thus, for each $k=1,\dots,K$ we have

	$\displaystyle\left\|\frac{1}{\|T_{k}\|}\sum_{g\in T_{k}}p((z_{ag,k})_{a})-\frac{1}{\|T_{k}\|}\sum_{g\in T_{k}c_{k}}p((z_{ag})_{a})\right\|$	$\displaystyle=\left\|\frac{1}{\|T_{k}\|}\sum_{g\in T_{k}c_{k}}p((z_{agc_{k}^{-1},k})_{a})-\frac{1}{\|T_{k}\|}\sum_{g\in T_{k}c_{k}}p((z_{ag})_{a})\right\|$
		$\displaystyle=\left\|\frac{1}{\|T_{k}\|}\sum_{g\in T_{k}c_{k}\cap\partial_{S}T_{k}c_{k}}\left(p((z_{agc_{k}^{-1},k})_{a})-p((z_{ag})_{a})\right)\right\|\leq 2\delta,$

the last inequality being because $\frac{|T_{k}c_{k}\cap\partial_{S}T_{k}c_{k}|}{|T_{k}|}<\frac{1}{M\delta}$ and the summands have norm $\leq 2M$ . Taking weighted averages over $k=1,\dots,K$ , we obtain

\left|\sum_{k=1}^{K}\frac{|T_{k}|}{\sum_{i=1}^{K}|T_{i}|}\cdot\left(\frac{1}{|T_{k}|}\sum_{g\in T_{k}}p((z_{ag,k})_{a})\right)-\sum_{k=1}^{K}\frac{|T_{k}|}{\sum_{i=1}^{K}|T_{i}|}\cdot\left(\frac{1}{|T_{k}|}\sum_{g\in T_{k}c_{k}}p((z_{ag})_{a})\right)\right|\leq 2\delta.

But applying 2.3 with $E=A$ , $E^{\prime}=\cup_{i}T_{i}c_{i}$ and the sequence $g\mapsto p((z_{ag})_{a})$ we also have

\left|\frac{1}{|A|}\sum_{g\in A}p((z_{ag})_{a})-\sum_{k=1}^{K}\frac{|T_{k}|}{\sum_{i=1}^{K}|T_{i}|}\cdot\left(\frac{1}{|T_{k}|}\sum_{g\in T_{k}c_{k}}p((z_{ag})_{a})\right)\right|\leq 2\delta.

So by the triangle inequality, we are done. ∎

The main tool we will need in the proof of 3 $\implies$ 1 is [DHZ, Theorem 5.2]; in order to state it we first recall some definitions:

Definition 2.7 (cf. [DHZ, Defs 3.1,3.2]).

A tiling $\mathcal{T}$ of a group $G$ consists of two objects:

•

A finite family $\mathcal{S}(\mathcal{T})$ (the shapes) of finite subsets of $G$ containing the identity $1_{G}$ .
•

A finite collection $C(\mathcal{T})=\{C(S);S\in\mathcal{S}(\mathcal{T})\}$ of subsets of $G$ , the center sets, such that the family of right translates of form $Sc$ with $S\in\mathcal{S}(\mathcal{T})$ and $c\in C(S)$ (such sets $Sc$ are the tiles of $\mathcal{T}$ ) form a partition of $G$ .

Definition 2.8 (cf. [DHZ, Page 17]).

We say that a sequence of tilings $(\mathcal{T}_{k})_{k\in\mathbb{N}}$ of a group $G$ is congruent if every tile of $\mathcal{T}_{k+1}$ equals a union of tiles of $\mathcal{T}_{k}$ .

Definition 2.9.

If $A$ and $B$ are finite subsets of a group $G$ , we say that $A$ is $(B,\varepsilon)$ -invariant if $\frac{|\partial_{B}A|}{|A|}<\varepsilon$ .⁶⁶6This is not the definition of $(B,\varepsilon)$ -invariant used in [DHZ], but it will be more convenient for our purposes.

We will need the following weak version of [DHZ, Theorem 5.2]:

Theorem 2.10 (cf. [DHZ, Theorem 5.2]).

For any countable amenable group $G$ there exists a congruent sequence of tilings $(\mathcal{T}_{k})_{k\in\mathbb{N}}$ such that, for every $A\subseteq G$ finite and $\delta>0$ , all the tiles of $\mathcal{T}_{k}$ are $(A,\delta)$ -invariant for big enough $k$ .

The following notation will be useful.

Definition 2.11.

For any tiling of a group $G$ and any finite set $B\subseteq G$ , we let $\partial_{\mathcal{T}}B$ denote the union of all tiles of $\mathcal{T}$ which intersect both $B$ and $G\setminus B$ .

Remark 2.12.

Let $G$ be a countable amenable group with a tiling $\mathcal{T}$ and a Følner sequence $(F_{N})_{N}$ . As $\partial_{\mathcal{T}}F_{N}\subseteq\cup_{S\in\mathcal{S}(\mathcal{T})}\partial_{S}F_{N}$ , Remark 2.5 implies that

\lim_{N\to\infty}\frac{|\partial_{\mathcal{T}}F_{N}|}{|F_{N}|}=0.

Lemma 2.13.

Let $G$ be a countably infinite amenable group with a Følner sequence $(F_{N})_{N}$ , and let $(\mathcal{T}_{k})_{k\in\mathbb{N}}$ be a congruent sequence of tilings of $G$ . Then there is a partition $\mathcal{P}$ of $G$ into tiles of the tilings $\mathcal{T}_{k}$ such that

•

For each $k\in\mathbb{N}$ , $\mathcal{P}$ contains only finitely many tiles of $\mathcal{T}_{k}$ .
•

If for each $N\in\mathbb{N}$ we let $A_{N}=\bigcup\{T\in\mathcal{P};T\subseteq F_{N}\}\subseteq F_{N}$ , then we have

$\lim_{N\to\infty}\frac{|A_{N}|}{|F_{N}|}=1.$

Proof.

We can assume that $\cup_{N}F_{N}=G$ , adding some Følner sets to the sequence if necessary. In the following, for each finite set $B\subseteq G$ and $k\in\mathbb{N}$ we will denote $\partial_{k}B:=\cup_{j=1}^{k}\partial_{\mathcal{T}_{j}}B$ , so that for all $k\in\mathbb{N}$ ,

\lim_{N\to\infty}\frac{|\partial_{k}F_{N}|}{|F_{N}|}=0.

Now for each $k\in\mathbb{N}$ let $N_{k}$ be a big enough number that $\frac{|\partial_{k+1}F_{N}|}{|F_{N}|}\leq\frac{1}{k+1}$ for all $N\geq N_{k}$ .

Let $D_{0}=\varnothing$ and for each $k\in\mathbb{N}$ let $D_{k}\subseteq G$ be the union of all tiles of $\mathcal{T}_{k+1}$ intersecting some element of $\bigcup_{N=1}^{N_{k}}F_{N}$ . Thus, $D_{k}\subseteq D_{k+1}$ for all $k$ , $G=\cup_{k}D_{k}$ and $D_{k}\setminus D_{k-1}$ is a union of tiles of $\mathcal{T}_{k}$ . We define $\mathcal{P}$ to be the partition of $G$ formed by all tiles of $\mathcal{T}_{k}$ contained in $D_{k}\setminus D_{k-1}$ , for all $k\in\mathbb{N}$ .

Now, fix $N$ and let $k$ be the smallest natural number such that $N\leq N_{k}$ (note that $k\to\infty$ when $N\to\infty$ ). Note that all tiles $T\in\mathcal{P}$ intersecting $F_{N}$ must be in $\mathcal{T}_{j}$ for some $j\leq k$ . Thus, the set $A_{N}$ of all tiles of $\mathcal{P}$ which are contained in $F_{N}$ must contain $|F_{N}\setminus\partial_{k}F_{N}|$ . But we have $N>N_{k-1}$ , so $\frac{|\partial_{k}F_{N}|}{|F_{N}|}\leq\frac{1}{k}$ . So $\frac{|A_{N}|}{|F_{N}|}\geq 1-\frac{1}{k}$ , and we are done.∎

Proof of Item 3 $\implies$ Item 1.

Let $S_{l}=\{h_{l,1},\dots,h_{l,j_{l}}\}$ and $M_{l}\geq\|p_{l}\|_{\infty}$ for all $l\in\mathbb{N}$ (we may assume $e\in S_{l}$ and $M_{l+1}\geq M_{l}$ for all $l$ ). We prove first that 3 implies the following:

3’.

Let $\delta>0$ and $L\in\mathbb{N}$ . Then for any sufficiently left-invariant ⁷⁷7By this we mean that there is a finite set $A\subseteq G$ and some $\varepsilon>0$ such that the property stated below is satisfied for all $(A,\varepsilon)$ -invariant sets. subset $B$ of $G$ there exists a sequence $(w_{g})_{g\in G}$ in $D$ such that, for all $l=1,\dots,L$ ,

$\left|\gamma(l)-\frac{1}{|B|}\sum_{g\in B}p_{l}(w_{h_{l,1}g},\dots,w_{h_{l,j_{l}}g})\right|<\delta.$

In order to prove 3’. from 3, fix $\delta,L$ and consider a tiling $\mathcal{T}$ of $G$ such that $\frac{|\partial_{S_{l}}S|}{|S|}<\frac{\delta}{10M_{L}}$ for all $l=1,\dots,L$ and $S\in\mathcal{S}(\mathcal{T})$ . For each $S\in\mathcal{S}(\mathcal{T})$ there is by 3 some $K_{S}\in\mathbb{N}$ and sequences $(z_{S,g,k})_{g\in G}$ in $D$ ( $k=1,\dots,K_{S}$ ) satisfying

\left|\gamma(l)-\frac{1}{K_{S}|S|}\sum_{k=1}^{K_{S}}\sum_{g\in S}p_{l}(z_{S,h_{l,1}g,k},\dots,z_{S,h_{l,j_{l}}g,k})\right|<\frac{\delta}{5}\text{ for }l=1,\dots,L.

