# An ergodic correspondence principle, invariant means and applications

Vitaly Bergelson  Department of Mathematics, Ohio State University, Columbus,
OH 43210, USA<EMAIL_ADDRESS>and  Andreu Ferré Moragues
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA
<EMAIL_ADDRESS>
###### Abstract.

A theorem due to Hindman states that if $E$ is a subset of $\mathbb{N}$ with
$d^{*}(E)>0$, where $d^{*}$ denotes the upper Banach density, then for any
$\varepsilon>0$ there exists $N\in\mathbb{N}$ such that
$d^{*}\left(\bigcup_{i=1}^{N}(E-i)\right)>1-\varepsilon$. Curiously, this
result does not hold if one replaces the upper Banach density $d^{*}$ with the
upper density $\bar{d}$. Originally proved combinatorially, Hindman’s theorem
allows for a quick and easy proof using an _ergodic_ version of Furstenberg’s
correspondence principle. In this paper, we establish a variant of the ergodic
Furstenberg’s correspondence principle for general amenable (semi)-groups and
obtain some new applications, which include a refinement and a generalization
of Hindman’s theorem and a characterization of countable amenable minimally
almost periodic groups.

##  1\. Introduction

Many results in additive combinatorics are of the form: If
$E\subseteq\mathbb{N}=\\{1,2,\dots,\\}$ is a "large" set, then $E$ is "highly
organized". For example, the celebrated Szemerédi theorem [Sz] states that if
$E$ has positive upper density,
$\bar{d}(E):=\limsup_{N\to\infty}\frac{|E\cap\\{1,\dots,N\\}|}{N}>0$, then $E$
is "AP-rich", meaning that $E$ contains arbitrarily long arithmetic
progressions. An equivalent form of Szemerédi’s theorem is the following: if
$E\subseteq\mathbb{N}$ has positive upper Banach density, i.e.
$d^{*}(E):=\limsup_{N-M\to\infty}\frac{|E\cap\\{M,\dots,N-1\\}|}{N-M}>0$, then
$E$ is AP-rich 111Indeed, one can show that both the $d^{*}$ and $\bar{d}$
versions of Szemerédi’s Theorem are equivalent to the original ”finitistic”
version in [Sz]. See also Theorem 1.5 in [B1]. In [F3], Furstenberg obtained a
proof of Szemerédi’s theorem via the ergodic Szemerédi theorem (EST), which
states that for any probability measure preserving system
$(X,\mathcal{B},\mu,T)$, any $A\in\mathcal{B}$ with $\mu(A)>0$, and any
$k\in\mathbb{N}$, there exists $n\in\mathbb{N}$ such that

$\mu(A\cap T^{-n}A\cap\dots\cap T^{-kn}A)>0.$

Furstenberg’s approach (see [F3]) to deriving Szemerédi’s theorem from his EST
can be described as follows. Suppose that $\bar{d}(E)>0$. Viewing the $0$-$1$
valued sequence $\xi(m)=\mathbb{1}_{E}(m)$, $m\in\mathbb{Z}$, as an element of
the symbolic space $\\{0,1\\}^{\mathbb{Z}}$, and denoting by $T$ the shift
transformation $Tx(n)=x(n+1)$ for all $n\in\mathbb{Z}$, Furstenberg
establishes the existence of a $T$-invariant measure $\mu$ on
$\\{0,1\\}^{\mathbb{Z}}$ as follows. First, by a diagonalization argument, we
can find a common subsequence $(N_{k})$ so that the following limit exists for
a countable dense subset of $C(\\{0,1\\}^{\mathbb{Z}})$

(1.1) $L(f):=\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n=1}^{N_{k}}f(T^{n}\xi).$

Applying a standard approximation argument, we see that formula (1.1) holds
for all $f\in C(\\{0,1\\}^{\mathbb{Z}})$. Now, $L$ is a positive, normalized
functional, and so by Riesz’s representation theorem, there is a Borel
probability measure $\mu$ on $\\{0,1\\}^{\mathbb{Z}}$ such that
$L(f)=\int_{\\{0,1\\}^{\mathbb{Z}}}f\ d\mu$. Let
$X:=\overline{\\{T^{n}\xi:n\in\mathbb{Z}\\}}$ be the orbit closure of $\xi$.
Observe that $\mu$ is supported on $X$. Letting
$A:=X\cap\\{x\in\\{0,1\\}^{\mathbb{Z}}:x(0)=1\\}$, we get
$\mu(A)=\int_{\\{0,1\\}^{\mathbb{Z}}}\varphi\ d\mu=L(\varphi)=\bar{d}(E)>0$,
for $\varphi(x):=x(0)$. (Note that the fact $L(\varphi)=\bar{d}(E)$ follows
from (1.1)).

By the EST there exists some $n\in\mathbb{N}$ such that $\mu(A\cap
T^{-n}A\cap\dots\cap T^{-kn}A)>0$. Then, for any $x\in A\cap
T^{-n}A\cap\dots\cap T^{-kn}A$ we have $x(0)=1,x(n)=1,\dots,x(kn)=1$. Since
$x\in X$, we can choose $l\in\mathbb{N}$ such that $T^{l}\xi$ and $x$ are as
close as we wish. This implies that for some $l\in\mathbb{N}$,
$\mathbb{1}_{E}(l)=\mathbb{1}_{E}(l+n)=\dots=\mathbb{1}_{E}(l+kn)=1$ and hence
$E$ contains the arithmetic progression $\\{l,l+n,\dots,l+kn\\}$ of length
$k+1$.

One can check that the functional $L$ satisfies the identity

(1.2)
$\lim_{k\to\infty}\frac{|E\cap(E-n)\cap\dots\cap(E-kn)\cap[1,N_{k}]|}{N_{k}}=\\\
L(\varphi\cdot T^{n}\varphi\cdot\dotso\cdot
T^{kn}\varphi)=\int_{\\{0,1\\}^{\mathbb{Z}}}\mathbb{1}_{A}(x)\cdot\mathbb{1}_{A}(T^{n}x)\cdot\dotso\cdot\mathbb{1}_{A}(T^{kn}x)\
d\mu.\\\ \textrm{ (see \cite[cite]{[\@@bibref{}{fsz}{}{}]} p. 210)}$

Now, from (1.2) we can derive the inequality

(1.3) $\bar{d}(E\cap(E-n)\cap\dots\cap(E-kn))\geq\mu(A\cap
T^{-n}A\cap\dots\cap T^{-kn}A).$

The foregoing discussion leads to the more general principle:

######  Theorem 1.1 (Furstenberg’s correspondence principle for $\bar{d}$.
(cf. [B1], Theorem 1.1)).

Let $E\subseteq\mathbb{N}$ with $\bar{d}(E)>0$. Then, there is an invertible
measure preserving system $(X,\mathcal{B},\mu,T)$ and a set $A\in\mathcal{B}$
with $\mu(A)=\bar{d}(E)>0$ satisfying

(1.4) $\bar{d}(E\cap(E-h_{1})\cap\dotso\cap(E-h_{r}))\geq\mu(A\cap
T^{-h_{1}}A\cap\dotso\cap T^{-h_{r}}A)$

for all $r\in\mathbb{N}$ and $h_{1},\dots,h_{r}\in\mathbb{N}$.

######  Remark 1.2.

All proofs of the above result known to the authors also give a version of
(1.4) for unions:

(1.5) $\bar{d}(E\cup(E-h_{1})\cup\dots\cup(E-h_{r}))\geq\mu(A\cup
T^{-h_{1}}A\cup\dots\cup T^{-h_{r}}A)$

for all $r\in\mathbb{N}$ and $h_{1},\dots,h_{r}\in\mathbb{N}$. See the
discussion and ramifications of this fact below.

A priori, one could expect that, given $E$ with $\bar{d}(E)>0$, it is possible
to judiciously choose the system $(X,\mathcal{B},\mu,T)$ to be ergodic in the
above construction. It turns out that, this is not always the case. To see
this, we will invoke the following interesting result of Hindman:

######  Theorem 1.3 (Hindman’s covering theorem [H2]).

Let $E\subseteq\mathbb{N}$ be a set with $d^{*}(E)>0$. Then, for every
$\varepsilon>0$ there is some $N\in\mathbb{N}$ such that

(1.6) $d^{*}\left(\bigcup_{i=1}^{N}(E-i)\right)>1-\varepsilon.$

Curiously enough, Theorem 1.3 fails to be true if one replaces $d^{*}$ with
$\bar{d}$. Consider, for example, the following set $E\subseteq\mathbb{N}$
provided by Hindman in [H2]:

(1.7) $E:=\bigcup_{n\in\mathbb{N}}[2^{2n},2^{2n+1}).$

Then $\bar{d}(E)=\frac{2}{3}$, and one can check that, moreover,
$\bar{d}(\bigcup_{i=0}^{N}(E-i))=\frac{2}{3}$ for all $N\in\mathbb{N}$. It
follows that for this set $E$ any measure preserving system
$(X,\mathcal{B},\mu,T)$ satisfying (1.5) cannot be ergodic. The reason is that
for an ergodic system $(X,\mathcal{B},\mu,T)$, if $\mu(A)>0$, then
$\lim_{N\to\infty}\mu\left(\bigcup_{i=1}^{N}T^{-i}A\right)=1$. Assuming the
inequality (1.5) is valid for the system in question, we would have

(1.8)
$\bar{d}\left(\bigcup_{i=1}^{N}(E-i)\right)\geq\mu\left(\bigcup_{i=1}^{N}T^{-i}A\right),$

and this cannot hold in our example since the left hand side is bounded away
from $1$.

However, by appropriately amplifying the proof of Theorem 1.1 discussed above,
one can obtain the following ergodic variant thereof. This amplification will
be carried out in the appropriate generality in Sections 2 and 6.

######  Theorem 1.4 (Ergodic Furstenberg’s correspondence principle. (cf.
[BHK], Proposition 3.1)).

Let $E\subseteq\mathbb{N}$ be such that $d^{*}(E)>0$. Then, there is an
ergodic measure preserving system $(X,\mathcal{B},\mu,T)$ and a set
$A\in\mathcal{B}$ with $\mu(A)=d^{*}(E)>0$ satisfying

(1.9) $d^{*}(E\cap(E-h_{1})\cap\dotso\cap(E-h_{r}))\geq\mu(A\cap
T^{-h_{1}}A\cap\dotso\cap T^{-h_{r}}A)$

for all $r\in\mathbb{N}$ and $h_{1},\dots,h_{r}\in\mathbb{N}$.

Similarly to the situation with Theorem 1.1, one can show that a version of
(1.9) holds for unions, i.e.

(1.10) $d^{*}(E\cup(E-h_{1})\cup\dots\cup(E-h_{r}))\geq\mu(A\cup
T^{-h_{1}}A\cup\dots\cup T^{-h_{r}}A)$

for all $r\in\mathbb{N}$ and $h_{1},\dots,h_{r}\in\mathbb{N}$. This
observation was made and utilized in [BBF] (cf. [BBF] Prop. 2.3).

The functions $\bar{d}$ and $d^{*}$ have very similar properties. For example,
$\bar{d}$ and $d^{*}$ satisfy $d^{*}(\mathbb{N})=1$ and
$\bar{d}(\mathbb{N})=1$ and are shift-invariant (i.e.
$\bar{d}(E-n)=\bar{d}(E)$ for all $n\in\mathbb{N}$ and $d^{*}(E-n)=d^{*}(E)$
for all $n\in\mathbb{N}$), which allows one to view $\mathbb{N}$ as a
generalized probability space with either $\bar{d}$ or $d^{*}$ serving as an
(admittedly vague) substitute for the probability measure. Moreover, either of
Theorems 1.1 and 1.4 can be used to derive Szemerédi’s theorem from EST.

While in certain investigations (see [HF1], [HF2], [Fra], [GLR]), it is of
interest to consider the non-ergodic measure preserving systems corresponding
to sets satisfying $\bar{d}(E)>0$ (or more generally, sets with the property
$\bar{d}_{(I_{N})}(E):=\limsup_{N\to\infty}\frac{|E\cap I_{N}|}{|I_{N}|}>0$,
where $(I_{N})$ is a sequence of intervals with increasing length), in some
other situations, $d^{*}$ allows for stronger/sharper results. One such
application was obtained in [BHK]. Also, observe that Theorem 1.4 immediately
implies (via (1.10)) Theorem 1.3. The ergodic approach to Theorem 1.3 has two
additional advantages. First, it will allow us to characterize sequences
$(n_{k})_{k\in\mathbb{N}}$ with the "Hindman property", i.e., sequences
$(n_{k})_{k\in\mathbb{N}}$ such that for any $E\subseteq\mathbb{Z}$ with
$d^{*}(E)>0$ one has that for all $\varepsilon>0$ there is some
$N\in\mathbb{N}$ such that

(1.11) $d^{*}\left(\bigcup_{k=1}^{N}(E-n_{k})\right)>1-\varepsilon,$

(see (1.6)). Second, the robustness of the ergodic approach will enable us to
formulate and prove with ease a Hindman-like result for any amenable group. (A
different proof of this result was obtained in [BBF] through combinatorial
methods).