(6)

Then any finite subset $B$ of $G$ such that $\frac{|\partial_{T}B|}{|B|}<\frac{\delta}{10M_{L}}$ and

|B|\geq\frac{10M_{L}}{\delta}\sum_{S\in\mathcal{S}(\mathcal{T})}|S|K_{S}

(7)

will satisfy 3’.; to see why, first note that the union $B_{0}$ of all tiles of $\mathcal{T}$ contained in $B$ satisfies $\frac{|B_{0}|}{|B|}>1-\frac{\delta}{10M_{L}}$ ; now obtain a set $B_{1}\subseteq B_{0}$ by removing finitely many tiles from $B_{0}$ in such a way that, for each $S\in\mathcal{S}(\mathcal{T})$ , the number of tiles of shape $S$ contained in $B_{1}$ is a multiple of $K_{S}$ . Note that, due to Equation 7, this can be done in such a way that $\frac{|B_{1}|}{|B|}\geq 1-\frac{\delta}{5M_{L}}$ .

So letting $\mathcal{P}$ be the set of tiles of $\mathcal{T}$ contained in $B_{1}$ , we can define a function $f:\mathcal{P}\to\mathbb{N}$ that associates for each translate $Sc$ some number in $\{1,2,\dots,K_{S}\}$ , and such that for each $S$ , the same number of right-translates of $S$ have value each of the numbers $1,2,\dots,K_{S}$ . Finally, define a sequence $(w_{g})_{g\in G}$ by

w_{g}=z_{S,gc^{-1},f(Sc)}\text{ if }g\in Sc\text{ and }Sc\text{ is a tile of }\mathcal{T}\text{ contained in }B_{1}.

The rest of values of $w_{g}$ are not important, we can let them be some fixed value of $D$ . Then, by 2.6 applied to the set $B$ and the tiles of $\mathcal{T}$ contained in $B_{1}$ , we obtain for all $l=1,\dots,L$ that

\left|\frac{1}{|B|}\sum_{g\in B}p(w_{h_{l,1}g},\dots,w_{h_{l,j_{l}}})-\sum_{Sc\in\mathcal{P}}\frac{|Sc|}{|B_{1}|}\cdot\left(\frac{1}{|S|}\sum_{g\in S}p(z_{S,h_{l,1}g,f(Sc)},\dots,z_{S,h_{l,j_{l}}g,f(Sc)})\right)\right|\leq 4\left(\frac{\delta}{5}\right).

But as the sequences $(z_{S,g,k})_{g\in G}$ in $D$ ( $k=1,\dots,K_{S}$ ) satisfy Equation 6, we have for all $l=1,\dots,L$ that

\left|\gamma(l)-\sum_{Sc\in\mathcal{P}}\frac{|Sc|}{|B_{1}|}\cdot\left(\frac{1}{|S|}\sum_{g\in S}p(z_{S,h_{l,1}g,f(Sc)},\dots,z_{S,h_{l,j_{l}}g,f(Sc)})\right)\right|\leq\frac{\delta}{5},

so we are done proving 3’. by the triangle inequality.

Now we use 3’. to prove 1. By 3’. and Theorem 2.10, there is a congruent sequence of tilings $(\mathcal{T}_{L})_{L\in\mathbb{N}}$ of $G$ (we may also assume $\mathcal{S}(\mathcal{T}_{L})\cap\mathcal{S}(\mathcal{T}_{L^{\prime}})=\varnothing$ if $L\neq L^{\prime}$ ) such that

1.

For all $S\in\mathcal{S}(\mathcal{T}_{L})$ and for $l=1,\dots,L$ we have $\frac{|\partial_{S_{l}}S|}{|S|}<\frac{1}{L}$ .
2.

For each $S\in\mathcal{S}(\mathcal{T}_{L})$ there is a sequence $(w_{S,g})_{g\in G}$ such that for all $l=1,\dots,L$ ,

$\left|\gamma(l)-\frac{1}{|S|}\sum_{g\in S}p_{l}(w_{S,h_{l,1}g},\dots,w_{S,h_{l,j_{l}}g})\right|<\frac{1}{L}.$ (8)

Now, letting $(F_{N})_{N}$ be the Følner sequence of Item 1, we let $\mathcal{P}$ and $(A_{N})_{N}$ be as in 2.13, with $\mathcal{P}_{L}\subseteq\mathcal{P}$ ( $L=1,2,\dots$ ) being finite sets of tiles of $\mathcal{T}_{L}$ such that $\mathcal{P}=\sqcup_{L}\mathcal{P}_{L}$ . We define the sequence $(z_{g})_{g\in G}$ by

z_{g}=w_{S,gc^{-1}}\text{ if }g\in Sc,\text{ where }S\in\mathcal{S}(\mathcal{T}_{L})\text{ for some $L$ and }Sc\in\mathcal{P}_{L}.

All that is left is proving that $(z_{g})_{g}$ satisfies 1 for all $l\in\mathbb{N}$ . So let $\varepsilon>0$ and $l\in\mathbb{N}$ . Note that for big enough $N$ we have $\frac{|A_{N}|}{|F_{N}|}>1-\frac{\varepsilon}{10M_{l}}$ . Fix some natural number $L>\frac{10}{\varepsilon}$ (also suppose $L\geq l$ ). Letting $A_{N}^{\prime}$ be obtained from $A_{N}$ by removing the (finitely many) tiles which are in $\mathcal{P}_{i}$ for some $i=1,\dots,L$ , then for big enough $N$ we have $\frac{|A_{N}^{\prime}|}{|F_{N}|}>1-\frac{\varepsilon}{5M_{l}}$ . Thus, applying 2.6 to the set $F_{N}$ and to all the tiles contained in $A_{N}^{\prime}$ , and for each $S\in\mathcal{S}(\mathcal{T}_{L})$ letting $n_{S}$ be the number of tiles of shape $S$ in $\mathcal{P}_{L}$ which are contained in $A_{N}^{\prime}$ , we obtain

\left|\frac{1}{|F_{N}|}\sum_{g\in F_{N}}p_{l}(z_{h_{l,1}g},\dots,z_{h_{l,j_{l}}g})-\sum_{L\in\mathbb{N}}\sum_{S\in\mathcal{S}(\mathcal{T}_{L})}\frac{n_{S}|S|}{|A_{N}^{\prime}|}\cdot\left(\frac{1}{|S|}\sum_{g\in S}p_{l}(w_{S,h_{l,1}g,L},\dots,w_{S,h_{l,j_{l}}g,L})\right)\right|\leq 4\frac{\varepsilon}{5}.

(9)

However, the double sum in Equation 9 is at distance $\leq\frac{\varepsilon}{10}$ of $\gamma(l)$ (this follows from taking an affine combination of Equation 8 applied to the tiles $S$ of $A_{N}^{\prime}$ , with constant $\frac{1}{L}<\frac{\varepsilon}{10}$ ). So by the triangle inequality, for big enough $N$ we have

\left|\gamma(l)-\frac{1}{|F_{N}|}\sum_{g\in F_{N}}p_{l}(z_{h_{l,1}g},\dots,z_{h_{l,j_{l}}g})\right|<\varepsilon.

As $\varepsilon$ is arbitrary, we are done. ∎

As promised in the introduction, we now prove a converse to the Furstenberg correspondence principle. Theorem 2.14 is a slight generalization of Theorem 1.16 in which we also allow intersections with complements of the sets $g_{i}A$ . This makes Theorem 2.14 a statement about densities of any sets in the algebra $\mathcal{A}\subseteq\mathcal{P}(G)$ generated by the family $\{gA;g\in G\}$ . Indeed, any element of $\mathcal{A}$ is a finite disjoint union of elements of the form $\left(\bigcap_{i=1}^{k}g_{i}A\right)\cap\left(\bigcap_{i=k+1}^{n}G\setminus g_{i}A\right)$ , where $n\in\mathbb{N}$ , $k\in\{0,1,\dots,n\}$ and $g_{1},\dots,g_{n}\in G$ .

Theorem 2.14.

Let $G$ be a countably infinite amenable group with a Følner sequence $(F_{N})_{N}$ . For every m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and every $B\in\mathcal{B}$ there exists a subset $A\subseteq G$ such that, for all $n\in\mathbb{N}$ , $k\in\{0,1,\dots,n\}$ and $g_{1},\dots,g_{n}\in G$ we have

d_{F}\left(\left(\bigcap_{i=1}^{k}g_{i}A\right)\cap\left(\bigcap_{i=k+1}^{n}G\setminus g_{i}A\right)\right)=\mu\left(\left(\bigcap_{i=1}^{k}T_{g_{i}}B\right)\cap\left(\bigcap_{i=k+1}^{n}X\setminus T_{g_{i}}B\right)\right).

(10)

Reciprocally, for any subset $A\subseteq G$ there is a m.p.s. $(X,\mathcal{B},\mu,T)$ and $B\in\mathcal{B}$ satisfying Equation 10 for all $n,k,g_{1},\dots,g_{n}$ as above such that the density in Equation 10 exists.

Proof.

Note that for all $n\in\mathbb{N}$ , $k\in\{0,1,\dots,n\}$ and $g_{1},\dots,g_{n}\in G$ we have

\displaystyle\mu\left(\left(\bigcap_{i=1}^{k}T_{g_{i}}B\right)\cap\left(\bigcap_{i=k+1}^{n}X\setminus T_{g_{i}}B\right)\right)=\int_{X}\left(\prod_{i=1}^{k}\chi_{B}\left(T_{g_{i}^{-1}}x\right)\right)\cdot\left(\prod_{i=k+1}^{n}\left(1-\chi_{B}\left(T_{g_{i}^{-1}}x\right)\right)\right)d\mu.

So by Theorem 1.14 applied to the polynomials of the form $p(x_{1},\dots,x_{j})=x_{1}x_{2}\dots x_{k}(1-x_{k+1})\dots(1-x_{n})$ and with $D=\{0,1\}$ , there exists some characteristic function $\chi_{A}:G\to\{0,1\}$ such that for all $n\in\mathbb{N}$ , $k\in\{0,1,\dots,n\}$ and $g_{1},\dots,g_{n}\in G$ we have

	$\displaystyle\mu\left(\left(\bigcap_{i=1}^{k}T_{g_{i}}B\right)\cap\left(\bigcap_{i=k+1}^{n}X\setminus T_{g_{i}}B\right)\right)$
	$\displaystyle=\lim_{N\to\infty}\frac{1}{\|F_{N}\|}\sum_{g\in F_{N}}\left(\prod_{i=1}^{k}\chi_{A}(g_{i}^{-1}g)\right)\cdot\left(\prod_{i=k+1}^{n}(1-\chi_{A}(g_{i}^{-1}g))\right)$
	$\displaystyle=d_{F}\left(\left(\bigcap_{i=1}^{k}g_{i}A\right)\cap\left(\bigcap_{i=k+1}^{n}G\setminus g_{i}A\right)\right).$

The other implication is proved similarly, applying Theorem 1.14 in the other direction. ∎

Remark 2.15.

It is possible to give a version of Theorem 1.16 that involves translates of several sets $B_{1},\dots,B_{l}\subseteq X$ , as in [BF2, Definition A.3], or even countably many sets $(B_{k})_{k\in\mathbb{N}}$ . But we will not do so as it is not the main theme of this article and it would involve proving a modified version of Theorem 1.14.

Proof of 1.12.

$\implies$

Suppose there is some set $E\subseteq G$ such that $\overline{d_{F}}(E)^{2}>\limsup_{h\in H}\overline{d_{F}}(E\cap hE)$ . Then there is a Følner subsequence $F^{\prime}$ of $F$ such that $d_{F^{\prime}}(E),d_{F^{\prime}}(E\cap hE)$ exist for all $h$ and $d_{F^{\prime}}(E)^{2}>\limsup_{h\in H}d_{F^{\prime}}(E\cap hE)$ . But by Theorem 1.14 there is some m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and some $B\in\mathcal{B}$ such that $\mu(B)=d_{F^{\prime}}(E)$ and $\mu(B\cap T_{h}B)=d_{F^{\prime}}(E\cap hE)$ for all $h\in G$ , thus $\mu(B)^{2}>\limsup_{h\in H}\mu(B\cap T_{h}B)$ .
$\impliedby$

Suppose there is a m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and some $B\in\mathcal{B}$ such that $\mu(B)^{2}>\limsup_{h\in H}\mu(B\cap T_{h}B)$ . By Theorem 1.14 there is some set $E\subseteq G$ such that $\mu(B)=d_{F}(E)$ and $\mu(B\cap T_{h}B)=d_{F}(E\cap hE)$ for all $h\in G$ , thus $d_{F^{\prime}}(E)^{2}>\limsup_{h\in H}d_{F^{\prime}}(E\cap hE)$ .∎

The proof of 1.13 is essentially the same as that of 1.12, using Theorem 1.14 to prove both directions.

Proof of Theorem 1.17.

We can change the set $\cup_{g\in A}gE$ by $\cap_{g\in A}gE$ by considering the complement of $E$ . Also, it is enough to prove Theorem 1.17 for just one Følner sequence $F$ in $G$ . Indeed, suppose there is $E\subseteq G$ such that for all finite $\varnothing\neq A\subseteq G$ we have $d_{F}(E)=d_{F}\left(\cap_{g\in A}gE\right)=\frac{1}{2}$ . Then for any other Følner sequence $F^{\prime}$ , by Theorem 1.16 applied two times we can find a set $E^{\prime}$ such that $d_{F^{\prime}}(E)=d_{F}(E)$ and $d_{F^{\prime}}(\cap_{g\in A}gE)=d_{F}(\cap_{g\in A}gE)$ for all finite $A\subseteq G$ .

So we just need to choose a Følner sequence $(F_{N})_{N}$ for which the theorem is true. Let $(A_{N})_{N}$ be a Følner sequence in $G$ and let $F_{N}=A_{N}g_{N,1}\cup A_{N}g_{N,2}$ , where $g_{N,1},g_{N,2}\in G$ are chosen so that the sets $A_{N}g_{N,i}$ , for $i=1,2$ and $N\in\mathbb{N}$ , are pairwise disjoint. And let $E=\cup_{N}A_{N}g_{N,1}$ . We then have $d_{F}(E)=d_{F}\left(\cap_{g\in A}gE\right)=\frac{1}{2}$ for all finite $A\subseteq G$ , as we wanted. ∎

3 Characterizations of vdC sets in amenable groups

In this section we prove Theorem 3.1 below, our main characterization theorem for vdC sets in countable amenable groups. Theorem 3.1 gives a characterization of $F$ -vdC sets analogous to Theorem 1.2 but for any Følner sequence. In particular, it implies that the notion of $F$ -vdC set is independent of the Følner sequence, thus answering the question in [BL, Section 4.2] of whether $F$ -vdC implies vdC.

Theorem 3.1.

Let $G$ be a countably infinite amenable group with a Følner sequence $(F_{N})_{N}$ . For a set $H\subseteq G$ , the following are equivalent:

1.

$H$ is an $F$ -vdC set.
2.

If a sequence $(z_{g})_{g\in G}$ of complex numbers in the unit disk $\mathbb{D}$ satisfies

$\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}z_{hg}\overline{z_{g}}=0\text{ for all }h\in H,$

then

$\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}z_{g}=0.$
3.

If a sequence $(z_{g})_{g\in G}$ of complex numbers in $\mathbb{S}^{1}$ satisfies

$\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}z_{hg}\overline{z_{g}}=0\text{ for all }h\in H,$

then

$\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}z_{g}=0.$

$H$ is vdC in $G$ : For any m.p.s. $(X,\mathcal{A},\mu,(T_{g})_{g\in G})$ and any $f\in L^{\infty}(\mu)$ ,

\int_{X}(f(T_{h}(x)))\cdot\overline{f(x)}d\mu(x)=0\text{ for all }h\in H\text{ implies }\int_{X}fd\mu=0.

(11)

5.

(Finitistic characterization) For every $\varepsilon>0$ there is $\delta>0$ and finite sets $A\subseteq G,H_{0}\subseteq H$ such that for any $K\in\mathbb{N}$ and sequences $(z_{a,k})_{a\in G}$ in $\mathbb{D}$ , for $k=1,\dots,K$ , such that

$\left|\frac{1}{K|A|}\sum_{k=1}^{K}\sum_{a\in A}z_{ha,k}\overline{z_{a,k}}\right|<\delta\text{ for all }h\in H_{0},$

we have

$\left|\frac{1}{K|A|}\sum_{k=1}^{K}\sum_{a\in A}z_{a,k}\right|<\varepsilon.$

Remark 3.2.

It is an interesting question whether, if we change ‘ $f\in L^{\infty}(\mu)$ ’ by ‘ $f\in L^{2}(\mu)$ ’ in Definition 1.6, the resulting definition of vdC set in $G$ is equivalent. A similar question is posed in [FT, Conjecture 3.7], where they call the sets defined by the $L^{2}$ definition ‘sets of operatorial recurrence’.

The relationship between equidistribution and Cesaro averages is explained by 3.4 below, which was introduced by Weyl in [We].

Definition 3.3.

Let $G$ be a countable amenable group with a Følner sequence $(F_{N})_{N}$ . We say that a sequence $(z_{g})_{g\in G}$ , with $z_{g}\in\mathbb{S}^{1}$ for all $g\in G$ , is $F$ -u.d. in $\mathbb{S}^{1}$ if for every continuous function $f:\mathbb{S}^{1}\to\mathbb{C}$ we have

\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}f(z_{g})=\int_{\mathbb{S}^{1}}fdm,

where $m$ is the uniform probability measure in $\mathbb{S}^{1}$ .

Thus, a sequence $(x_{g})_{g\in G}$ in $\mathbb{T}$ is u.d. mod $1$ iff $(e^{2\pi ix_{g}})_{g\in G}$ is $F$ -u.d. in $\mathbb{S}^{1}$ , where $F=(\{1,\dots,N\})_{N}$ .

Proposition 3.4 (Weyl’s criterion for uniform distribution).

Let $G$ be a countable amenable group with a Følner sequence $(F_{N})_{N}$ . A sequence $(z_{g})_{g\in G}$ in $\mathbb{S}^{1}$ is $F$ -u.d. in $\mathbb{S}^{1}$ if and only if for all $l\in\mathbb{Z}\setminus\{0\}$ (or equivalently, for all $l\in\mathbb{N}$ ) we have

\lim_{N}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}z_{g}^{l}=0.

For a proof of 3.4 see e.g. [KN, Theorem 2.1] (the proof works for any Følner sequence).

Proof of Theorem 3.1.

We prove 2 $\implies$ 2 $\implies$ 2 $\implies$ 2 $\implies$ 2 $\implies$ 2

2 $\implies$ 3

Obvious.
3 $\implies$ 1

If $H$ is not $F$ -vdC then there is a sequence of complex numbers $(z_{g})_{g\in G}$ in $\mathbb{S}^{1}$ which is not $F$ -u.d. in $\mathbb{S}^{1}$ but such that $(z_{hg}\overline{z_{g}})_{g}$ is $F$ -u.d. in $\mathbb{S}^{1}$ for all $h\in H$ . By Weyl’s criterion, that means that for all $h\in H$ we have

$\lim_{N}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}z_{hg}^{l}\overline{z_{g}^{l}}=0\text{ for all }l\in\mathbb{Z}\setminus\{0\},$

but there is some $l_{0}\in\mathbb{Z}\setminus\{0\}$ such that

$\limsup_{N}\left|\frac{1}{|F_{N}|}\sum_{g\in F_{N}}z_{g}^{l_{0}}\right|>0.$

The sequence $(z_{g}^{l_{0}})_{g\in G}$ contradicts 3, so we are done.