While the fact that Furstenberg’s correspondence principle works equally well
for unions was not observed/utilized in the early papers in ergodic Ramsey
theory, in hindsight this observation is quite natural if one takes into
account the algebraic nature of this principle. The versatility of
Furstenberg’s correspondence principle can be perhaps best of all perceived
via Gelfand’s representation theorem. The possibility of using Gelfand’s
representation theorem was mentioned in [F4] and was explicitly implemented in
[B1], [BF], [BCRZ], and [B2]. See also the appendix to Section I in [F2]. We
would like to point out that the correspondence principle was not born with
the ergodic theoretical proof of Szemerédi’s theorem; indeed, a form of it
appears already in Furstenberg’s thesis [F1], where it was used as a tool to
reconstruct a stationary process from its past. More concretely, the seminal
idea which was put to action in [F1] was to replace the approximate measure
space $\mathbb{Z}$ together with the density preserving transformation
$x\mapsto x+1$ by a genuine measure preserving system, namely: the orbit
closure of the sequence $(\mathbb{1}_{E}(n))_{n\in\mathbb{Z}}$ in
$\\{0,1\\}^{\mathbb{Z}}$, where $\mathbb{1}_{E}$ corresponds to the given time
series.

It follows from Gelfand’s representation theorem that any commutative, unital,
countably generated $C^{*}$-algebra $\mathcal{A}$ is topologically and
algebraically isomorphic to the algebra of continuous functions $C(X)$, where
$X$ is a compact metric space. In our situation, such a $C^{*}$-algebra can be
naturally generated by the family $(\mathbb{1}_{E-n})_{n\in\mathbb{Z}}$, where
$E\subseteq\mathbb{Z}$ satisfies
$d_{(I_{N})}(E):=\lim_{N\to\infty}\frac{|E\cap I_{N}|}{|I_{N}|}>0$ for some
sequence of intervals $(I_{N})$ with $|I_{N}|\to\infty$. Let us denote this
$C^{*}$-algebra by $\mathcal{A}_{E}$.

One can then refine $(I_{N})$ to obtain a subsequence $(I_{N_{k}})$ so that
for any set $F$ in the Boolean algebra $\mathcal{B}_{E}$ generated by the
family $(E-n)_{n\in\mathbb{Z}}$, $d_{(I_{N_{k}})}(F)$ is well defined. In
other words, $d_{(I_{N_{k}})}(\cdot)$ is a _shift-invariant density_ on
$\mathcal{B}_{E}$. In turn, $d_{(I_{N_{k}})}(\cdot)$ induces a shift-invariant
mean on $\mathcal{A}_{E}$, i.e., a positive functional
$L:\mathcal{A}_{E}\rightarrow\mathbb{C}$ such that $L(1)=1$ and for any
$F\in\mathcal{B}_{E}$, $L(\mathbb{1}_{F})=d_{(I_{N_{k}})}(F)$. By Gelfand’s
representation theorem, there exists a compact metric space $X$ such that
$\mathcal{A}_{E}$ is algebraically and topologically isomorphic to $C(X)$. Let
$\tilde{L}:C(X)\rightarrow\mathbb{C}$ be the linear functional on $C(X)$
induced by $L$. By Riesz’s representation theorem, the functional $\tilde{L}$
is given by a Borel probability measure on $X$. Let
$\Gamma:\mathcal{A}_{E}\rightarrow C(X)$ denote the Gelfand isomorphism. Then,
for all $\varphi\in\mathcal{A}_{E}$ we have

$L(\varphi)=\tilde{L}(\Gamma(\varphi))=\int_{X}\Gamma(\varphi)\ d\mu.$

Since $\mathbb{1}_{E}$ is an idempotent, and since $\Gamma$ is, in particular,
an algebraic isomorphism, the image, $\Gamma(\mathbb{1}_{E})$, is again an
idempotent, and hence of the form $\mathbb{1}_{A}$, where
$A\in\textrm{Borel}(X)$ and satisfies
$\mu(A)=\tilde{L}(\mathbb{1}_{A})=\tilde{L}(\Gamma(\mathbb{1}_{E}))=d_{(I_{N})}(E)$.

The $L$-invariant shift operator given by $\varphi(n)\mapsto\varphi(n+1)$,
$\varphi\in\mathcal{A}_{E}$ induces a $\mathbb{Z}$-action on $C(X)$, which is,
by a theorem of Banach, induced by a measure preserving homeomorphism
$T:X\rightarrow X$. Let $\varphi=\mathbb{1}_{E-h_{1}\cap\dots\cap
E-h_{r}}=\prod_{i=1}^{r}\mathbb{1}_{E-h_{i}}$,
$h_{1},\dots,h_{r}\in\mathbb{Z}$. Since $\mathcal{A}_{E}$ is an algebra, it is
clear that $\varphi\in\mathcal{A}_{E}$. This leads us to the equality

(1.12) $d_{(I_{N_{k}})}(E\cap E-h_{1}\cap\dots\cap
E-h_{r})=L(\mathbb{1}_{E\cap E-h_{1}\cap\dots\cap E-h_{r}})=\mu(A\cap
T^{-h_{1}}A\cap\dots\cap T^{-h_{r}}A).$

Linearity and the inclusion-exclusion principle imply that functions of the
form $\varphi=\mathbb{1}_{E\cup E-h_{1}\cup\dots\cup E-h_{r}}$ are also in
$\mathcal{A}_{E}$. Since $\Gamma$ is an algebraic isomorphism, we see that a
formula similar to (1.12) holds for unions as well:

(1.13) $d_{(I_{N_{k}})}(E\cup E-h_{1}\cup\dots\cup
E-h_{r})=L(\mathbb{1}_{E\cup E-h_{1}\cup\dots\cup E-h_{r}})=\mu(A\cup
T^{-h_{1}}A\cup\dots\cup T^{-h_{r}}A).$

As we already mentioned above, all known proofs of Furstenberg’s
correspondence principle are such that they allow for (1.5) and (1.10). We
explained above how one can get (1.5) with the help of Gelfand’s transform,
but, as it will be seen in Sections 2 and 6 (where we will prove and juxtapose
general versions of Theorems 1.1 and 1.4 in the setup of amenable groups), the
other methods also have an algebraic aspect which is adequate for our
purposes. In any case, whatever method is used, it still has to be properly
modified and amplified to allow for ergodicity. The approach that we choose in
Section 2 uses symbolic dynamics and goes along the lines of the proof of the
correspondence principle in [F3], which is sketched above. As we will see, it
allows to conveniently localize the place in the proof where the amplification
leading to ergodicity has to be made.

At this point, we would like to make a comment which explains why utilizing
$d^{*}$ allows one to establish the ergodic version of Furstenberg’s
correspondence principle. The key advantage of dealing with $d^{*}$ is that,
in the course of proving the correspondence principle, one can keep changing
the sequence of intervals along which the averaging scheme is applied, which
is essential for applicability of the ergodic decomposition. On the other
hand, the proof of the $\bar{d}$ version of Furstenberg’s correspondence
priniciple is based on refining the averaging scheme given by the sequence of
intervals along which $\bar{d}_{(I_{N})}(E)>0$. Theorems 1.1 and 1.4 will be
generalized, juxtaposed and exploited in Section 2.

The structure of the paper is as follows. In Section 2 we give a proof of a
general version of Theorem 1.4, which works for any combination of unions,
intersections and complements of shifts of $E$ for countable amenable groups.
Moreover, we also give a proof of a general version of Theorem 1.1, comparing
the two methods of proof, and pinpointing what exactly in the proof allows us
to have ergodicity.

In Section 3 we give a quick proof of a general form of Hindman’s theorem 1.3
for countable cancellative amenable semigroups in Section 3. We also
generalize the example (1.7) to countable abelian groups and to finitely
generated virtually nilpotent groups. In Section 3 we prove a few more
combinatorial results which make use of the "ergodic" nature of $d^{*}$.

Section 4 is devoted to the characterization of sequences $(n_{k})$ for which
(1.11) holds and is followed by Section 5, where we give a characterization of
countable amenable WM groups via the “Hindman property” with the help of a
general version of Theorem 1.4 proved in Section 2. A group $G$ is called
_weakly mixing_ or WM if any measure preserving ergodic action of $G$ on a
probability space is automatically weakly mixing 222Such groups also appear in
the literature under the name _minimally almost periodic groups_.. For example
$A(\mathbb{N})$, the group of finite even permutations of $\mathbb{N}$ is a
countable amenable WM group. It is worth noting that in [BCRZ] a different
characterization of WM groups is obtained via another result due to Hindman,
namely his "finite sums" theorem [H1].

Lastly, in Section 6 we establish a general form of an ergodic Furstenberg
correspondence principle for discrete amenable semigroups and derive as a
corollary a general form of Hindman’s theorem.

######  Remark 1.5.

Throughout this paper, all the measures used are normalized so that
$\mu(X)=1$.

##  2\. Furstenberg’s correspondence principle for amenable groups: $\bar{d}$
and $d^{*}$ versions

The goal of this section is two-fold. First, we will formulate and prove
"amenable" versions of Theorems 1.1 and 1.4 (Theorems 2.3 and 2.8) that (i)
encompass not only intersections but unions of sets and their complements, and
(ii) are valid for general discrete countable amenable groups. Second, we will
pinpoint the distinction between $\bar{d}$ and $d^{*}$ which allows for the
stronger, ergodic version of Furstenberg’s correspondence principle (see
Theorem 2.8 below).

A definition of amenability which is convenient for our purposes uses the
notion of Følner sequence. A sequence $(F_{N})$ of finite non-empty subsets of
a countable group $G$ is a _(left) Følner sequence_ if

$\lim_{N\to\infty}\frac{|F_{N}\Delta gF_{N}|}{|F_{N}|}=0.$

for all $g\in G$. A countable group $G$ is _amenable_ if it admits a (left)
Følner sequence 333One can show that every amenable group $G$ admits also
right- and indeed two-sided analogues of left Følner sequences (see Corollary
5.3 in [N]). Throughout this paper we deal only with left Følner sequences and
will routinely omit the adjective ”left”..

To facilitate the discussion, we will present first the versions of Theorems
1.1 and 1.4 (see Theorems 2.3 and 2.8) for general countable amenable groups.
The proofs are in essence the same as those of Theorems 1.1 and 1.4, but we
will need this generality for the applications in the forthcoming sections.

We will then juxtapose the proofs of Theorems 2.3 and 2.8 which will allow us
to explain what exactly in the proof of Theorem 2.8 leads to the ergodicity of
the system $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$.

To formulate Theorems 2.3 and 2.8 we need a few definitions:

######  Definition 2.1.

Let $E\subseteq G$ and let $(F_{N})$ be a Følner sequence. We denote by
$\bar{d}_{(F_{N})}(E)$ the upper density of the set $E$ along $(F_{N})$. This
notion of largeness is given by the formula

$\bar{d}_{(F_{N})}(E):=\limsup_{N\to\infty}\frac{|F_{N}\cap E|}{|F_{N}|}$

We are now in a position to define upper Banach density in this more general
context:

######  Definition 2.2.

Let $E\subseteq G$. We denote by $d^{*}(E)$ the upper Banach density of the
set $E$, which is given by

$d^{*}(E):=\sup\\{\bar{d}_{(F_{N})}(E):(F_{N})\textrm{ is a F\o lner
sequence}\\}.$

We begin with a short proof (based on the idea of the proof of Theorem 1.1 in
[F3] and [F4] 555We could also use for our goals the proofs of the amenable
version of Theorem 1.1 given in [BMc] (Theorem 2.1) and [B2] (Theorem 4.11).
(see also [FKO])) of a generalization of Theorem 1.1 for countable amenable
groups. Note that Theorem 1.1 corresponds to the special case $G:=\mathbb{Z}$
and $F_{N}:=[a_{N},b_{N}]$ with $b_{N}-a_{N}\to\infty$.

In what follows we will use the notation $A^{1}=A$ and $A^{0}=A^{c}$.

######  Theorem 2.3 (Furstenberg Correspondence Principle for countable
amenable groups, $\bar{d}_{(F_{N})}$ version).

Let $(F_{N})$ be a Følner sequence in a countable amenable group $G$ and let
$E$ be a subset of $G$ with $\bar{d}_{(F_{N})}(E)>0$. Then there exist a
probability measure preserving system $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$
and a set $A\in\mathcal{B}$ with $\mu(A)=\bar{d}_{(F_{N})}(E)$ such that

(2.1)
$\mu((T_{g_{1}})^{-1}A^{w_{1}}\star\dots\star(T_{g_{k}})^{-1}A^{w_{k}})\leq\bar{d}_{(F_{N})}(g_{1}^{-1}E^{w_{1}}\star\dots\star
g_{k}^{-1}E^{w_{k}}),$

for all $k\in\mathbb{N}$, all $\\{g_{1},\dots,g_{k}\\}\subset G$ and all
$(w_{1},\dots,w_{k})\in\\{0,1\\}^{k}$, and where each of the stars denotes
either union or intersection with the understanding that

  * (i)

for all $1\leq i\leq k-1$, the operation represented by $\star$ which stands
between $E^{w_{i}}$ and $E^{w_{i+1}}$ is the same as the operation appearing
between $A^{w_{i}}$ and $A^{w_{i+1}}$.

  * (ii)

the choices of parentheses which are needed to make the expressions on both
sides of formula (2.1) well defined also match. 666For example, we have
$\mu(((T_{g_{1}})^{-1}A\cup(T_{g_{2}})^{-1}A^{c})\cap(T_{g_{3}})^{-1}A)\leq\bar{d}_{(F_{N})}((g_{1}^{-1}E\cup
g_{2}^{-1}E^{c})\cap g_{3}^{-1}E)\textrm{ and }$
$\mu((T_{g_{1}})^{-1}A\cup((T_{g_{2}})^{-1}A^{c}\cap(T_{g_{3}})^{-1}A))\leq\bar{d}_{(F_{N})}(g_{1}^{-1}E\cup(g_{2}^{-1}E^{c}\cap
g_{3}^{-1}E)).$

###### Proof.

Let $X=\\{0,1\\}^{G}$ (viewed as a compact metric space with the usual product
topology). Let $(T_{g})_{g\in G}$ be the action of $G$ on $X$ by
homeomorphisms defined by the formula $(T_{g}x)_{g_{0}}=x_{gg_{0}}$ for all
$g,g_{0}\in G$. Define $\omega\in X$ by setting $\omega(g)=1$ if $g\in E$ and
$\omega(g)=0$ otherwise. Put $A=\\{x\in X:x(e)=1\\}$ (Here and elsewhere, $e$
denotes the neutral element of the group $G$). Note that $A$ is a clopen set
in $X$ (and hence $\mathbb{1}_{A}$ is a continuous function). Moreover, we
have that $T_{g}\omega\in A$ if and only if $g\in E$. Let $(F_{N_{k}})$ be a
subsequence such that

$\bar{d}_{(F_{N})}(E)=\lim_{k\to\infty}\frac{|E\cap F_{N_{k}}|}{|F_{N_{k}}|}.$

Let $\mu$ be any weak* limit point of the sequence of measures

$\frac{1}{|F_{N_{k}}|}\sum_{g\in F_{N_{k}}}\delta_{T_{g}\omega}.$

Moreover, since $C(Y)$ is separable, there exists a further subsequence of
$(F_{N_{k}})$, which we will, in order not to overload the notation, still
denote by $(F_{N_{k}})$, such that, in the weak* topology,
$\mu=\lim_{k\to\infty}\frac{1}{|F_{N_{k}}|}\sum_{g\in
F_{N_{k}}}\delta_{T_{g}\omega}$. Clearly, $\mu$ is a $G$-invariant probability
measure on $X$. We claim that $\mu(A)=\bar{d}_{(F_{N})}(E)$. Indeed, since
$\mathbb{1}_{A}$ is a continuous function, we can write

(2.2) $\mu(A)=\int_{X}\mathbb{1}_{A}\
d\mu=\lim_{k\to\infty}\frac{1}{|F_{N_{k}}|}\sum_{g\in
F_{N_{k}}}\mathbb{1}_{A}(T_{g}\omega)=d_{(F_{N_{k}})}(E)=\bar{d}_{(F_{N})}(E).$

Now let $g_{1},\dots,g_{k}\in G$ and consider the set
$(T_{g_{1}})^{-1}A^{w_{1}}\star\dotso\star(T_{g_{k}})^{-1}A^{w_{k}}$, where
the stars denote an arbitrary fixed choice of either union or intersection.
This is a clopen set in $X$, so its indicator function is, again, continuous.
We have

$\mu\left((T_{g_{1}})^{-1}A^{w_{1}}\star\dotso\star(T_{g_{k}})^{-1}A^{w_{k}}\right)=\lim_{k\to\infty}\frac{1}{|F_{N_{k}}|}\sum_{g\in
F_{N_{k}}}\mathbb{1}_{(T_{g_{1}})^{-1}A^{w_{1}}\star\dotso\star(T_{g_{k}})^{-1}A^{w_{k}}}(T_{g}\omega)$
$=\lim_{k\to\infty}\frac{1}{|F_{N_{k}}|}|(g_{1}^{-1}E^{w_{1}}\star\dotso\star
g_{k}^{-1}E^{w_{k}})\cap
F_{N_{k}}|\leq\bar{d}_{(F_{N})}(g_{1}^{-1}E^{w_{1}}\star\dotso\star
g_{k}^{-1}E^{w_{k}}).$

We are done. ∎

######  Remark 2.4.

Observe that when $G:=\mathbb{Z}$ and $F_{N}:=[a_{N},b_{N}]$ (where
$b_{N}-a_{N}\to\infty$) inequality (2.1) implies the inequalities (1.4) and
(1.5). As we saw in the Introduction with the help of Theorem 1.3, for some
sets $E\subseteq\mathbb{Z}$ no system $(X,\mathcal{B},\mu,T)$ satisfying
inequality (1.5) can be ergodic. The general version of Theorem 1.1 which also
involves complements of sets, allows us to arrive at the same conclusion
without invoking Theorem 1.3. Indeed, we see that (for $G:=\mathbb{Z}$ and
$F_{N}:=[1,N]$) inequality (2.1) implies

(2.3) $\bar{d}(E^{c}\cap(E-h))\geq\mu(A^{c}\cap T^{-h}A)$

for all $h\in\mathbb{Z}$. Take $E$ as in the Introduction (see (1.7)). Then,
$\mu(A)=\bar{d}(E)=\frac{2}{3}$, so $\mu(A^{c})>0$. However, one can easily
check (see also Section 3), that $\bar{d}(E^{c}\cap(E-h))=0$ for all
$h\in\mathbb{Z}$. This contradicts inequality (2.3).

######  Remark 2.5.

Let $G$ be a countable amenable group. Call a set $E$ a _Hindman set_ if there
exists some Følner sequence $(F_{N})$ such that

$0<\bar{d}_{(F_{N})}(E)<1\textrm{ and }\bar{d}_{(F_{N})}\left(\bigcup_{g\in
F}g^{-1}E\right)<\frac{3}{4}$

for all finite sets $F\subseteq G$. One can show (see Proposition 3.3) that
any countable abelian group has a Hindman set.

It is interesting to observe that if our countable amenable group $G$ admits a
Hindman set, then any measure preserving system
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ satisfying inequalities (2.1) cannot be
ergodic. This can be seen in two ways. First, using the special case of
inequality (2.3) for unions, i.e.

(2.4) $\bar{d}_{(F_{N})}\left(\bigcup_{g\in
F}g^{-1}E\right)\geq\mu\left(\bigcup_{g\in F}g^{-1}A\right)\textrm{ for all
finite sets }F\subseteq G,$

and arguing, as in the Introduction, that inequality (2.4) together with the
fact that $\bar{d}_{(F_{N})}\left(\bigcup_{g\in
F}g^{-1}E\right)=\bar{d}_{(F_{N})}(E)<1$ contradict ergodicity.

Alternatively, we can use another special case of inequality (2.3), namely:

$\bar{d}_{(F_{N})}(E^{c}\cap g^{-1}E)\geq\mu(A^{c}\cap g^{-1}A)\textrm{ for
all }g\in G,$

together with the fact that if $E$ is a Hindman set then
$\bar{d}_{(F_{N})}(E^{c}\cap g^{-1}E)=0$ for all $g\in G$. (This discussion
will be completed in Section 3).

We move now to an ergodic version of Theorem 1.4 for general countable
amenable groups. The ergodic amplification of Theorem 2.3 hinges on two
additional tools: the ergodic decomposition for amenable group actions, and a
result about quasi-generic points (see Proposition 2.7 below).

######  Definition 2.6.

Let $X$ be a compact metric space on which $G$ acts by homeomorphisms. Let
$\mu$ be a $G$-invariant measure. We say that $x_{0}\in X$ is _quasi-generic_
for $\mu$ if there exists a Følner sequence $(F_{N})$ such that

$\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}f(T_{g}x_{0})=\int_{X}f\
d\mu$

for every $f\in C(X)$.

The following Proposition is an amenable version of Proposition 3.9 in [F5].
We include the proof for the reader’s convenience.

######  Proposition 2.7.

Let $(T_{g})_{g\in G}$ be an action of $G$ by homeomorphisms on a compact
metric space $X$. Let $x_{0}\in X$, and let $Y:=\overline{\\{T_{g}x_{0}:g\in
G\\}}$. Suppose that $\mu\in\mathcal{M}(Y)$ is an ergodic $G$-invariant
measure. Then $x_{0}$ is quasi-generic for $\mu$.

###### Proof.

Since $\mu$ is an ergodic measure, it follows by the mean ergodic theorem
(which can be proved in the same way as the classical mean ergodic theorem for
isometries of Hilbert spaces, see, for example, Theorem 4.15 in [B2]), that
for any Følner sequence $(F_{N})$, and any $f\in L^{2}(\mu)$

$\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}f(T_{g}x)=\int_{Y}f\
d\mu,$

where the convergence is in the $L^{2}(\mu)$-norm. Thus, there exists some
subsequence $(F_{N_{k}})$ along which we have pointwise convergence for all
$f$ in a countable dense subset of $C(Y)$, which in turn, by a simple triangle
inequality argument, implies that, for any $f\in C(Y)$ we have

$\lim_{N\to\infty}\frac{1}{|F_{N_{k}}|}\sum_{g\in
F_{N_{k}}}f(T_{g}x)=\int_{X}f\ d\mu,$

for a.e. $x\in Y$ (and so a.e. $x\in Y$ is quasi-generic for $\mu$).

Let $x_{1}\in Y$ be quasi-generic for $\mu$ along the Følner sequence
$(F_{N_{k}})$. Take a countable dense set $\\{f_{k}:k\in\mathbb{N}\\}$ in
$C(Y)$ such that

(2.5) $\left|\frac{1}{|F_{N_{k}}|}\sum_{g\in
F_{N_{k}}}f_{j}(T_{g}x_{1})-\int_{Y}f_{j}\ d\mu\right|<\frac{1}{k}$

for all $k\in\mathbb{N}$ and all $j=1,\dots,k$. Since the functions $f_{j}$
are continuous, we can pick $(g_{k})\subseteq G$ such that the inequality
(2.5) holds if we replace $x_{1}$ by $T_{g_{k}}x_{0}$, which after a change of
variables in the sum yields

$\left|\frac{1}{|F_{N_{k}}|}\sum_{g\in
F_{N_{k}}g_{k}}f_{j}(T_{g}x_{0})-\int_{Y}f_{j}\ d\mu\right|<\frac{1}{k},$

which implies that

$\lim_{k\to\infty}\frac{1}{|F_{N_{k}}g_{k}|}\sum_{g\in
F_{N_{k}}g_{k}}f(T_{g}x_{0})=\int_{Y}f\ d\mu$

for all $f\in C(Y)$. In other words, $x_{0}$ is a quasi-generic point for
$\mu$ with respect to the Følner sequence $(F_{N_{k}}g_{k})$. Indeed, observe
that for all $g\in G$ we have that $|gF_{N_{k}}g_{k}\Delta
F_{N_{k}}g_{k}|=|gF_{N_{k}}\Delta F_{N_{k}}|$, which implies that
$(F_{N_{k}}g_{k})$ is still a Følner sequence ∎

We are now ready to formulate and prove the amenable ergodic Furstenberg’s
Correspondence Principle, adapting arguments from both [BHK] and [BBF]. We
note that Theorem 1.4 corresponds to the special case $G:=\mathbb{Z}$ in
Theorem 2.8.

######  Theorem 2.8 (Ergodic Furstenberg Correspondence Principle for
countable amenable groups; $d^{*}$ version).

Let $E$ be a subset of a countable amenable group $G$ with positive upper
Banach density. Then there exists an ergodic probability measure preserving
system $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and a set $A\in\mathcal{B}$ with
$\mu(A)=d^{*}(E)$ such that

(2.6)
$\mu((T_{g_{1}})^{-1}A^{w_{1}}\star\dots\star(T_{g_{k}})^{-1}A^{w_{k}})\leq
d^{*}(g_{1}^{-1}E^{w_{1}}\star\dots\star g_{k}^{-1}E^{w_{k}})$

for all $k\in\mathbb{N}$, all $\\{g_{1},\dots,g_{k}\\}\subset G$ and all
$(w_{1},\dots,w_{k})\in\\{0,1\\}^{k}$, and where each of the stars denotes
either union or intersection with the understanding that

  * (i)

for all $1\leq i\leq k-1$, the operation represented by $\star$ which stands
between $E^{w_{i}}$ and $E^{w_{i+1}}$ is the same as the operation appearing
between $A^{w_{i}}$ and $A^{w_{i+1}}$.