Suppose a sequence $(z_{g})_{g\in G}$ does not satisfy 2. Taking a Følner subsequence $(F_{N}^{\prime})_{N}$ if necessary, we can assume that we have

	$\displaystyle\lim_{N\to\infty}\frac{1}{\|F_{N}^{\prime}\|}\sum_{g\in F_{N}^{\prime}}z_{g}=\lambda\neq 0$
	$\displaystyle\lim_{N\to\infty}\frac{1}{\|F_{N}^{\prime}\|}\sum_{g\in F_{N}^{\prime}}z_{hg}\overline{z_{g}}=0\text{ for all }h\in H.$

This contradicts 4 by Theorem 1.14 applied with $D=\mathbb{D}$ to the Cesaro averages of $(z_{g})_{g\in G}$ and $(z_{hg}\overline{z_{g}})_{g\in G}$ .

The contrapositive of this implication follows from Theorem 1.14 applied to Cesaro averages of the functions $(z_{g})_{g\in G}$ and $(z_{hg}\overline{z_{g}})_{g\in G}$ , with $D=\mathbb{D}$ . Indeed, if $H\subseteq G$ does not satisfy 5, then there is a m.p.s. $(X,\mathcal{A},\mu,(T_{g})_{g\in G})$ and $f\in L^{\infty}(\mu)$ such that, for some $\lambda\neq 0$

\int_{X}(f(T_{h}(x)))\cdot\overline{f(x)}d\mu(x)=0\text{ for all }h\in H\text{ but }\int_{X}fd\mu=\lambda.

(12)

So by Theorem 1.14 Item 1 $\implies$ Item 3, for any finite sets $A\subseteq G,H_{0}\subseteq H$ there is $K\in\mathbb{N}$ and sequences $(z_{a,k})_{a\in G}$ in $\mathbb{D}$ , for $k=1,\dots,K$ , such that

\left|\frac{1}{K|A|}\sum_{k=1}^{K}\sum_{a\in A}z_{ha,k}\overline{z_{a,k}}\right|<\delta\text{ for all }h\in H_{0},

but

\left|\frac{1}{K|A|}\sum_{k=1}^{K}\sum_{a\in A}z_{a,k}-\lambda\right|<\delta,

contradicting 4.

Suppose that $\neg$ 5 holds for some $\varepsilon>0$ . So for any finite sets $A\subseteq G$ and $H_{0}\subseteq H$ and for any $\delta>0$ there exist $K\in\mathbb{N}$ and sequences $(z_{a,k})_{a\in G}$ in $\mathbb{D}$ , for $k=1,\dots,K$ , such that

\left|\frac{1}{K|A|}\sum_{k=1}^{K}\sum_{a\in A}z_{ha,k}\overline{z_{a,k}}\right|<\delta\text{ for all }h\in H_{0},

but (we can assume that the following average is a positive real number)

\frac{1}{K|A|}\sum_{k=1}^{K}\sum_{a\in A}z_{a,k}>\varepsilon.

In fact, we can assume the even stronger

\left|\frac{1}{K|A|}\sum_{k=1}^{K}\sum_{a\in A}z_{a,k}-\varepsilon\right|<\delta;

This can be achieved by adding some sequences $(z_{a,k})_{a\in G}$ ( $k=K+1,\dots,K+K^{\prime}$ ) with $z_{a,k}=0$ for all $a\in G$ ⁸⁸8It may also be necessary to change $K$ by a multiple of $K$ , by repeating each sequence $(z_{a,k})_{a}$ ( $k=1,\dots,K$ ) several times. We will now prove the following:

\neg

4’.

There is $\varepsilon>0$ such that, for any finite sets $A\subseteq G$ and $H_{0}\subseteq H$ and for any $L\in\mathbb{N},\delta>0$ there exist $J\in\mathbb{N}$ and sequences $(w_{a,j})_{a\in G}$ in $\mathbb{S}^{1}$ , for $j=1,\dots,J$ , such that

\left|\frac{1}{J|A|}\sum_{j=1}^{J}\sum_{a\in A}w_{ha,j}^{l}\overline{w_{a,j}^{l}}\right|<\delta\text{ for all }h\in H_{0},l=1,\dots,L.

(13)

but

\left|\frac{\varepsilon}{2}-\frac{1}{J|A|}\sum_{j=1}^{J}\sum_{a\in A}w_{a,j}\right|<\delta.

(14)

First we note that $\neg$ 4’. implies $\neg$ 1, due to Theorem 1.14 applied to the Cesaro averages of $(z_{g})_{g}$ and $(z_{hg}^{l}\overline{z_{g}^{l}})_{g}$ , for $h\in H$ and $l\in\mathbb{Z}$ , and Weyl’s criterion. Let us now prove $\neg$ 4’. using a probabilistic trick from [Ru, Section 6]

Let $(z_{a,k})_{a\in G}$ , $k=1,\dots,K$ , be as above, and let $\delta_{1}>0$ . We will consider a family of independent random variables $(\xi_{a,k})_{a\in G,k=1,\dots,K}$ supported in $\mathbb{S}^{1}$ , with $\xi_{a,k}$ having density function $d_{a,k}:\mathbb{S}^{1}\to[0,1];\;z\mapsto 1+\text{Re}(z\overline{z_{a,k}})$ . Then,

\mathbb{E}(\xi_{a,k})=\frac{z_{a,k}}{2}\text{ and }\mathbb{E}(\xi_{a,k}^{n})=0\text{ if }n=\pm 2,\pm 3,\dots.

(15)

Equation 15 can be proved when $z_{a,k}=1$ integrating, as we have $\int_{0}^{1}e^{2\pi ix}(1+\cos(2\pi x))=\frac{1}{2}$ and for $n=\pm 2,\pm 3,\dots$ , $\int_{0}^{1}e^{2\pi inx}(1+\cos(2\pi x))=0$ . For other values of $z_{a,k}$ one can change variables to $w=z\overline{z_{a,k}}$ . So we have

\mathbb{E}(\xi_{ha,k}\overline{\xi_{a,k}})=\frac{z_{ha,k}\overline{z_{a,k}}}{4}\text{ and }\mathbb{E}(\xi_{ha,k}^{l}\overline{\xi_{a,k}^{l}})=0\text{ if }l=\pm 2,\pm 3,\dots.

(16)

Now, for each $m\in\mathbb{N}$ we define a sequence $(z_{a,k,m})_{a\in G}$ by choosing, independently for all $k=1,\dots,K,a\in G$ and $m\in\mathbb{N}$ , a complex number $z_{a,k,m}\in\mathbb{S}^{1}$ according to the distribution of $\xi_{a,k}$ . Then the strong law of large numbers implies that with probability $1$ we will have, for all $k=1,\dots,K,l\in\mathbb{Z},h\in H$ and $a\in G$ ,

\lim_{M\to\infty}\left|\mathbb{E}\left(\xi_{ha,k}^{l}\overline{\xi_{a,k}^{l}}\right)-\frac{1}{M}\sum_{m=1}^{M}z_{ha,k,m}^{l}\overline{z_{a,k,m}^{l}}\right|=0.

(17)

So we can fix a family $(z_{a,k,m})_{a\in G;k=1,\dots,K;m\in\mathbb{N}}$ such that Equation 17 holds for all $k,l,h,a$ as above. Then, taking averages over all $a\in A$ and $k=1,\dots,K$ , we obtain that for all $h\in H_{0}$ and $l\in\mathbb{Z}$

\lim_{M\to\infty}\left|\frac{1}{K|A|}\sum_{k=1}^{K}\sum_{a\in A}\mathbb{E}\left(\xi_{ha,k}^{l}\overline{\xi_{a,k}^{l}}\right)-\frac{1}{MK|A|}\sum_{k=1}^{K}\sum_{a\in A}\sum_{m=1}^{M}z_{ha,k,m}^{l}\overline{z_{a,k,m}^{l}}\right|=0.

Notice that, due to Equation 16, for $l>1$ the expression in the LHS is $0$ , and for $l=1$ it is $\frac{1}{K|A|}\sum_{k=1}^{K}\sum_{a\in A}\frac{z_{ha,k}\overline{z_{a,k}}}{4}$ , which has norm $<\frac{\delta}{4}$ . So taking some big enough value $M$ , taking the sequences $(w_{a,j})_{a\in G}$ to be the sequences $(z_{a,k,m})_{a\in G}$ , for $k=1,\dots,K$ and $m=1,\dots,M$ , Equation 13 will be satisfied with $J=KM$ for all $h\in H_{0}$ and $l=1,\dots,L$ . We can check similarly that, for a big enough value of $M$ , Equation 14 will be satisfied by the sequences $(z_{a,k,m})_{a\in G}$ , for $k=1,\dots,K$ and $m=1,\dots,M$ , so we are done. ∎

As a consequence of Theorem 3.1 2 $\iff$ 3, we see that there are two equivalent definitions of $F$ -vdC sets, one using sequences of complex numbers in $\mathbb{S}^{1}$ and one using sequences in $\mathbb{D}$ . This can be seen as a consequence of the fact that $\mathbb{D}$ is the convex hull of $\mathbb{S}^{1}$ . In fact, using the same idea of the proof of 1 $\implies$ 5 in Theorem 3.1, we prove in 3.5 below that there is a close relationship between Cesaro averages of sequences taking values in a compact set $D\subseteq\mathbb{C}$ , and Cesaro averages of sequences in the convex hull of $D$ .

Proposition 3.5.