  * (ii)

the choices of parentheses which are needed to make the expressions on both
sides of formula (2.1) well defined also match.

###### Proof.

We start as in the proof of Theorem 2.3. Let $X=\\{0,1\\}^{G}$ and
$(T_{g})_{g\in G}$ be the action of $G$ on $X$ by homeomorphisms defined by
the formula $(T_{g}x)_{g_{0}}=x_{gg_{0}}$ for all $g,g_{0}\in G$. Define, as
before, $\omega\in X$ by setting $\omega(g)=1$ if $g\in E$ and $\omega(g)=0$
otherwise.

Let $Y=\overline{\\{T_{g}\omega:g\in G\\}}$. Put $A=\\{x\in Y:x(e)=1\\}$.
Then, $A$ is a clopen set in $Y$. Moreover, we have that $T_{g}\omega\in A$ if
and only if $g\in E$. Let $(F_{N})$ be a Følner sequence such that
$d^{*}(E)=\bar{d}_{(F_{N})}(E)$. Let $\nu$ be any weak* limit point of the
sequence of measures

$\frac{1}{|F_{N}|}\sum_{g\in F_{N}}\delta_{T_{g}\omega}.$

Then, as in Theorem 2.3, $\nu$ is a $G$-invariant probability measure on $Y$
such that $\nu(A)=d^{*}(E)$.

By the ergodic decomposition theorem (see Theorem 4.2 in [V]), there is a
probability measure $\lambda$ on the set of ergodic normalized measures
$\mathcal{M}_{G}(X)$ such that

(2.7) $\nu(C)=\int_{\mathcal{M}_{G}(X)}\mu_{z}(C)\ d\lambda(z)$

for all $C\in\mathcal{B}$. It follows from the equality (2.7) that there
exists some $z$ such that $\mu_{z}(A)\geq\nu(A)=d^{*}(E)$.

We show that the measure $\mu_{z}$, which we will now denote by $\mu$, works.
Let $g_{1},\dots,g_{k}\in G$ and observe that the set
$(T_{g_{1}})^{-1}A^{w_{1}}\star\dots\star(T_{g_{k}})^{-1}A^{w_{k}}$ is a
clopen set in $Y$, so its indicator function is continuous.

By Proposition 2.7, there exists a Følner sequence $(G_{N})$, with respect to
which the point $\omega$ is quasi-generic for the measure $\mu$. This implies
that

(2.8)
$\mu((T_{g_{1}})^{-1}A^{w_{1}}\star\dots\star(T_{g_{k}})^{-1}A^{w_{k}})=\lim_{N\to\infty}\frac{1}{|G_{N}|}\sum_{g\in
G_{N}}\mathbb{1}_{T_{g_{1}}^{-1}A^{w_{1}}\star\dots\star
T_{g_{k}}^{-1}A^{w_{k}}}(T_{g}\omega)=\\\
\lim_{N\to\infty}\frac{1}{|G_{N}|}\left|(g_{1}^{-1}E^{w_{1}}\star\dots\star
g_{k}^{-1}E^{w_{k}})\cap G_{N}\right|\leq
d^{*}(g_{1}^{-1}E^{w_{1}}\star\dots\star g_{k}^{-1}E^{w_{k}}).$

In particular, letting $k=1,w_{1}=1$ and $g_{1}=e$ in inequality (2.8) we
obtain $\mu(A)\leq d^{*}(E)$. Recalling the previous inequality
$\mu(A)\geq\nu(A)=d^{*}(E)$ we see that $\mu(A)=d^{*}(E)$, so we are done. ∎

We conclude this section with some comments on why the utilization of $d^{*}$
allows us to achieve in Theorem 2.8 the goal of ergodicity (whereas, as we saw
above, there are sets $E$ for which Theorem 2.3 cannot guarantee it). In both
proofs, we start with a Følner sequence $(F_{N})$ wich satisfies
$\bar{d}_{(F_{N})}(E)>0$ (in Theorem 2.3) or $d^{*}(E)=\bar{d}_{(F_{N})}(E)$
(in Theorem 2.8). Then we consider weak* limits of the sequences of measures
$\frac{1}{|F_{N}|}\sum_{g\in F_{N}}\delta_{T_{g}\omega}$
($\omega=(\mathbb{1}_{E}(g))_{g\in G}$) along a subsequence of $(F_{N})$. The
advantage of $d^{*}$ comes in handy when, after invoking the ergodic
decomposition in the proof of Theorem 2.8, we start using shifts
$(F_{N_{k}}g_{k})$ of a relevant subsequence $(F_{N_{k}})$ in order to use
quasi-genericity of $\omega$, as guaranteed by Proposition 2.7. Unlike the
proof of Theorem 2.3 where we are restricted to a given Følner sequence
$(F_{N})$ and its subsequences, in the proof of Theorem 2.8, we are
conveniently passing to a different Følner sequence without affecting the
value of $d^{*}$ for expressions of the form
$g_{1}^{-1}E^{w_{1}}\star\dotso\star g_{k}^{-1}E^{w_{k}}$ 777This property of
$d^{*}$, when applied to unions, is also behind Hindman’s proof of Theorem
1.3..

##  3\. Hindman’s Theorem via ergodic theory and some other consequences of
the ergodic version of Furstenberg’s correspondence principle

In this section we will give a short proof of a natural generalization of
Hindman’s theorem to the context of countable amenable groups. Strictly
speaking, Hindman’s theorem, as formulated in the introduction (Theorem 1.3)
deals with $(\mathbb{N},+)$, which is a semigroup rather than a group, but the
results of this section are easily adjusted so that they hold for countable
cancellative amenable semigroups. See the explanatory remark at the end of the
section (see also Section 6 for a general semigroup version). We also deduce
other ergodic-flavored corollaries which provide additional evidence to our
claim that $d^{*}$ is better suited for applications.

We begin with showing that a general version of Hindman’s theorem (see Theorem
1.6 in the introduction) is an immediate corollary of Theorem 2.8:

######  Theorem 3.1.

Let $G$ be a countable amenable group, and let $E$ be a subset of $G$ with
$d^{*}(E)>0$. Then, for every $\varepsilon>0$, there exists $k\in\mathbb{N}$
and $g_{1},\dots,g_{k}\in G$ such that $d^{*}(g_{1}^{-1}E\cup\dots\cup
g_{k}^{-1}E)>1-\varepsilon$.

###### Proof.

By Theorem 2.8 there exists an ergodic system
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and a set $A\in\mathcal{B}$ such that
$\mu(A)=d^{*}(E)$ and for which

$d^{*}(h_{1}^{-1}E\cup\dots\cup
h_{k}^{-1}E)\geq\mu((T_{h_{1}})^{-1}A\cup\dots\cup(T_{h_{k}})^{-1}A)$

for all $k\in\mathbb{N}$ and $h_{1},\dots,h_{k}\in G$. By ergodicity of the
action $(T_{g})_{g\in G}$ we have

$\mu\left(\bigcup_{g\in G}(T_{g})^{-1}A\right)=1,$

and so since $G$ is countable, the result follows. ∎

Another corollary that can be obtained from Theorem 2.8, is an ergodicity-like
statement for the group $G$. Namely:

######  Corollary 3.2.

Let $G$ be a countable amenable group, and let $E\subseteq G$ be such that
$d^{*}(E)\in(0,1)$. Then there exists some $g\in G$ such that $d^{*}(E^{c}\cap
g^{-1}E)>0$.

###### Proof.

Let $E\subseteq G$ be such that $d^{*}(E)\in(0,1)$. By Theorem 2.8 we can find
an ergodic measure preserving system $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$
and a set $A\in\mathcal{B}$ such that $\mu(A)=d^{*}(E)$ and for all $h\in G$,

$d^{*}(E^{c}\cap h^{-1}E)\geq\mu(A^{c}\cap(T_{h})^{-1}A).$

Since $\mu(A)>0$ and $\mu(A^{c})>0$ and the action $(T_{g})_{g\in G}$ is
ergodic, there is some $g\in G$ such that $\mu(A^{c}\cap(T_{g})^{-1}A)>0$, so
we are done. ∎

We proceed to show that an example similar to (1.7) exists in the context of
countable abelian groups.

######  Proposition 3.3.

Let $G$ be a countably infinite abelian group. Let $(F_{N})\subseteq G$ be a
Følner sequence. Then, there exists a set $E\subseteq G$ such that

(3.1) $\bar{d}_{(F_{N})}(E)>0,\textrm{ and
}\bar{d}_{(F_{N})}\left(\bigcup_{g\in F}g^{-1}E\right)<\frac{3}{4},$

for all finite sets $F\subseteq G$.

###### Proof.

We will show first that the assertion of the proposition holds for a
particular Følner sequence, and then upgrade it to an arbitrary Følner
sequence.

Assume first that $G$ is finitely generated and $\\{g_{1},\dots,g_{k}\\}$
generate $G$. Then, one of the generators of $G$, say $g_{1}$ has infinite
order. Consider the Følner sequence
$G_{N}:=\\{g_{1}^{a_{1}}\dotso g_{k}^{a_{N}}:0\leq a_{i}\leq 2^{2N}\textrm{
for }1\leq i\leq N\\}$. Define
$A_{N}=\\{g_{1}^{a_{1}}\dotso g_{k}^{a_{N}}:2^{2N-1}\leq a_{1}\leq
2^{2N}\textrm{ and }0\leq a_{i}\leq 2^{2N}\textrm{ otherwise }\\}$ and put
$E=\bigcup_{N\in\mathbb{N}}A_{N}$. It is not hard to check that the set $E$
satisfies (3.3).

Assume now that $G$ is infinitely generated, and let
$\\{g_{n}:n\in\mathbb{N}\\}$ be a set of generators for the group $G$. We
distinguish two cases. In the first case, one of the generators, say $g_{1}$
has infinite order. Consider the Følner sequence

$G_{N}:=\\{g_{1}^{a_{1}}g_{2}^{a_{2}}\cdot\dotso\cdot g_{N}^{a_{N}}:0\leq
a_{i}\leq 2^{2N},\textrm{ for }1\leq i\leq N\\},$

and set

(3.2) $A_{N}:=\\{g_{1}^{a_{1}}g_{2}^{a_{2}}\cdot\dotso\cdot
g_{N}^{a_{N}}:2^{N}\leq a_{1}\leq 2^{2N},\textrm{ and }0\leq a_{i}\leq
2^{2N}\textrm{ for }2\leq i\leq N\\}.$

Letting $E:=\bigcup_{N\in\mathbb{N}}A_{N}$, we get

$\bar{d}_{(G_{N})}(E)=\bar{d}_{(G_{N})}\left(\bigcup_{g\in
F}g^{-1}E\right)<\frac{3}{4},$

for all finite sets $F\subseteq G$.

Now assume that all elements of $G$ have finite order and that the enumeration
of $(g_{n})$ is such that $\textrm{ord}(g_{n+1})\geq\textrm{ord}(g_{n})$ for
all $n\in\mathbb{N}$. Consider the Følner sequence

$G_{N}:=\\{g_{1}^{a_{1}}\cdot\dotso\cdot g_{2^{2N}}^{a_{N}}:0\leq
a_{i}\leq\textrm{ord}(g_{i})\textrm{ for all }1\leq i\leq 2^{2N}\\}$

and the sets

$A_{N}:=\\{g_{1}^{a_{1}}\cdot\dotso\cdot g_{2^{2N}}^{a_{N}}:0\leq a_{i}\leq
b_{i}\textrm{ for all }1\leq i\leq 2^{2N},\ \textrm{with }i\textrm{ even;
}a_{i}=0\textrm{ with }i\textrm{ odd }\\},$

where $b_{i}$ is chosen so that $\frac{\left|(\bigcup_{j=1}^{i}A_{j})\cap
G_{i}\right|}{|G_{i}|}\in(\frac{1}{4},\frac{1}{2})$. Letting
$E:=\bigcup_{N\in\mathbb{N}}A_{N}$, we get that

$\bar{d}_{(G_{N})}(E)\geq\frac{1}{4},\quad\textrm{ and
}\quad\bar{d}_{(G_{N})}\left(\bigcup_{g\in F}g^{-1}E\right)<\frac{3}{4},$

for all finite subsets $F\subseteq G$.

Let now $(F_{N})$ be an arbitrary Følner sequence. We indicate next how to
construct the set $E$ that satisfies (3.1) for $(F_{N})$. To do this, we need
the following general fact, which follows from Lemma 4.1 in [DHZ] (we thank
Tomasz Downarowicz for providing this information).

######  Fact 3.4.

Let $\varepsilon>0$. We say that a family of finite sets
$\\{A_{n}:n\in\mathbb{N}\\}$ is $\varepsilon$-disjoint if each set $A_{n}$
admits a subset $A_{n}^{\prime}$ such that
$|A_{n}^{\prime}|\geq(1-\varepsilon)|A_{n}|$, and such that the new family
$\\{A_{n}^{\prime}:n\in\mathbb{N}\\}$ is disjoint. Fix a Følner sequence
$(F_{N})$ and let $(\varepsilon_{N})$ be a sequence of positive numbers with
$\varepsilon_{N}\to 0$ as $N\to\infty$. Given another Følner sequence
$(G_{N})$, we can find a Følner sequence $(G_{N}^{\prime})$ that is equivalent
to $(G_{N})$ (i.e., satisfying $\frac{|G_{N}\Delta
G_{N}^{\prime}|}{|G_{N}|}\to 0$ as $N\to\infty$) such that for each
$N\in\mathbb{N}$, $G_{N}^{\prime}$ is a union of $\varepsilon_{N}$-disjoint
subsets of the form $\\{F_{N}g:g\in G\\}$.

It is not hard to see that Fact 3.4 implies that the set $E$ in question can
be constructed with the help of the same argument utilized above for the
special Følner sequence $(G_{N})$. ∎

######  Remark 3.5.

For given $0<a\leq b<1$, one can construct sets $E\subseteq\mathbb{Z}$ with
$\bar{d}(E)=a$ such that $\bar{d}(\bigcup_{i=1}^{N}(E-i))=b$, for all large
enough $N$. This can be done as follows. Start with a sequence of disjoint
intervals $[a_{n},b_{n}]$ such that the set
$A:=\bigcup_{n\in\mathbb{N}}[a_{n},b_{n}]$ satisfies $\bar{d}(A)=b$ and
$\bar{d}(\bigcup_{i=1}^{N}(A-i))=b$ for all $N\in\mathbb{N}$ (this can be done
by imitating Hindman’s construction for $b=\frac{2}{3}$). Now let
$\beta=\frac{b}{a}$ and let $E:=A\cap\\{\lfloor
n\beta\rfloor:n\in\mathbb{N}\\}$. One easily checks that $E$ satisfies the
required properties.

######  Remark 3.6.

It is worth noting that the set $E$ constructed in Proposition 3.3 also
satisfies (as in Hindman’s original example)

(3.3) $\bar{d}_{(F_{N})}(E^{c}\cap g^{-1}E)=0$

for all $g\in G$.

It is of interest to know whether the phenomenon exhibited in Proposition 3.3
takes place in non-commutative amenable groups. We cannot show this in
complete generality, but for virtually nilpotent groups, we have the following
theorem:

######  Theorem 3.7.

Let $G$ be a finitely generated, countable virtually nilpotent group. Let
$(F_{N})$ be a Følner sequence. Then, there exists a set $E\subseteq G$ with

(3.4) $\bar{d}_{(F_{N})}(E)>0\textrm{ and
}\bar{d}_{(F_{N})}\left(\bigcup_{g\in B}g^{-1}E\right)<\frac{3}{4},$

for all finite subsets $B\subseteq G$.

###### Sketch of the proof.

First, by Fact 3.4, it suffices to show the result for a particular Følner
sequence $(F_{N})$ of our choosing. Let $F:=\\{x_{1},\dots,x_{k}\\}$ be a set
of generators for $G$. Then, the sequence
$\left(\bigcup_{j=1}^{n}F^{j}\right)$ (i.e. words of length at most $n$
generated by $F$) is a Følner sequence of polynomial growth.

Let $F_{N}:=\bigcup_{j=1}^{2^{2N}}F^{j}$, and let $A_{N}$ the set of words in
$F$ of length between $2^{2N-1}$ and $2^{2N}$. Then the set
$E:=\bigcup_{N\in\mathbb{N}}A_{N}$ satisfies (3.4). ∎

We will describe now another interesting application of Theorem 2.8

######  Definition 3.8.

Let $E\subseteq G$. We denote by $\mathcal{A}_{E}$ the algebra of sets
generated by shifts of the set $E$, i.e. sets of the form $\\{g^{-1}E:g\in
G\\}$ with the help of the operations of union, intersection and complement.

######  Theorem 3.9.

Let $E\subseteq G$ be such that $d^{*}(E)>0$. Then, there exists a Følner
sequence $(G_{N})_{N\in\mathbb{N}}$ in $G$, such that for all Følner sequences
$(F_{N})_{N\in\mathbb{N}}$, for all $E_{1},E_{2}\in\mathcal{A}_{E}$, we have

(3.5) $\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}d_{(G_{K})}(E_{1}\cap
g^{-1}E_{2})=d_{(G_{K})}(E_{1})d_{(G_{K})}(E_{2}).$

Moreover, if we let $E_{1}=E_{2}=E$ in (3.5), we get

(3.6) $\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}d_{(G_{K})}(E\cap
g^{-1}E)=d_{(G_{K})}(E)^{2}=d^{*}(E)^{2}.$

###### Proof.

Let $E\subseteq G$ with $d^{*}(E)>0$ and $E_{1},E_{2}\in\mathcal{A}_{E}$. By
Theorem 2.8 there exists an ergodic measure preserving system
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and a set $A\in\mathcal{B}$ with
$\mu(A)=d^{*}(E)$ satisfying (2.6). Let $A_{1},A_{2}\in\mathcal{B}$ be the
sets corresponding to the sets $E_{1},E_{2}$. By the proof of Theorem 2.8, the
functions $\mathbb{1}_{A_{1}}$ and $\mathbb{1}_{A_{2}}$ are continuous. Let
$(G_{K})$ be a Følner sequence such that
$\mu=\textrm{w*-}\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}\delta_{T_{g}\omega}$, where $\omega=(\mathbb{1}_{E}(g))_{g\in G}$.
Thus, we can write

(3.7) $\mu(A_{1}\cap g^{-1}A_{2})=\lim_{N\to\infty}\frac{1}{|G_{N}|}\sum_{g\in
G_{N}}\mathbb{1}_{A_{1}\cap g^{-1}A_{2}}(T_{g}\omega)=d_{(G_{K})}(E_{1}\cap
g^{-1}E_{2}).$

Taking an average over $g\in G$ in (3.7) along any left Følner sequence
$(F_{N})$ in $G$ immediately yields (3.5), given that the action
$(T_{g})_{g\in G}$ is ergodic. By construction of $\mu$, we have that
$\mu(A)=d_{(G_{K})}(E)=d^{*}(E)$ whence (3.6) also follows. ∎

We remark that (3.5) and (3.6) do not hold for arbitrary sequences $(G_{N})$:
let $E$ be as in (1.7), take $E_{1}=E_{2}=E$ and put $G_{N}=[1,2^{2N}]$.
Nonetheless, we have the following Proposition:

######  Proposition 3.10.

Let $E\subseteq G$. Let $(G_{N})$ be a Følner sequence in $G$. Then, there
exists a Følner subsequence $(G_{N_{k}})$ such that for all Følner sequences
$(F_{N})$ and all sets $F\in\mathcal{A}_{E}$ we have

(3.8) $\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}d_{(G_{N_{k}})}(F\cap g^{-1}F)\geq d_{(G_{N_{k}})}(F)^{2}.$

If we let $F=E$ in (3.8) we have the following variant of (3.8):

(3.9) $\liminf_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}\bar{d}_{(G_{N})}(E\cap g^{-1}E)\geq\bar{d}_{(G_{N})}(E)^{2}.$

###### Proof.

Given that $G$ is countable, the family of sets $\mathcal{A}_{E}$ is
countable. Thus, via a diagonal procedure, we can take a subsequence
$(G_{N_{k}})$ of our given Følner sequence $(G_{N})$ so that

(3.10) $\mu:=\textrm{w*-}\lim_{k\to\infty}\frac{1}{|G_{N_{k}}|}\sum_{g\in
G_{N_{k}}}\delta_{T_{g}\omega}$

exists, where $\omega=(\mathbb{1}_{E}(g))_{g\in G}$.

Letting $A:=\\{x:x(e)=1\\}$ we see that $F(x)=x(0)=\mathbb{1}_{A}(x)\in C(X)$,
so the function $F_{1}$ representing the set $E$ is also in $C(X)$. Thus, we
can write

(3.11) $\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}d_{(G_{N_{k}})}(F\cap
g^{-1}F)=\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}\int F_{1}\cdot
T_{g}F_{1}\ d\mu.$

Using the ergodic decomposition for $\mu$, we see that the last term in
Equation (3.11) can be rewritten as

(3.12) $\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}\int\left(\int_{X}F_{1}\cdot T_{g}F_{1}\ d\mu_{t}\right)d\lambda(t),$

where each $\mu_{t}$ is an ergodic measure. Thus, by von Neumann’s Mean
Ergodic Theorem and Jensen’s inequality, we have

(3.13) $\int\left(\int_{X}F_{1}\
d\mu_{t}\right)^{2}d\lambda(t)\geq\left(\int_{X}F_{1}\ d\mu\right)^{2}.$

By construction, the right hand side in (3.13) is equal to
$d_{(G_{N_{k}})}(F)^{2}$. To prove (3.9), we use Theorem 2.3 to obtain a
measure preserving system $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and a set
$A\in\mathcal{B}$ with $\mu(A)=\bar{d}_{(G_{N})}(E)$ satisfying inequality
(2.1). In particular, this means that for all $g\in G$ we have

(3.14) $\bar{d}_{(G_{N})}(E\cap g^{-1}E)\geq\mu(A\cap(T_{g})^{-1}A).$

Using the same argument that leads to inequality (3.13), we get that for any
Følner sequence $(F_{N})$

(3.15) $\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}\mu(A\cap(T_{g})^{-1}A)\geq\mu(A)^{2}.$

Combining (3.14) and (3.15) we obtain

$\liminf_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in F_{N}}\bar{d}_{(G_{N})}(E\cap
g^{-1}E)\geq\mu^{2}(A)=\bar{d}_{(G_{N})}(E)^{2},$

as desired. ∎

######  Remark 3.11.

As was mentioned at the beginning of the section, the results contained herein
can be effortlessly carried over to countable cancellative amenable
semigroups. Indeed, if $S$ is such a semigroup, then one can embed it into
$G:=\\{st^{-1}:s,t\in S\\}$, which is now going to be a countable amenable
group (see Proposition 1.17 in [P]). It is straightforward to check that a
Følner sequence in $S$ becomes a Følner sequence in $G$, and this is all that
is needed to push the results to this more general context.

##  4\. Ergodic sequences and Hindman’s theorem

The goal of this section is to extend Hindman’s covering theorem (Theorem 1.3)
to unions of the form $\bigcup_{n=1}^{N}(E-k_{n})$. In particular, we will use
Ergodic Theory to characterize (and provide numerous examples of) the
sequences $(k_{n})$ with the property that for any $E\subseteq\mathbb{Z}$ with
$d^{*}(E)>0$ one has $d^{*}\left(\bigcup_{n=1}^{N}(E-k_{n})\right)\to 1$ as
$N\to\infty$. The ergodic approach can easily be extended to amenable groups.
We will discuss this after the proof of Theorem 4.8.

######  Definition 4.1.

We say that a sequence of integers $(k_{n})_{n\in\mathbb{N}}$ has the
_combinatorial sweeping out property_ if for every $E\subseteq\mathbb{Z}$ with
$d^{*}(E)>0$ we have

(4.1)
$d^{*}\left(\bigcup_{n=1}^{N}(E-k_{n})\right)\xrightarrow[N\to\infty]{}1$

The class of sequences satisfying (4.1) is quite wide. For example, as we will
see below, ergodic sequences have the combinatorial sweeping out property.

######  Definition 4.2.

We say that a sequence of positive integers $(k_{n})$ is an _ergodic sequence_
if for every ergodic measure preserving system $(X,\mathcal{B},\mu,T)$ we have

(4.2) $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(A\cap
T^{-k_{n}}B)=\mu(A)\mu(B).$

for all $A,B\in\mathcal{B}$.

######  Remark 4.3.

One can show (see Theorem 4.9 below) that a sequence is ergodic if and only if
for all $f\in L^{2}(\mu)$ we have

(4.3) $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}T^{k_{n}}f=\int_{X}f\ d\mu,$

where the convergence is with respect to the $L^{2}(\mu)$ norm.

The following definition deals with an ergodic counterpart of the notion of
combinatorial sweeping out:

######  Definition 4.4.

We say that a sequence of integers $(k_{n})_{n\in\mathbb{N}}$ has the _ergodic
sweeping out property_ if for every ergodic measure preserving system
$(X,\mathcal{B},\mu,T)$ and for every $A\in\mathcal{B}$ with $\mu(A)>0$ we
have

(4.4) $\mu\left(\bigcup_{n\in\mathbb{N}}T^{-k_{n}}A\right)=1.$

As we will see in Theorem 4.8, the notions of ergodic sweeping out and
combinatorial sweeping out, in fact, coincide, so one can use the term
sweeping out unambiguously. But first we will show, as promised, that ergodic
sequences have the combinatorial sweeping out property.