Let $G$ be a countably infinite amenable group with a Følner sequence $F=(F_{N})_{N}$ . Let $D\subseteq\mathbb{C}$ be compact and let $C\subseteq\mathbb{C}$ be the convex hull of $D$ . Then for any sequence $(z_{g})_{g\in G}$ of complex numbers in $C$ there is a sequence $(w_{g})_{g\in G}$ in $D$ such that, for any $k\in\mathbb{N}$ and any pairwise distinct elements $h_{1},\dots,h_{k}\in G$ , we have

\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}w_{h_{1}g}\cdots w_{h_{k}g}=\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}z_{h_{1}g}\cdots z_{h_{k}g},

whenever the right hand side is defined.

Proof.

Consider a list $(h_{l,1},\dots,h_{l,j_{l}})$ , $l\in\mathbb{N}$ , of all the finite sequences of pairwise distinct elements of $G$ , and let $p_{l}:\mathbb{C}^{j_{l}}\to\mathbb{C}$ ; $p_{l}(z_{1},\dots,z_{j_{l}})=z_{1}z_{2}\cdots z_{j_{l}}$ .

Consider a Følner subsequence $(F_{N_{i}})_{i}$ of $F$ such that for all $l\in\mathbb{N}$ , the limit

\gamma(l):=\lim_{i\to\infty}\frac{1}{|F_{N_{i}}|}\sum_{g\in F_{N_{i}}}p_{l}(z_{h_{l,1}g},\dots,z_{h_{l,j_{l}}g})

is defined. Then by 2.1, for all $A\subseteq G$ finite and for all $L\in\mathbb{N},\delta>0$ there exist some $K\in\mathbb{N}$ and some sequences $(z_{g,k})_{g\in G}$ in $C$ , for $k=1,\dots,K$ , such that for all $l=1,\dots,L$ we have

\left|\gamma(l)-\frac{1}{K|A|}\sum_{k=1}^{K}\sum_{g\in A}z_{h_{l,1}g,k}\dots z_{h_{l,j_{l}}g,k}\right|<\delta.

(18)

Claim 3.6.

For every sequence $\mathbf{x}=(x_{g})_{g\in G}$ taking values in $C$ , any finite $A\subseteq G$ and any $L\in\mathbb{N},\delta>0$ there is some $K_{\mathbf{x}}\in\mathbb{N}$ and sequences $(x_{g,k})_{g\in G}$ in $D$ , $k=1,\dots,K_{\mathbf{x}}$ , such that for all $g\in A$ and all $l=1,\dots,L$ we have

\left|\frac{1}{|A|}\sum_{g\in A}x_{h_{l,1}g}\cdots x_{h_{l,j_{l}}g}-\frac{1}{K_{\mathbf{x}}|A|}\sum_{k=1}^{K_{\mathbf{x}}}x_{h_{l,1}g,k}\cdots x_{h_{l,j_{l}}g,k}\right|<\delta.

Suppose that 3.6 is true. It then follows easily from 3.6 and Equation 18 that for all $A\subseteq G$ finite and for all $L\in\mathbb{N},\delta>0$ there exist some $K\in\mathbb{N}$ and some sequences $(z_{g,k})_{g\in G}$ in $D$ , for $k=1,\dots,K$ , such that for all $l=1,\dots,L$ we have

\left|\gamma(l)-\frac{1}{K|A|}\sum_{k=1}^{K}\sum_{g\in A}z_{h_{l,1}g,k}\dots z_{h_{l,j_{l}}g,k}\right|<\delta.

Thus, by 2.1 there exists a sequence $(w_{g})_{g\in G}$ of elements of $D$ such that, for all $l\in\mathbb{N}$ ,

\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}w_{h_{l,1}g}\cdots w_{h_{l,j_{l}}g}=\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}p_{l}(w_{h_{l,1}g},\dots,w_{h_{l,j_{l}}g})=\gamma(l),

so we are done.

It only remains to prove 3.6, so let $\mathbf{x}=(x_{g})_{g\in G}$ take values in $C$ . Note that the set of extreme points of $C$ is contained in $D$ , so by Choquet’s theorem, for every $x\in C$ there is a probability measure $\mu$ supported in $D$ and with average $x$ . So we can consider for each $g\in G$ a random variable $\xi_{g}$ supported in $D$ and satisfying $\mathbb{E}(\xi_{g})=x_{g}$ . Note that, if the variables $(\xi_{g})_{g\in G}$ are pairwise independent and $h_{1},\dots,h_{k}\in G$ are distinct, then we have

\mathbb{E}(\xi_{h_{1}}\cdots\xi_{h_{k}})=\mathbb{E}(\xi_{h_{1}})\cdots\mathbb{E}(\xi_{h_{k}})=x_{h_{1}}\cdots x_{h_{k}}.

Now for each $k\in\mathbb{N}$ we will choose a sequence $(x_{g,k})_{g\in G}$ , where the variables $x_{g,k}$ are chosen pairwise independently and with distribution $\xi_{g}$ . Then, for all $A\subseteq G$ finite, $L\in\mathbb{N}$ and $\delta>0$ , by the central limit theorem we will have with probability $1$ that, for big enough $K$ ,

\left|\frac{1}{|A|}\sum_{g\in A}x_{h_{l,1}g}\cdots x_{h_{l,j_{l}}g}-\frac{1}{K|A|}\sum_{k=1}^{K}x_{h_{l,1}g,k}\cdots x_{h_{l,j_{l}}g,k}\right|<\delta,

concluding the proof.∎

Applying 3.5 to the set $D=\{0,1\}$ and Theorem 1.14, one obtains the following:

Corollary 3.7.

Let $G$ be a countably infinite amenable group with a Følner sequence $(F_{N})_{N}$ . For every m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and every measurable $f:X\to[0,1]$ there exists a subset $A\subseteq G$ such that for all $k\in\mathbb{N}$ and all distinct $h_{1},\dots,h_{k}\in G$ we have

d_{F}\left(h_{1}^{-1}A\cap\dots\cap h_{k}^{-1}A\right)=\int_{X}f(T_{h_{1}}(x))\cdots f(T_{h_{k}}(x))d\mu.\qed

4 Spectral Characterization of vdC sets in abelian groups

In this section we prove a spectral criterion for vdC sets in countable abelian groups (Theorem 1.7), which is a direct generalization of the spectral criterion obtained in [Ru, Theorem 1]; we state it in a fashion similar to [BL, Theorem 8]. Theorem 1.7 implies that the notion of vdC set in $\mathbb{Z}^{d}$ defined in [BL] is equivalent to our notion of vdC set, even if it is not defined in terms of a Følner sequence (they use a Følner net instead). Also see [FT, Theorem 4.3] for a different proof of Theorem 1.7, shorter than the one included here.

Theorem 1.7.

Let $G$ be a countable abelian group. A set $H\subseteq G$ is vdC in $G$ iff any Borel probability measure $\mu$ in $\widehat{G}$ with $\widehat{\mu}(h)=0\;\forall h\in H$ satisfies $\mu(\{0\})=0$ .

As in [BL, Theorem 1.8] it follows from Theorem 1.7 that, if $H$ is vdC in $G$ and a Borel probability measure $\mu$ in $\widehat{G}$ satisfies $\widehat{\mu}(h)=0$ for all $h\in H$ , then $\mu(\{\gamma\})=0$ for all $\gamma\in\widehat{G}$ .

Proof.

\implies

Suppose there is a probability measure $\mu$ in $\widehat{G}$ such that $\widehat{\mu}(h)=0$ for all $h\in H$ but $\mu(\{0\})=\lambda>0$ .

Consider a sequence of finitely supported measures $(\mu_{N})_{N}$ which converge weakly to $\mu$ ; we can suppose $\mu_{N}(\{1\})\to\lambda$ . Specifically, $\mu_{N}$ will be an average of Dirac measures

\mu_{N}:=\frac{1}{u_{N}}\sum_{i=1}^{u_{N}}\delta_{x_{N,i}},

for some natural numbers $(u_{N})_{N}\to\infty$ and $x_{N,1},\dots,x_{N,u_{N}}\in\widehat{G}$ , so that $x_{N,i}=1\in\widehat{G}$ iff $i\leq u_{N}\lambda$ . The fact that $\mu_{N}\to\mu$ weakly implies that for all $h\in H$ (seeing $h$ as a map $h:\widehat{G}\to\mathbb{S}^{1}$ ) we have

\lim_{N}\frac{1}{u_{N}}\sum_{i=1}^{u_{N}}x_{N,i}(h)=\lim_{N}\int_{\widehat{G}}hd\mu_{N}=\int_{\widehat{G}}hd\mu=\widehat{\mu}(h)=0.

(19)

We will prove that, letting $\varepsilon<\lambda$ , Item 5 of Theorem 3.1 is not satisfied. So let $A\subseteq G$ and $H_{0}\subseteq H$ be finite and let $(F_{N})_{N}$ be a Følner sequence in $G$ . For each $N$ we consider the sequences $(x_{N,i}(ag))_{a\in A}$ for all $i=1,\dots,u_{N}$ and $g\in F_{N}$ . It will be enough to prove that

\lim_{N\to\infty}\frac{1}{u_{N}|F_{N}|\cdot|A|}\sum_{a\in A,g\in F_{N},i=1,\dots,u_{N}}x_{N,i}(ag)=\lambda,

(20)

and for all $h\in H_{0}$ ,

\lim_{N\to\infty}\frac{1}{u_{N}|F_{N}|\cdot|A|}\sum_{a\in A,g\in F_{N},i=1,\dots,u_{N}}x_{N,i}(hag)\overline{x_{N,i}(ag)}=0.

(21)

Equation 21 is a direct consequence of Equation 19 and the fact that $x_{N,i}(hag)\overline{x_{N,i}(ag)}=x_{N,i}(h)$ . To prove Equation 20 first note that, as $(F_{N})_{N}$ is a Følner sequence, Equation 20 is equivalent to

\lim_{N\to\infty}\frac{1}{u_{N}|F_{N}|}\sum_{g\in F_{N},i=1,\dots,u_{N}}x_{N,i}(g)=\lambda.