######  Proposition 4.5.

Let $(k_{n})$ be an ergodic sequence. Then $(k_{n})$ has the combinatorial
sweeping out property.

###### Proof.

We first note that if $(k_{n})$ is an ergodic sequence,
$(X,\mathcal{B},\mu,T)$ is an ergodic measure preserving system and
$A\in\mathcal{B}$ is such that $\mu(A)>0$, then

(4.5) $\mu\left(\bigcup_{n\in\mathbb{N}}T^{-k_{n}}A\right)=1.$

(Otherwise taking $B=X\setminus\bigcup_{n\in\mathbb{N}}T^{-k_{n}}A$ we would
get a contradiction with (4.2).)

Now let $E\subseteq\mathbb{Z}$ be such that $d^{*}(E)>0$ and take
$\varepsilon>0$. By Theorem 2.8 there exists an ergodic measure preserving
system $(X,\mathcal{B},\mu,T)$ and a set $A\in\mathcal{B}$ with
$\mu(A)=d^{*}(E)$ satisfying

(4.6)
$d^{*}\left(\bigcup_{n=1}^{N}(E-k_{n})\right)\geq\mu\left(\bigcup_{n=1}^{N}T^{-k_{n}}A\right)$

for all $N\in\mathbb{N}$. Using (4.5) and continuity of $\mu$ we see that
$(k_{n})$ satisfies (1.8). ∎

We would like to note that in the proof of Proposition 4.5 the ergodic
Furstenberg correspondence principle, Theorem 2.8, was utilized. Theorem 2.8
will also play an instrumental role in the proof of the equivalence of ergodic
sweeping out and combinatorial sweeping out (see Theorem 4.8).

There are sequences with the combinatorial sweeping out property that are not
ergodic. A rather cheap example is provided by the sequence $k_{n}:=[\log n]$.
While this sequence does not satisfy (4.2), it takes on all nonnegative
integer values and hence is sweeping out. A more interesting example is given
by the sequence $k_{n}:=[n^{2}+\log n]$. By [BKQW], $(k_{n})$ is not ergodic.
However, one can show that for any ergodic measure preserving system
$(X,\mathcal{B},\mu,T)$ and any sets $A,B\in\mathcal{B}$ with $\mu(A)>0$ and
$\mu(B)>0$ there is some $n\in\mathbb{N}$ such that $\mu(A\cap
T^{-([n^{2}+\log n])}B)>0$. This implies that
$\mu\left(\bigcup_{n\in\mathbb{N}}T^{-[n^{2}+\log n]}A\right)=1$. This fact
together with Theorem 4.8 below imply that $(k_{n})$ is sweeping out.

######  Definition 4.6.

We say that an invertible measure preserving system $(X,\mathcal{B},\mu,T)$
has a topological model if there exists a measure-theoretically isomorphic
system $(\hat{X},\hat{\mathcal{B}},\hat{\mu},\hat{T})$, where $\hat{X}$ is a
compact metric space and $T$ is a homeomorphism from $\hat{X}$ to itself.

######  Theorem 4.7 (Jewett-Krieger Theorem, [J], [Kr]).

Every ergodic invertible measure preserving system $(X,\mathcal{B},\mu,T)$ has
a
uniquely ergodic 888A measure preserving system $(X,\mathcal{B},\mu,T)$ is
called _uniquely ergodic_ if $X$ is a compact metric space, $T:X\rightarrow X$
is a homeomorphism and $\mu$ is the unique $T$-invariant normalized Borel
measure on $X$.  topological model.

The following theorem characterizes sequences which are "good" for Hindman’s
covering theorem (see Theorem 1.3) and establishes the equivalence between
measurable sweeping out and combinatorial sweeping out:

######  Theorem 4.8.

A sequence of integers has the combinatorial sweeping out property if and only
if it has the ergodic sweeping out property.

###### Proof.

In one direction, assume that $(k_{n})$ has the ergodic sweeping out property.
Let $E\subseteq\mathbb{Z}$ with $d^{*}(E)>0$. Let $(X,\mathcal{B},\mu,T)$ be a
measure preserving system and $A\in\mathcal{B}$ which are guaranteed by
Theorem 2.8 and satisfy the following special case of (2.6):

$d^{*}\left(\bigcup_{n=1}^{N}(E-k_{n})\right)\geq\mu\left(\bigcup_{n=1}^{N}T^{-k_{n}}A\right)\textrm{
for all }N\in\mathbb{N}.$

Since $\mu(A)=d^{*}(E)>0$ and the system $(X,\mathcal{B},\mu,T)$ is ergodic,
equality (4.4) holds. Continuity of the measure $\mu$ and equality (4.4)
together imply that

$d^{*}\left(\bigcup_{n=1}^{N}(E-k_{n})\right)\xrightarrow[N\to\infty]{}1$

and so $(k_{n})$ is combinatorially sweeping out.

In the other direction, we will show the contrapositive. Assume that there
exists an ergodic system $(X,\mathcal{B},\mu,T)$ and a set $A\in\mathcal{B}$
with $\mu(A)>0$ such that
$\mu\left(\bigcup_{n\in\mathbb{N}}T^{-k_{n}}A\right)=1-\delta$ for some
$\delta>0$. Without loss of generality, one can assume that
$(X,\mathcal{B},\mu,T)$ is invertible. Indeed, otherwise we can work with the
invertible extension of $(X,\mathcal{B},\mu,T)$.

In view of the Jewett-Krieger theorem (Theorem 4.7), we can also assume that
$(X,T)$ is a uniquely ergodic topological dynamical system.

Since $X$ is a compact metric space, the probability measure $\mu$ is regular.
Let $K_{1}\subseteq A$ be a compact set such that $\mu(K_{1})>0$ and
$K_{2}\subseteq X\setminus\bigcup_{n\in\mathbb{N}}T^{-k_{n}}A$ a compact
subset such that $\mu(K_{2})\geq\frac{\delta}{2}$. Since $T$ is ergodic, von
Neumann’s mean ergodic theorem implies

$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}T^{n}f=\int_{X}f\ d\mu,$

where $f=\mathbb{1}_{K_{1}}$ and the convergence is in the $L^{2}(\mu)$-norm.
Thus, there is a subsequence $(N_{k})$ and $X_{0}\subseteq X$ with
$\mu(X_{0})=1$ such that for all $x\in X_{0}$ we have

$\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n=1}^{N_{k}}f(T^{n}x)=\int_{X}f\ d\mu.$

Let $x_{0}\in X_{0}$ and consider the set

$E:=\\{n\in\mathbb{Z}:T^{n}x_{0}\in K_{1}\\}.$

Note that by the choice of $x_{0}$ and $E$

$\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n=1}^{N_{k}}\mathbb{1}_{E}(n)=\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n=1}^{N_{k}}\mathbb{1}_{K_{1}}(T^{n}x_{0})=\int_{X}\mathbb{1}_{K_{1}}\
d\mu=\mu(K_{1})>0$

which implies that $d^{*}(E)>0$. We claim that for all $N\in\mathbb{N}$, we
have

$d^{*}\left(\bigcup_{n=1}^{N}(E-k_{n})\right)\leq 1-\frac{\delta}{2}.$

Indeed, let $N\in\mathbb{N}$. By our choice of $K_{2}$, the set
$T^{-k_{1}}K_{1}\cup\dots\cup T^{-k_{N}}K_{1}$ is a compact set disjoint from
$K_{2}$, so by Urysohn’s lemma there is a continuous function
$f:X\rightarrow[0,1]$ such that $f(x)=1$ if $x\in T^{-k_{1}}K_{1}\cup\dots\cup
T^{-k_{N}}K_{1}$ and $f(x)=0$ if $x\in K_{2}$. Thus,

(4.7)
$d^{*}\left(\bigcup_{n=1}^{N}(E-k_{n})\right)=\limsup_{N-M\to\infty}\frac{1}{N-M}\sum_{n=M}^{N-1}\mathbb{1}_{T^{-k_{1}}K_{1}\cup\dots\cup
T^{-k_{N}}K_{1}}(T^{n}x_{0})\\\
\leq\lim_{N-M\to\infty}\frac{1}{N-M}\sum_{n=M}^{N-1}f(T^{n}x_{0})=\int_{X}f\
d\mu\leq 1-\frac{\delta}{2}.$

(Note that we used the fact that for uniquely ergodic systems,
$\lim_{N-M\to\infty}\frac{1}{N-M}\sum_{n=M}^{N-1}f(T^{n}x_{0})$ exists for any
continuous function $f$ and any $x_{0}\in X$). ∎

While we have chosen to stick with $\mathbb{Z}$ for the sake of clarity, it is
worth mentioning that one can establish a version of Theorem 4.8 for general
countable amenable groups. Indeed, it is not hard to define in total analogy
with Definition 4.1 and 4.2 the notions of combinatorial and measurable
sweeping out for general countable amenable groups. To carry out the amenable
generalization of Theorem 4.8, one has to invoke a general form of Jewett-
Krieger’s theorem due to Rosenthal (see [R]). Note that Definition 4.2
naturally extends to $\mathbb{Z}^{d}$ and even to any amenable group, and can
be used to provide numerous examples of ergodic (and hence sweeping out)
sequences. One can show, for example, that every subset of positive density of
a minimally almost periodic group is ergodic.

We give now some examples of ergodic sequences both in $\mathbb{Z}$ and
$\mathbb{Z}^{d}$. (See [BLes] and [BKQW].)

  * (1)

$\\{[bn^{c}]:n\in\mathbb{N}\\}$, where $c\notin\mathbb{Q}$, $c>1$ and $b\neq
0$.

  * (2)

$\\{[bn^{c}+dn^{a}]:n\in\mathbb{N}\\}$, where $b,d\neq 0$,
$b/d\notin\mathbb{Q}$, $c\geq 1$, $a>0$ and $a\neq c$.

  * (3)

$\\{[bn^{c}(\log n)^{d}]:n\in\mathbb{N}\\}$, where $b\neq 0$,
$c\notin\mathbb{Q}$, $c>1$ and $d$ is any number

  * (4)

$\\{[bn^{c}(\log n)^{d}]:n\in\mathbb{N}\\}$, where $b\neq 0$,
$c\in\mathbb{Q}$, $c>1$ and $d\neq 0$.

  * (5)

$\\{[bn^{c}+d(\log n)^{a}]:n\in\mathbb{N}\\}$, where $b,d\neq 0$, $c\geq 1$
and $a>1$.

Another class of examples of ergodic sequences is provided by sequences of the
form $[g(n)]$, where $g$ is any _tempered function_ 999Let $k$ be a non-
negative integer. A real-valued function $g$ which is $(k+1)$ times
continuously differentiable on $[x_{0},\infty)$, where $x_{0}\geq 0$, is
called a _tempered function of order_ $k$ if (a) $g^{(k+1)}(x)$ tends
monotonically to zero as $x\to\infty$, and (b)
$\lim_{x\to\infty}x|g^{(k+1)}(x)|=+\infty$. (see [BK], Theorem 7.1)

The paper [BKS] provides a class of examples of ergodic sequences involving
primes for $\mathbb{Z}^{d}$ actions. It is clear that these sequences will be
sweeping out for $\mathbb{Z}^{d}$ due to a straightforward generalization of
Lemma 4.5. Namely, sequences of the form
$\\{([\xi_{1}(p_{n})],\dots,[\xi_{d}(p_{n})]):n\in\mathbb{N}\\}$, where
$p_{n}$ denotes the $n$-th prime with the standard order, and where
$\xi_{1},\dots,\xi_{d}$ are functions in a Hardy field with subpolynomial
growth such that either

(4.8)
$\lim_{x\to\infty}\frac{\xi(x)}{x^{l+1}}=\lim_{x\to\infty}\frac{x^{l}}{\xi(x)}=0\textrm{
for some }l\in\mathbb{N},\textrm{ or
}\lim_{x\to\infty}\frac{\xi(x)}{x}=\lim_{x\to\infty}\frac{\log x}{\xi(x)}=0,$

and such that any combination of the form $\sum_{i=1}^{d}b_{i}\xi_{i}$ also
satisfies (4.8) for all
$(b_{1},\dots,b_{d})\in\mathbb{R}^{d}\setminus\\{\vec{0}\\}$. A particular
case of this would be sequences of the form
$\\{([p_{n}^{c_{1}}],\dots,[p_{n}^{c_{d}}]):n\in\mathbb{N}\\}$, where $p_{n}$
denotes the $n$-th prime, and where $c_{1},\dots,c_{d}$ are distinct positive
real numbers such that $c_{i}\notin\mathbb{N}$ for all $i$. (This special case
was obtained in [BKMST]).