(22)

Now, let $\delta>0$ and consider a neighborhood $U$ of $1\in\widehat{G}$ with $\mu\left(\overline{U}\right)<\lambda+\delta$ . After a reordering, we can assume that for big enough $N$ the points $x_{N,i}$ are in $\widehat{G}\setminus U$ for all $i\geq u_{N}(\lambda+2\delta)$ .

Claim 4.1.

There exists $M\in\mathbb{N}$ such that, for all $x\in\widehat{G}\setminus U$ and for all $N\geq M$ , $\left|\frac{1}{|F_{N}|}\sum_{g\in F_{N}}x(g)\right|<\delta$ .

4.1 implies Equation 22, because $\delta$ is arbitrary and by 4.1 we have that for all $i\leq u_{N}\lambda$ , the average $\frac{1}{|F_{N}|}\sum_{g\in F_{N}}x_{N,i}(g)$ is exactly $1$ (as $x_{N,i}=1$ ), and for all $i\geq u_{N}(\lambda+2\delta)$ and big enough $N$ , $\frac{1}{|F_{N}|}\sum_{g\in F_{N}}x_{N,i}(g)$ has norm at most $\delta$ .

To prove 4.1 let $\varepsilon>0$ and $g_{1},\dots,g_{k}\in G$ be such that for all $x\in\widehat{G}\setminus U$ we have $|x(g_{i})-1|>\varepsilon$ for some $i\in\{1,\dots,k\}$ . Now note that for all $N\in\mathbb{N}$ , $x\in\widehat{G}\setminus U$ and $i=1,\dots,k$ ,

|x(g_{i})-1|\cdot\left|\frac{1}{|F_{N}|}\sum_{g\in F_{N}}x(g)\right|=\frac{1}{|F_{N}|}\left|\sum_{g\in F_{N}}x(g_{i}g)-\sum_{g\in F_{N}}x(g)\right|\leq\frac{1}{|F_{N}|}\sum_{g\in F_{N}\Delta g_{i}F_{N}}\left|x(g)\right|.

As $\lim_{N}\frac{|F_{N}\Delta g_{i}F_{N}|}{|F_{N}|}=0$ for all $i$ , there is some $M$ such that for all $N\geq M$ and for all $i$ , the right hand side is $<\delta\varepsilon$ for all $x\in\widehat{G}\setminus U$ . Thus, $M$ satisfies 4.1.

\impliedby

The proof of [BL, Theorem 1.8, S2 $\implies$ S1] can be adapted to any countable abelian group; instead of [BL, Lemma 1.9] one needs to prove a statement of the form

Lemma 4.2.

Let $(u_{g})_{g\in G}$ be a sequence of complex numbers in $\mathbb{D}$ and let $(F_{N})_{N}$ be a Følner sequence such that, for all $h\in G$ , $\gamma(h)=\lim_{N}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}u_{hg}\overline{u_{g}}$ is defined. Then there is a positive measure $\sigma$ on $\widehat{G}$ such that $\widehat{\sigma}(h)=\gamma(h)$ for all $h$ , and

\limsup_{N}\frac{1}{|F_{N}|}\left|\sum_{g\in F_{N}}u_{g}\right|\leq\sqrt{\sigma(\{0\})}.

And the lemma can also be proved similarly to [BL, Lemma 1.9], changing the functions $g_{N},h_{N}$ from [BL] by

g_{N}(x)=\frac{1}{|F_{N}|}\left|\sum_{g\in F_{N}}z_{g}x(g)\right|^{2}\text{ and }h_{N}(x)=\frac{1}{|F_{N}|}\left|\sum_{g\in F_{N}}g(x)\right|^{2},

and one also needs to check that [CKM, Theorem 2] works for any countable abelian group:

Lemma 4.3 (Cf. [CKM, Theorem 2]).

Let $G$ be a countable abelian group, let $\widehat{G}$ be its dual. Let $(\mu_{n})_{n},(\nu_{n})_{n},\mu,\nu$ be Borel probability measures in $\widehat{G}$ such that $\mu_{n}\to\mu$ weakly, $\nu_{n}\to\nu$ weakly. Then

\rho(\mu,\nu)\geq\limsup_{n}\rho(\mu_{n},\nu_{n}).

(23)

The same proof of [CKM, Theorem 2] is valid; the proof uses the existence of Radon-Nikodym derivatives and a countable partition of unity $f_{j}:T\to\mathbb{R}$ , for $j\in\mathbb{Z}$ . These partitions of unity always exist for outer regular Radon measures (see [RuRC, Theorem 3.14]), so they exist for any Borel probability measure in $\widehat{G}$ .∎

5 Properties of vdC sets

In [BL] and [Ru], several properties of the family of vdC subsets of $\mathbb{Z}^{d}$ were proved. In this section we check that many of these properties hold for vdC sets in any countable group. Some of the statements about vdC sets follow from statements about sets of recurrence and the fact that any vdC set is a set of recurrence. Other properties can be proved in the same way as their analogs for sets of recurrence, even if they are not directly implied by them.

We also prove 1.8, a finitistic criterion for the notion of vdC set which will be needed to prove 5.4 below.

Remark 5.1.

All the properties below are also satisfied for sets of operatorial recurrence, as defined in [FT] (see [FT, Section 5]), with the proofs of the properties being essentially the same as for vdC sets.

Proposition 5.2 (Ramsey Property of vdC sets, cf. [Ru, Cor. 1]).

Let $G$ be a countably infinite group and let $H,H_{1},H_{2}\subseteq G$ satisfy $H=H_{1}\cup H_{2}$ . If $H$ is a vdC set in $G$ , then either $H_{1}$ or $H_{2}$ are vdC sets in $G$ .

Proof.

Suppose that $H_{1},H_{2}$ are not vdC sets in $G$ . Then for $i=1,2$ there are measure preserving systems $(X_{i},\mathcal{B}_{i},\mu_{i},(T_{i}^{g})_{g\in G})$ , and functions $f_{i}\in L^{\infty}(\mu_{i})$ such that

\int_{X_{i}}f_{i}(T_{i}^{h}x)\cdot\overline{f_{i}(x)}d\mu_{i}(x)=0\text{ for all }h\in H_{i},

but

\int_{X_{i}}f_{i}(x)d\mu_{i}(x)\neq 0.

But then, considering the function $f:X_{1}\times X_{2}\to\mathbb{C};f(x_{1},x_{2})=f_{1}(x_{1})\cdot f_{2}(x_{2})$ , we have by Fubini’s theorem

\int_{X_{1}\times X_{2}}f_{1}(T_{1}^{h}x_{1})f_{2}(T_{2}^{h}x_{2})\overline{f_{1}(x_{1})}\overline{f_{2}(x_{2})}d(\mu_{1}\times\mu_{2})(x_{1},x_{2})=0\text{ for all }h\in H_{1}\cup H_{2},

but

\int_{X_{1}\times X_{2}}f_{1}(x_{1})f_{2}(x_{2})d(\mu_{1}\times\mu_{2})(x_{1},x_{2})\neq 0,

so $H_{1}\cup H_{2}$ is not vdC in $G$ . ∎

We will need the following in order to prove 5.4.

Proposition 5.3.

Let $G$ be a countably infinite group, and let $H\subseteq G\setminus\{1\}$ be finite. Then $H$ is not a set of recurrence in $G$ , so it is not vdC.

Proof.

Consider the $\left(\frac{1}{2},\frac{1}{2}\right)$ -Bernoulli scheme in $\{0,1\}^{G}$ with the action $(T_{g})_{g\in G}$ of $G$ given by $T_{g}((x_{a})_{a\in G})=(x_{ag})_{a\in G}$ . Let $A=\{(x_{a})_{a};x_{1}=0\}\subseteq\{0,1\}^{G}$ and let $B=A\setminus\cup_{h\in H}T_{-h}A$ . As $H$ is finite, $B$ has positive measure, but clearly $\mu(B\cap T_{-h}B)=0$ for all $h\in H$ , so we are done. ∎

The following generalizes [Ru, Cor. 3], and has a similar proof.

Proposition 5.4.

Let $G$ be a countably infinite group and let $H\subseteq G\setminus\{1\}$ be a vdC set in $G$ . Then we can find infinitely many disjoint vdC subsets of $H$ .

Proof of 5.4.

It will be enough to prove that there are two disjoint vdC subsets $H^{\prime},H^{\prime\prime}$ of $H$ .

We will define a disjoint sequence $H_{1},H_{2},\dots$ of finite subsets of $H$ by recursion. Suppose $H_{1},\dots,H_{n-1}$ are given; notice that $\bigcup_{i=1}^{n-1}H_{i}$ is finite, so it is not a vdC set. So by 5.2, $H\setminus\bigcup_{i=1}^{n-1}H_{i}$ is a vdC set. Then by 1.8 we can then define $H_{n}$ to be a finite subset of $H\setminus\bigcup_{i=1}^{n-1}H_{i}$ such that for some constant $\delta_{n}>0$ and for any m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ , $f\in L^{\infty}(X,\mu)$ we have

\left|\int_{X}f(T_{h}x)\overline{f(x)}d\mu(x)\right|<\delta_{n}\;\forall h\in H_{n}\text{ implies }\left|\int_{X}fd\mu\right|<\frac{1}{n}.

Now let $H^{\prime}=\bigcup_{n\in\mathbb{N}}H_{2n-1}$ and $H^{\prime\prime}=\bigcup_{n\in\mathbb{N}}H_{2n}$ . It is then clear that both $H^{\prime}$ and $H^{\prime\prime}$ satisfy the definition of vdC set in $G$ , so we are done. ∎

The following generalizes [BL, Corollary 1.15.2] to countable amenable groups.

Theorem 5.5.

Let $S$ be a subgroup of a countable amenable group $G$ , let $H\subseteq S$ . Then $H$ is vdC in $S$ iff it is vdC in $G$ .

Proof.

If $H$ is not vdC in $G$ , then it is not vdC in $S$ , as any measure preserving action $(T_{g})_{g\in G}$ on a measure space restricts to a measure preserving action $(T_{g})_{g\in S}$ on the same measure space.

The fact that if $H$ is not vdC in $S$ then it is not vdC in $G$ can be deduced from Item 5 of Theorem 3.1: indeed, let $\varepsilon$ be as in Item 5 (applied to the group $S$ ) and consider any $A\subseteq G$ and $H_{0}\subseteq H$ finite and any $\delta>0$ .

We can express $A=A_{1}\cup\dots\cup A_{m}$ , where the $A_{i}$ are pairwise disjoint and of the form $A_{i}=A\cap(Sg_{i})$ , for some $g_{i}\in G$ . Thus, $Ag_{i}^{-1}\subseteq S$ for all $i$ .

Now, for each $i$ we know that there exist $K_{i}\in\mathbb{N}$ and $G$ -sequences $(z_{i,a,k})_{a\in S}$ in $\mathbb{D}$ , for $k=1,\dots,K_{i}$ , such that

\left|\frac{1}{K_{i}|A_{i}|}\sum_{k=1}^{K_{i}}\sum_{a\in A_{i}g_{i}^{-1}}z_{i,ha,k}\overline{z_{i,a,k}}\right|<\delta\text{ for all }h\in H_{0},

(24)

but

\left|\frac{1}{K_{i}|A_{i}|}\sum_{k=1}^{K_{i}}\sum_{a\in A_{i}g_{i}^{-1}}z_{i,a,k}\right|>\varepsilon.

(25)

Note that we can assume $K_{1},K_{2},\dots,K_{m}$ are all equal to some number $K$ (e.g. taking $K$ to be the least common multiple of all of them) and that $\frac{1}{K|A_{i}|}\sum_{k=1}^{K}\sum_{a\in A_{i}g_{i}^{-1}}z_{i,a,k}$ is a positive real number for all $i=1,\dots,m$ (multiplying the sequences $(z_{i,a,k})$ by some complex number of norm $1$ if needed).

Finally, define for each $k=1,\dots,K$ a sequence $(z_{a,k})_{a\in G}$ by $z_{a,k}=z_{i,ag_{i}^{-1},k}$ for $a\in Sg_{i}$ and by $z_{g}=0$ elsewhere. This sequence will satisfy Item 5 of Theorem 3.1 (by taking averages of Equations 24 and 25 for $k=1,\dots,K$ ), so we are done. ∎

The following is proved in [BL, Cor. 1.15.1] for vdC sets in $\mathbb{Z}^{d}$ using the spectral criterion; using Definition 1.6 instead we prove it for any countable group.

Proposition 5.6.

Let $\pi:G\to S$ be a group homomorphism, let $H$ be a vdC set in $G$ . Then $\pi(H)$ is a vdC set in $S$ .

Proof.

The contra-positive of the claim follows easily from the fact that any measure preserving action $(T_{s})_{s\in S}$ on a mps $(X,\mathcal{A},\mu)$ induces a measure preserving action $(S_{g})_{g\in G}$ on $(X,\mathcal{A},\mu)$ by $S_{g}=T_{\pi(g)}$ . ∎

Remark 5.7.

In 5.6, $\pi(H)$ may be a vdC set even if $H$ is not. Indeed, the set $\{(n,1);n\in\mathbb{N}\}$ is not a set of recurrence in $\mathbb{Z}^{2}$ (this follows from 5.8 below), but its projection to the first coordinate is a vdC set.

The following generalizes [BL, Corollary 1.16].

Corollary 5.8.

Let $G$ be a countable group and let $S$ be a finite index subgroup of $G$ . Then $G\setminus S$ is not a set of recurrence in $G$ , so it is not vdC in $G$ .

Proof.

Consider the action of $G$ on the finite set $G/S$ of left-cosets of $S$ , where we give $G/S$ the uniform probability measure $\mu$ . The set $\{S\}\subseteq G/S$ has positive measure, but for all $g\in G\setminus S$ we have $g\{S\}\cap\{S\}=\varnothing$ . ∎

We finally prove that difference sets are nice vdC (see e.g. [Fa, Lemma 5.2.8] for the case $G=\mathbb{Z}$ ). The proof of 5.9 is just the proof that any set of differences is a set of recurrence⁹⁹9A slight modification of this proof shows that $AA^{-1}$ is a set of nice recurrence (see Definition 1.9), which is already found in [Fu, Page 74] for the case $G=\mathbb{Z}$ .

Proposition 5.9.

Let $G$ be a countable group and let $A\subseteq G$ be infinite. Then the difference set $AA^{-1}=\{ba^{-1};a,b\in A,a\neq b\}$ is a nice vdC set in $G$ .

Proof.

Suppose that for some m.p.s. $(X,\mathcal{A},\mu,(T_{g})_{g\in G})$ and some function $f\in L^{\infty}(\mu)$ we have

\left|\int_{X}fd\mu\right|^{2}>\limsup_{h\in AA^{-1}}\left|\int_{X}f(T_{h}x)\overline{f(x)}d\mu(x)\right|.

(26)

So for some $\lambda<\left|\int_{X}fd\mu\right|^{2}$ and some finite subset $B\subseteq AA^{-1}$ we have

\left|\int_{X}f(T_{a}(x))\cdot\overline{f(T_{b}(x))}d\mu(x)\right|<\lambda\text{ for all }a,b\in A\text{ with }ab^{-1}\not\in B.

Then for any $N\in\mathbb{N}$ , letting $A_{0}$ be a subset of $A$ with $N$ elements, we have

N^{2}\left|\int_{X}fd\mu\right|^{2}=\frac{\left|\left\langle\sum_{a\in A_{0}}T_{a}f,1\right\rangle\right|^{2}}{\langle 1,1\rangle}\leq\left\langle\sum_{a\in A_{0}}T_{a}f,\sum_{a\in A_{0}}T_{a}f\right\rangle\leq N\cdot|B|\cdot\|f\|^{2}_{L^{2}(X,\mu)}+N^{2}\lambda.

This is a contradiction for big enough $N$ , because $\lambda<|\int_{X}fd\mu|^{2}$ . ∎

The corollary below was suggested by V. Bergelson. Before stating it, recall that a subset $H$ of a countable group $G$ is thick when for any finite $A\subseteq G$ , $H$ contains a right translate of $A$ . In particular, a subset of a countable amenable group has upper Banach density $1$ iff it is thick. Also note that a set $H$ is vdC in $G$ if and only if $H\cup H^{-1}$ is vdC in $G$ ; this follows from the fact that for any m.p.s. $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and any $f\in L^{\infty}(X,\mu)$ we have $\int_{X}f(T_{h}x)\overline{f(x)}d\mu=\overline{\int_{X}f(T_{h^{-1}}x)\overline{f(x)}d\mu}$ .

Corollary 5.10.

If $G$ is a countable group and a subset $H\subseteq G$ is thick, then $H$ is a nice vdC set.

Proof.

If $H\subseteq G$ is thick, then there is an infinite set $A=\{a_{1},a_{2},\dots\}$ such that $H$ contains the set $\{a_{n}a_{m}^{-1};m>n\}$ . Indeed, we can define $(a_{n})_{n}$ be recursion by letting $a_{n}\neq a_{1},\dots,a_{n-1}$ be such that $A_{n}a_{n}^{-1}\subseteq H$ , where $A_{n}:=\{a_{1},\dots,a_{n-1}\}$ .

Thus $H\cup H^{-1}$ contains the set $AA^{-1}$ , so $H\cup H^{-1}$ is vdC, so $H$ is vdC. ∎

Proof of 1.8.

The most natural proof of this fact uses Loeb measures. In order to avoid using non-standard analysis, we will adapt the proof of [Fo, Lemma 6.4].

Let $H_{n}$ be an increasing sequence of finite sets with $H=\cup_{n}H_{n}$ . Suppose we have $\varepsilon>0$ and a sequence of m.p.s.’s $(X_{n},\mathcal{B}_{n},\mu_{n},(T_{n,g})_{g\in G})$ and measurable functions $f_{n}:X\to\mathbb{D}$ such that

\left|\int_{X_{n}}f_{n}(T_{n,h}x)\overline{f_{n}(x)}d\mu_{n}(x)\right|<\frac{1}{n}\;\forall h\in H_{n},\text{ but }\left|\int_{X_{n}}f_{n}(x)d\mu_{n}(x)\right|>\varepsilon.

We will then prove that $H$ is not vdC by constructing a m.p.s. $(Y,\mathcal{C},\nu,(S_{g})_{g\in G})$ and some measurable $f_{\infty}:Y\to\mathbb{D}$ such that

\left|\int_{Y}f_{\infty}(S_{h}y)\overline{f_{\infty}(y)}d\nu(y)\right|=0\;\forall h\in H,\text{ but }\left|\int_{Y}f_{\infty}(y)d\nu(y)\right|\geq\varepsilon.

(27)

To do it, first consider the (infinite) measure space $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ with $X$ being the disjoint union $\sqcup_{n\in\mathbb{N}}X_{n}$ , $\mathcal{B}:=\left\{\sqcup_{n}B_{n};B_{n}\in\mathcal{B}_{n}\;\forall n\right\}$ , $\mu(\sqcup_{n}B_{n}):=\sum_{n}\mu_{n}(B_{n})$ and $T_{g}(x)=T_{n,g}(x)$ if $x\in X_{n}$ . And let $f:X\to\mathbb{D}$ be given by $f|_{X_{n}}=f_{n}$ .

Let $K$ be the smallest sub- $C^{*}$ -algebra of $L^{\infty}(X)$ containing $1$ and $T_{g}f$ for all $g\in G$ . Then by the Gelfand representation theorem there is a $C^{*}$ -algebra isomorphism $\Phi:K\to C(Y)$ , where $Y$ is the spectrum of $K$ , a compact metrizable space whose elements are non-zero $*$ -homomorphisms $y:K\to\mathbb{C}$ . The isometry $\Phi$ is given by $\Phi(\varphi)(y)=y(\varphi)$ .