We conclude with showing that strong and weak convergence lead to the same
definition of an ergodic sequence. We did not need this fact for this section,
but we believe it is of independent interest. Note that Theorem 4.9 can be
viewed as a generalization of the well-known fact that the $L^{2}$ version of
the mean ergodic theorem follows from its weak convergence version.

######  Theorem 4.9.

For a sequence $(k_{n})\subseteq\mathbb{Z}$, the following are equivalent:

  * (1)

For every ergodic measure preserving system $(X,\mathcal{B},\mu,T)$ and for
all $A,B\in\mathcal{B}$

(4.9) $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(A\cap
T^{-k_{n}}B)=\mu(A)\mu(B).$

  * (2)

For every ergodic measure preserving system $(X,\mathcal{B},\mu,T)$ and for
all $f\in L^{2}(\mu)$

(4.10) $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}T^{k_{n}}f=\int_{X}f\ d\mu.$

where the convergence is with respect to the $L^{2}(\mu)$ norm.

###### Proof.

Strong convergence implies weak convergence, so one direction is trivial. In
the other direction, notice that for any $x\in\mathbb{T}$, (4.9) implies that
$(k_{n}x)_{n\in\mathbb{N}}$ is uniformly distributed in
$\overline{\\{nx:n\in\mathbb{Z}\\}}\subseteq\mathbb{T}$. To see this we argue
as follows. First, observe that, by a standard approximation argument, (4.9)
implies that for all $f,g\in L^{2}(\mu)$

(4.11) $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\int_{X}f\cdot
T^{k_{n}}\bar{g}\ d\mu=\int_{X}f\ d\mu\int_{X}\bar{g}\ d\mu.$

Let $X=\mathbb{T}$, $x\in\mathbb{T}\setminus\\{0\\}$ and put $T:X\rightarrow
X$ the rotation by $x$, i.e. $Tz=z+x$ for all $z\in\mathbb{T}$. Let $\chi$ be
a non-trivial character of the compact abelian group
$G_{x}:=\overline{\\{nx:n\in\mathbb{Z}\\}}$ (note that either
$G_{x}=\mathbb{T}$ or it is a finite subgroup of $\mathbb{T}$), and set
$f(y)=\bar{\chi}(y)$ and $g(y)=f(y)=\bar{\chi}(y)$. With these choices,
equation (4.11) becomes

(4.12)
$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\int_{X}\bar{\chi}(y)\chi(y+k_{n}x)\
d\nu(y)=\int_{X}\chi(y)\ d\nu(y)\int_{X}\bar{\chi}(y)\ d\nu(y),$

where $\nu$ is the Haar measure on $G_{x}$. Now, simplifying (4.12) we get

(4.13) $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\chi(k_{n}x)=0,$

so $(k_{n}x)$ is equidistributed on $G_{x}$, as desired. Then, for any $f\in
L^{2}(\mu)$, we have

(4.14) $\left|\left|\frac{1}{N}\sum_{n=1}^{N}T^{k_{n}}f-\int_{X}f\
d\mu\right|\right|^{2}=\left|\left|\frac{1}{N}\sum_{n=1}^{N}T^{k_{n}}f\right|\right|^{2}-\left|\int_{X}f\
d\mu\right|^{2}$

after expanding the inner product and recombining. Next we are going to invoke
the classical Herglotz’s theorem, which states that the positive definite
sequence $a(n)=\langle T^{n}f,f\rangle$ has a representation
$a(n)=\int_{\mathbb{T}}e^{2\pi inx}\ d\nu_{f}(x)$, for some finite positive
measure $\nu_{f}$ on $\mathbb{T}$. Thus, the right hand side of (4.14) becomes

(4.15) $\frac{1}{N^{2}}\sum_{n,m=1}^{N}\langle
T^{k_{n}-k_{m}}f,f\rangle-\left|\int_{X}f\
d\mu\right|^{2}=\frac{1}{N^{2}}\sum_{n,m=1}^{N}\int_{\mathbb{T}}e^{2\pi
i(k_{n}-k_{m})x}\ d\nu_{f}(x)-\left|\int_{X}f\ d\mu\right|^{2}=\\\
\int_{\mathbb{T}}\left|\frac{1}{N}\sum_{n=1}^{N}e^{2\pi ik_{n}x}\right|^{2}\
d\nu_{f}(x)-\left|\int_{X}f\ d\mu\right|^{2}.$

Using (4.13) for $\chi(x)=e^{2\pi ix}$, and invoking ergodicity of $T$ we get

(4.16)
$\lim_{N\to\infty}\int_{\mathbb{T}}\left|\frac{1}{N}\sum_{n=1}^{N}e^{2\pi
ik_{n}x}\right|^{2}\ d\nu_{f}(x)-\left|\int_{X}f\
d\mu\right|^{2}=\nu_{f}(\\{0\\})-\left|\int_{X}f\ d\mu\right|^{2}=0.$

(In the last step we used the fact that
$\nu_{f}(\\{0\\})=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\int_{\mathbb{T}}e^{2\pi
inx}\ d\nu_{f}(x)=\\\ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\langle
T^{n}f,f\rangle=\left|\int_{X}f\ d\mu\right|^{2}$.) ∎

######  Remark 4.10.

Theorem 4.9 can be extended to the context of locally compact abelian groups.
We omit the details.

##  5\. A characterization of countable amenable weakly mixing groups

In this section, we will establish a characterization of countable weakly
mixing groups with the help of the amenable version of Hindman’s covering
theorem (Theorem 3.1). Recall that a group $G$ is _weakly mixing_ (or
minimally almost periodic) if any ergodic measure preserving action of $G$ on
a probability space $\mathds{X}=(X,\mathcal{B},\mu)$ is automatically weakly
mixing i.e., the diagonal action of $G$ on $\mathds{X}\times\mathds{X}$ is
ergodic. (In this case, the diagonal action on an arbitrary finite product
$\mathds{X}\times\dots\times\mathds{X}$ is also ergodic).

We begin with the following proposition:

######  Proposition 5.1.

Let $G$ be a countable amenable WM group. Let $E\subseteq G$ with $d^{*}(E)>0$
and let $d\in\mathbb{N}$. Then, for all $\varepsilon>0$, there exist
$g_{1},\dots,g_{k}\in G$ such that

(5.1)
$d^{*}\left(\bigcup_{i=1}^{k}(g_{i},\dots,g_{i})^{-1}\underbrace{(E\times\dots\times
E)}_{d\ \textrm{times}}\right)>1-\varepsilon.$

###### Proof.

By Theorem 2.8 there exists an ergodic measure preserving system
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ and a set $A\in\mathcal{B}$ with
$\mu(A)=d^{*}(E)$ satisfying (2.6). Since $G$ is a WM group, the measure
$\mu\otimes\mu$ will be ergodic for the diagonal action on $X\times X$.

Let $(G_{N})$ be a Følner sequence such that
$\mu=\textrm{w*-}\lim_{N\to\infty}\frac{1}{|G_{N}|}\sum_{g\in
G_{N}}\delta_{T_{g}\omega}$, where $\omega=(\mathbb{1}_{E}(g))_{g\in G}$, as
in Section 2. Notice that, for each $k\in\mathbb{N}$, we have

(5.2) $d^{*}\left(\bigcup_{i=1}^{k}(g_{i},\dots,g_{i})^{-1}(E\times\dots\times
E)\right)\geq d_{(G_{N}\times\dots\times
G_{N})}\left(\bigcup_{i=1}^{k}(g_{i},\dots,g_{i})^{-1}(E\times\dots\times
E)\right),$

for all $g_{1},\dots,g_{k}\in G$. Applying the inclusion-exclusion principle
we can change the unions in (5.2) for intersections. The density of the
typical intersection can be computed with a product of densities (note that
$d_{(G_{N}\times G_{N})}(E_{1}\cap E_{2}\times F_{1}\cap
F_{2})=d_{(G_{N})}(E_{1}\cap E_{2})d_{(G_{N})}(F_{1}\cap F_{2})$). Finally,
using the definition of $\mu$ we see that the quantity on the right hand side
in (5.2) equals

$\mu\otimes\dots\otimes\mu\left(\bigcup_{i=1}^{k}(T_{g_{i}}\times\dots\times
T_{g_{i}})(A\times\dots\times A)\right)$

Since $G$ is a WM group, the measure $\mu\otimes\dots\otimes\mu$ is ergodic
for the diagonal $G$-action on the product space $X\times\dots\times X$. Since
$\mu(A)>0$ we have $(\mu\otimes\dots\otimes\mu)(A\times\dots\times A)>0$,
whence
$(\mu\otimes\dots\otimes\mu)\left(\bigcup_{g\in G}(T_{g}\times\dots\times
T_{g})^{-1}(A\times\dots\times A)\right)=1$, so the result follows by
continuity of $\mu$. ∎

We will show next that the covering property (5.1) characterizes weakly mixing
groups:

######  Theorem 5.2.

Let $G$ be a countable amenable group that is not WM. Then there exists a set
$E\subseteq G$ with $d^{*}(E)\in(0,1)$ such that for all $r\in\mathbb{N}$ and
any finite subset $\\{h_{1},\dots,h_{r}\\}$ of $G$ we have

(5.3) $d^{*}\left(\bigcup_{i=1}^{r}(h_{i},h_{i})^{-1}(E\times E)\right)\leq
C<1,$

where the constant $C$ in (5.3) is independent of $\\{h_{1},\dots,h_{r}\\}$.

###### Proof.

Before constructing the set $E$, we need to do some preparatory work. First,
we observe that by a Corollary of Lemma 3.3 in [BBF], (see equation (5) in
[BBF]) for any $B\subseteq G$, and for any Følner sequence $(F_{n})\subseteq
G$, there is a sequence $(t_{n})$ such that

$d^{*}(B)=d_{F_{n}t_{n}}(B).$

Hence, it suffices to show that for any pair of Følner sequences
$(I_{N})\subseteq G$ and $(J_{N})\subseteq G$ there exists $0<C<1$ such that

(5.4) $\bar{d}_{(I_{N}\times
J_{N})}\left(\bigcup_{i=1}^{r}(h_{i},h_{i})^{-1}(E\times E)\right)\leq C<1.$

for all $r\in\mathbb{N}$ and all finite sets $\\{h_{1},\dots,h_{r}\\}$. Note
that this step is essential because general Følner sequences of $G\times G$
need not be of the form $(I_{N}\times J_{N})$.

Next, we observe that since $G$ is not WM it must admit a non-trivial finite
dimensional representation $\pi:G\rightarrow U(k)$, where $U(k)$ is the
unitary group of $k\times k$ complex matrices (see for example [S], Theorem
3.4). Thus, $H=\\{\pi(g):g\in G\\}$ is a non-trivial subgroup of $U(k)$.

Let $X=\overline{H}$, the closure of $H$ in $U(k)$. Then, $X$ is a compact
metric group with the topology inherited from $\mathbb{C}^{k^{2}}$. As such,
it carries a unique normalized Haar measure (fully supported on $X$) which we
denote by $\mu$.

Moreover, $G$ acts on $X$ by translations as follows: let $R_{g}x:=\pi(g)\cdot
x$. Note that the action $(R_{g})_{g\in G}$ is minimal because $\pi(G)=H$ and
$H$ is dense in $X$. Clearly, $(X,\textrm{Borel}(X),\mu,(R_{g})_{g\in G})$ is
a uniquely ergodic measure preserving system.

Let $d$ be a bi-invariant metric on $X$ (such a metric exists by the Birkhoff-
Kakutani Theorem, see [Bi] and [K]). Let $g_{0}\in X$ be such that
$d(e,g_{0})=b=\max\\{d(e,g):g\in G\\}>0$ (since $X$ is compact and non-
trivial, $0<b<\infty$). Let $U:=B(e,\frac{b}{16})$ (the open ball of radius
$\frac{b}{16}$ centered at $e$). Clearly $\mu(U)>0$ since $\mu$ is fully
supported on $X$. Let $U^{\prime}:=B(e,r)$ where $0<r<\frac{b}{16}$ is such
that $\mu(\partial U^{\prime})=0$. Such an $r$ must exist (see for example
Lemma 2.21 in [BCRZ]). At this point we are ready to define the set $E$.
Namely, choose an arbitrary $x_{0}\in X$ and put

(5.5) $E:=\\{g\in G:R_{g}x_{0}\in U^{\prime}\\}.$

Since $(X,\mu,(R_{g})_{g\in G})$ is uniquely ergodic and $\mu(\partial
U^{\prime})=0$, the function $\mathbb{1}_{U^{\prime}}$ can be approximated by
continuous functions. From these two facts, it follows that
$\textrm{w*-}\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}\delta_{T_{g}x_{0}}$ converges to $\mu$.