We let $\mathcal{C}$ be the Borel $\sigma$ -algebra of $Y$ . Now consider a Banach limit $L:l^{\infty}(\mathbb{N})\to\mathbb{C};(a_{n})_{n}\mapsto L-\lim_{n}a_{n}$ and define a norm $1$ positive functional $F:K\to\mathbb{C}$ by

F(\varphi)=L-\lim_{n}\int_{X_{n}}\varphi(x)d\mu_{n}(x).

This induces a probability measure $\nu$ on $Y$ by $\int_{Y}\Phi(\varphi)d\nu=F(\varphi)$ . Also, the action of $G$ on $K$ induces an action $(S_{g})_{g\in G}$ of $G$ on $Y$ by homeomorphisms by $(S_{g}y)(\varphi)=y(\varphi\circ T_{g})$ . $S_{g}$ is $\nu$ -preserving for all $g\in G$ , as for any $\varphi\in K$ we have

\int_{Y}\Phi(\varphi)(S_{g}(y))d\nu(y)=\int_{Y}\Phi(\varphi\circ T_{g})(y)d\nu(y)=F(\varphi\circ T_{g})=F(\varphi)=\int_{Y}\Phi(\varphi)(y)d\nu(y).

We now let $f_{\infty}:=\Phi(f)\in C(Y)$ and we check Equation 27: for all $h\in H$ ,

\int_{Y}\Phi(f)(S_{h}y)\overline{\Phi(f)(y)}d\mu(y)=\int_{Y}\Phi(f\circ T_{h})(y)\overline{\Phi(f)(y)}d\mu(y)\\ =F((f\circ T_{h})\cdot\overline{f})=L-\lim_{n}\int_{X_{n}}f(T_{n,h}(x))\overline{f_{n}(x)}d\mu_{n}(x)=0.

\left|\int_{Y}\Phi(f)(y)d\nu(y)\right|=\left|F(f)\right|=\left|L-\lim_{n}\int_{X_{n}}f(x)d\mu_{n}(x)\right|=L-\lim_{n}\left|\int_{X_{n}}f(x)d\mu_{n}(x)\right|\geq\varepsilon.

∎

References

[Av] J. Avigad. Inverting the Furstenberg correspondence. Discrete and Continuous Dynamical Systems, 32(10), 3421–3431, 2012.
[Be2] V. Bergelson. A Density Statement Generalizing Schur’s Theorem. Journal of Combinatorial Theory, Series A 43, pp. 338-343, 1986.
[Be5] V. Bergelson. Ergodic Ramsey Theory - An update. Ergodic Theory and $\mathbb{Z}^{d}$ Actions. London Mathematical Society Lecture Note Series. Cambridge University Press; 1996:1-62.
[BF] V. Bergelson, A. Ferré Moragues. An ergodic correspondence principle, invariant means and applications. Israel J. Math. 245 (2021), pp. 921–962.
[BF2] V. Bergelson, A. Ferré Moragues. Uniqueness of a Furstenberg system. Proceedings of the American Mathematical Society, 149(7), 2983–2997 (2021).
[BL] V. Bergelson, E. Lesigne. Van der Corput sets in $\mathbb{Z}^{d}$ . Colloq. Math., 110 (1) (2008), pp. 1-49.
[Bo] J. Bourgain. Ruzsa’s problem on sets of recurrence. Israel J. Math. 59 (1987), 151–166.
[CKM] J. Coquet, T. Kamae, M. Mendès France Sur la mesure spectrale de certaines suites arithmétiques. Bull. Soc. Math. France 105 (1977), 369–384.
[DHZ] T. Downarowicz, D. Huczek, G. Zhang. Tilings of amenable groups. J. Reine Angew. Math. 747 277–98, 2019.
[Fa] S. Farhangi. Topics in Ergodic Theory and Ramsey Theory. Ph.D thesis, The Ohio State University, 2022.
[Fo] A. H. Forrest. Recurrence in Dynamical Systems: A Combinatorial Approach. Ph.D thesis, The Ohio State University, 1990.
[FS] A. Fish, S. Skinner. An inverse of Furstenberg’s correspondence principle and applications to nice recurrence. arXiv preprint, arXiv:2407.19444.
[FT] S. Farhangi, R. Tucker-Drob. Van der Corput sets in amenable groups and beyond. To be uploaded to arXiv.
[Fu] H. Furstenberg. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton Univ Press (1981).
[KM] T. Kamae, M. Mendès France. Van der Corput’s difference theorem. Israel Journal of Mathematics, Vol. 31, Nos. 3-4, 1978.
[KN] L. Kuipers, H. Niederreiter. Uniform Distribution of Sequences. John Wiley (1974).
[Lin] E. Lindenstrauss. Pointwise theorems for amenable groups. Electronic Research Announcements of the American Mathematical Society, vol. 5, no. 12, June 1999, pp. 82–90.
[Mou] A. Mountakis. Distinguishing sets of strong recurrence from van der Corput Sets . Israel Journal of Mathematics TBD, 2024.
[RuFA] W. Rudin. Fourier Analysis on Groups. Interscience, New York, 1962.
[RuRC] W. Rudin. Real and Complex Analysis. 3rd edn.(McGraw-Hill Book Co., New York,1987).
[Ru] Ruzsa. Connections between the uniform distribution of a sequence and its differences. Topics in classical number theory Vol. 2 pg. 1419-1443 (1984).
[We] H. Weyl. Über die Gleichverteilung von Zahlen mod. Eins. Mathematische Annalen 77, 313-352 (1916).

		$\displaystyle\lim_{N}\frac{1}{\|K_{N}\|\cdot\|A\|}\sum_{k\in K_{N}}\sum_{g\in A}p_{l}(z_{h_{l,1}g,k},\dots,z_{h_{l,j_{l}}g,k})$
	$\displaystyle=$	$\displaystyle\lim_{N}\frac{1}{\|A\|}\sum_{g\in A}\int_{D^{G}}p_{l}(z_{h_{l,1}g},\dots,z_{h_{l,j_{l}}g})d\nu_{N}((z_{a})_{a})$
	$\displaystyle=$	$\displaystyle\frac{1}{\|A\|}\sum_{g\in A}\int_{D^{G}}p_{l}(z_{h_{l,1}g},\dots,z_{h_{l,j_{l}}g})d\nu((z_{a})_{a})$
	$\displaystyle=$	$\displaystyle\frac{1}{\|A\|}\sum_{g\in A}\int_{X}p_{l}(f(T_{h_{l,1}g}x),\dots,f(T_{h_{l,j_{l}}g}x))d\mu(x)$
	$\displaystyle=$	$\displaystyle\frac{1}{\|A\|}\sum_{g\in A}\int_{X}p_{l}(f(T_{h_{l,1}}x),\dots,f(T_{h_{l,j_{l}}}x))d\mu=\frac{1}{\|A\|}\sum_{g\in A}\gamma(l)=\gamma(l).$

An inverse of Furstenberg’s correspondence principle and applications to van der Corput sets

Abstract

1 Introduction

Definition 1.1 ([KM, Page 1]).

Theorem 1.2 (cf. [Ru, Theorem 1]).

Definition 1.3.

Definition 1.4 (cf. [BL, Page 44]).

Theorem 1.5 (See Theorem 3.1).

Definition 1.6.

Theorem 1.7 (cf. [BL, Theorem 1.8]).

Proposition 1.8.

Definition 1.9 (cf. [Be2, Definition 2.2]).

Definition 1.10 (cf. [BL, Definition 10]).

Definition 1.11 ([BF, Definitions 2.1, 2.2]).

Proposition 1.12.

Proposition 1.13.

Theorem 1.14.

Theorem 1.15 (Furstenberg correspondence principle, cf. [Be5, Theorem 1.8]).

Theorem 1.16.

Theorem 1.17.

Acknowledgements.

2 Correspondence between Cesaro and integral averages

Proposition 2.1.

Remark 2.2.

Proof of Theorem 1.14 from 2.1.

Proof of 1⟹\implies2.

Proof of 2⟹\implies3.

Proposition 2.3.

Proof.

Definition 2.4.

Remark 2.5.

Lemma 2.6.

Proof.

Definition 2.7 (cf. [DHZ, Defs 3.1,3.2]).

Definition 2.8 (cf. [DHZ, Page 17]).

Definition 2.9.

Theorem 2.10 (cf. [DHZ, Theorem 5.2]).

Definition 2.11.

Remark 2.12.

Lemma 2.13.

Proof.

Proof of Item 3⟹\impliesItem 1.

Theorem 2.14.

Proof.

Remark 2.15.

Proof of 1.12.

Proof of Theorem 1.17.

3 Characterizations of vdC sets in amenable groups

Theorem 3.1.

Remark 3.2.

Definition 3.3.

Proposition 3.4 (Weyl’s criterion for uniform distribution).

Proof of Theorem 3.1.

Proposition 3.5.

Proof.

Claim 3.6.

Corollary 3.7.

4 Spectral Characterization of vdC sets in abelian groups

Theorem 1.7.

Proof.

Claim 4.1.

Lemma 4.2.

Lemma 4.3 (Cf. [CKM, Theorem 2]).

5 Properties of vdC sets

Remark 5.1.

Proposition 5.2 (Ramsey Property of vdC sets, cf. [Ru, Cor. 1]).

Proof.

Proposition 5.3.

Proof.

Proposition 5.4.

Proof of 5.4.

Theorem 5.5.

Proof.

Proposition 5.6.

Proof.

Remark 5.7.

Corollary 5.8.

Proof.

Proposition 5.9.

Proof.

Proof of 1 $\implies$ 2.

Proof of 2 $\implies$ 3.

Proof of Item 3 $\implies$ Item 1.