Next, observe that for all $g\in X$, the open set
$W:=B(g_{0},\frac{b}{16})\times B(e,\frac{b}{16})$ satisfies

(5.6) $(g,g)^{-1}(U\times U)\cap W=\emptyset,$

because otherwise, by the triangle inequality we would get that $d(e,g)\leq
d(e,g_{1}u)+d(g_{1}u_{1},g_{1}u_{2})+d(g_{1}u_{2},g_{0})<2\cdot\frac{b}{16}+d(u_{1},u_{2})<\frac{b}{8}+\frac{b}{8}<b$,
where $g_{1}\in G$ and $u_{1},u_{2}\in U$ are such that
$d(e,g_{1}u_{1})<\frac{b}{16}$ and $d(g_{0},g_{1}u_{2})<\frac{b}{16}$, a
contradiction with our choice of $e,g_{0}$.

It follows from (5.6) and from the fact that $G$ acts minimally on $X$ that
$\overline{\Delta\cdot(U\times U)}\neq X\times X$, where $\Delta$ is the
diagonal in $X^{2}$. Thus,

$(\mu\otimes\mu)(\Delta\cdot(U\times U)):=C<1.$

Note also that since $\mu$ is fully supported on $X$, $\mu\otimes\mu$ is fully
supported on $X\times X$, and so we have $(\mu\otimes\mu)(\Delta\cdot(U\times
U)>0$. Let $(I_{N}),(J_{N})$ be two Følner sequences in $G$, and
$\\{h_{1},\dots,h_{r}\\}$ a finite set of elements of $G$. Then,

(5.7) $\bar{d}_{(I_{N}\times
J_{N})}\left(\bigcup_{i=1}^{r}(h_{i},h_{i})^{-1}(E\times
E)\right)=\limsup_{N\to\infty}\frac{1}{|I_{N}||J_{N}|}\sum_{g\in
F_{N}}\sum_{h\in J_{N}}\mathbb{1}_{\bigcup_{i=1}^{r}(h_{i},h_{i})^{-1}(E\times
E)}(g,h).$

By (5.5), the last term in equation (5.7) is equal to

(5.8) $\limsup_{N\to\infty}\frac{1}{|I_{N}||J_{N}|}\sum_{g\in F_{N}}\sum_{h\in
J_{N}}\mathbb{1}_{\bigcup_{i=1}^{r}(T_{h_{i}}\times
T_{h_{i}})^{-1}(U^{\prime}\times U^{\prime})}(T_{g}x_{0},T_{h}x_{0}).$

We now use the inclusion-exclusion principle to separate the variables in each
summand. Let $\prod_{i=1}^{r}\mathbb{1}_{T_{h_{i}}U^{\prime}}$, $1\leq t\leq
r$, be a typical term obtained through this process. By unique ergodicity of
the action $(R_{g})_{g\in G}$ we see that for any Følner sequence $(F_{N})$

$\lim_{N\to\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}\mathbb{1}_{\bigcap_{i=1}^{t}T_{h_{i}}U^{\prime}}(T_{g}x_{0})=\mu\left(\bigcap_{i=1}^{t}T_{h_{i}}U^{\prime}\right).$

Indeed, since $U^{\prime}$ is an open set with $\mu(\partial U^{\prime})=0$,
we also have that $\mu(\partial(T_{g}U^{\prime}))=0$ for all $g\in G$ given
that $T_{g}(\partial U^{\prime})=\partial(T_{g}U^{\prime})$ as $T_{g}$ is a
measure preserving homeomorphism. Consequently, functions of the form
$\prod_{i=1}^{t}\mathbb{1}_{T_{g_{i}}U^{\prime}}$, $1\leq t\leq r$, can be
approximated by continuous functions. Thus, the limit in equation (5.8) in
fact exists and (after performing the inclusion-exclusion principle backwards)
is equal to

(5.9) $(\mu\otimes\mu)\left(\bigcup_{i=1}^{r}(T_{h_{i}}\times
T_{h_{i}})^{-1}(U^{\prime}\times
U^{\prime})\right)\leq(\mu\otimes\mu)(\Delta\cdot(U\times U))=C<1,$

which completes the proof. ∎

##  6\. A general form of Hindman’s covering theorem

One may wonder if a version of Hindman’s covering theorem (Theorem 3.1) is
valid for discrete amenable semigroups which are not necessarily countable or
cancellative. In this section we will show that this is indeed the case.

Recall that a discrete semigroup $G$ is left amenable if there exists a left
invariant mean $m:\ell^{\infty}(G)\rightarrow\mathbb{C}$ 101010We say that
$m\in\ell^{\infty}(G)^{*}$ is a left _invariant mean_ if it is a continuous
linear functional from $\ell^{\infty}(G)$ to $\mathbb{C}$ such that (i) for
every $f\in\ell^{\infty}(G)$ and for every $g\in G$ we have $m({}_{g}f)=m(f)$,
where ${}_{g}f(x):=f(gx)$ for all $x\in G$, (ii) $m(f)\geq 0$ for any bounded
function $f:G\rightarrow[0,\infty)$, and (iii) $m(\mathbb{1}_{G})=1$.. (Note
that, for discrete countable groups, this is equivalent to the definition of
left amenability given in Section 2.)

In the context of means, a notion of largeness presents itself. Let us denote
by $\mathfrak{M}(G)$ the space of left invariant means on $G$. We say a subset
$E$ of a discrete left amenable semigroup $G$ is large if
$m(\mathbb{1}_{E})>0$ for some mean $m\in\mathfrak{M}(G)$. This leads to the
following definition of a notion of upper Banach density that is valid in all
amenable semigroups (see Definition 2.7 [BG]):

######  Definition 6.1.

Let $G$ be an amenable semigroup, and let
$E\subseteq G$. The _upper Banach density_ of $E$ is

$d^{*}(E):=\sup\\{m(\mathbb{1}_{E}):m\in\mathfrak{M}(G).\\}$

Notice that since $\mathfrak{M}(G)$ is weak* compact there is some
$m\in\mathfrak{M}(G)$ that achieves the supremum in Definition 6.1. If $G$ is
a discrete countable amenable group and $E\subseteq G$, then the upper Banach
density of $E$ as given in Definition 2.2 agrees with the one in Definition
6.1. Moreover, in this case, we have
$d^{*}(E)=\max\\{m(\mathbb{1}_{E}):m\in\mathfrak{M}(G)\\}$. We are now in a
position to formulate a general version of Hindman’s theorem.

######  Theorem 6.2.

Let $G$ be an amenable semigroup, and let $E$ be a subset of $G$ with
$d^{*}(E)>0$. Then, for every $\varepsilon>0$, there exists $k\in\mathbb{N}$
and $g_{1},\dots,g_{k}\in G$ such that $d^{*}(g_{1}^{-1}E\cup\dots\cup
g_{k}^{-1}E)>1-\varepsilon$.

The proof of Theorem 6.2 requires a few preliminary results which will be
given next. Recall that the space of _weakly almost periodic_ functions on
$G$, denoted by $\textrm{WAP}(G)$ is comprised of functions
$f\in\ell^{\infty}(G)$ such that the weak closure of their shifts, i.e.
$\overline{\\{_{g}f:g\in G\\}}$ is weakly compact (i.e., compact with respect
to the weak topology induced by functionals on $\ell^{\infty}(G)$) .

######  Theorem 6.3 (Ryll-Nardzewski, cf. [P], page 86).

There is a unique left invariant mean on $\textrm{WAP}(G)$.

We will be using the fact (due to Eberlein, [E]) that for a discrete semigroup
$G$, functions of the form $f(g)=\langle h_{1},T_{g}h_{2}\rangle$, for
$h_{1},h_{2}\in L^{2}(\mu)$ are in $\textrm{WAP}(G)$ (see, for example,
Theorem 3.1 in [Bu]).

######  Lemma 6.4.

Let $G$ be a discrete semigroup and let $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$
be an ergodic measure preserving system. Let $m$ be the unique left invariant
mean on $\textrm{WAP}(G)$ and $f_{1},f_{2}\in L^{2}(\mu)$. Then,

$m(\langle f_{1},T_{g}f_{2}\rangle)=\int_{X}f_{1}\ d\mu\int_{X}\bar{f}_{2}\
d\mu$

###### Proof.

Let $F_{2}\in L^{2}(\mu)$ be defined via

$\langle F_{1},F_{2}\rangle=m(\langle F_{1},T_{g}f_{2}\rangle)$

for all $F_{1}\in L^{2}(\mu)$. We will show that $F_{2}=\int_{X}\bar{f}_{2}\
d\mu$, which implies the result. First observe that since the action
$(T_{g})_{g\in G}$ is ergodic, the only invariant functions are the constants.
Next, we show that $T_{h}F_{2}=F_{2}$ for all $h\in G$, which implies $F_{2}$
is a constant by ergodicity. Indeed,

$\langle
F_{1},T_{h}F_{2}\rangle=\langle(T_{h})^{*}F_{1},F_{2}\rangle=m(\langle(T_{h})^{*}F_{1},T_{g}f_{2}\rangle=m(\langle
F_{1},(T_{hg})f_{2}\rangle)=$ $m(\langle F_{1},T_{g}f_{2}\rangle)=\langle
F_{1},F_{2}\rangle.$

Hence, $F_{2}$ is a constant, so $F_{2}=\langle F_{2},\mathbb{1}_{X}\rangle$.
Thus, for all $f_{1}\in L^{2}(\mu)$ we have

$\int_{X}f_{1}\cdot\bar{F}_{2}\ d\mu=\langle f_{1},F_{2}\rangle=\langle
f_{1},\langle F_{2},\mathbb{1}_{X}\rangle\rangle=\langle\langle
f_{1},\mathbb{1}_{X}\rangle,F_{2}\rangle=$ $m(\langle\langle
f_{1},\mathbb{1}_{X}\rangle,(T_{g})f_{2}\rangle)=m\left(\int_{X}f_{1}\
d\mu\int_{X}T_{g}\bar{f}_{2}\ d\mu\right)=\int_{X}f_{1}\
d\mu\int_{X}\bar{f}_{2}\ d\mu,$

so we are done. ∎

######  Remark 6.5.

It follows from Lemma 6.4 that for any (not necessarily ergodic) measure
preserving action of a discrete semigroup $G$ on a probability space
$(X,\mathcal{B},\mu)$ and any $A\in\mathcal{B}$ we have

$m(\mu(A\cap(T_{g})^{-1}A))\geq\mu^{2}(A).$

Indeed, invoking a general form of the ergodic decomposition we have

$m(\mu(A\cap(T_{g})^{-1}A))=m\left(\int_{\Omega}\mu_{t}(A\cap(T_{g})^{-1}A)\
d\nu(t)\right),$

where $\mu_{t}$ is an ergodic measure for $\nu$-a.e. $t\in\Omega$. Then, since
$m$ is linear, Lemma 6.4 implies

$m\left(\int_{\Omega}\mu_{t}(A\cap(T_{g})^{-1}A)\
d\nu(t)\right)=\int_{\Omega}m(\mu_{t}(A\cap(T_{g})^{-1}A))\
d\nu(t)=\int_{\Omega}\mu_{t}(A)^{2}\ d\nu(t).$

Applying Jensen’s inequality we get

$\int_{\Omega}\mu_{t}(A)^{2}\ d\nu(t)\geq\left(\int_{\Omega}\mu_{t}(A)\
d\nu(t)\right)^{2}=\mu^{2}(A),$

as desired.

######  Lemma 6.6.

Let $G$ be a discrete semigroup and let $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$
an ergodic measure preserving system. Let $A,B\in\mathcal{B}$ with $\mu(A)>0$
and $\mu(B)>0$. Then,

(6.1) $R_{A,B}:=\left\\{g\in
G:\mu(B\cap(T_{g})^{-1}A)>\frac{\mu(A)\mu(B)}{2}\right\\}\neq\emptyset.$

###### Proof.

We proceed by contradiction. Assume that for all $g\in G$ we have
$\mu(B\cap(T_{g})^{-1}A)\leq\frac{\mu(A)\mu(B)}{2}$. Let $m$ be the unique
left invariant mean on $\textrm{WAP}(G)$. Let $A,B\in\mathcal{B}$ with
$\mu(A)>0$ and $\mu(B)>0$. By Lemma 6.4, we have

(6.2) $m(\mu(B\cap(T_{g})^{-1}A))=\mu(A)\mu(B).$

Equation (6.2) contradicts the assumption that, for all $g\in G$,
$\mu(B\cap(T_{g})^{-1}A)\leq\frac{\mu(A)\mu(B)}{2}$, (given that $m(f)\geq 0$
if $f\geq 0$) so we are done. ∎

######  Lemma 6.7.

Let $G$ be a discrete semigroup and $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ an
ergodic measure preserving system. Let $A\in\mathcal{B}$ with $\mu(A)>0$.
Then, for all $\varepsilon>0$, there are $g_{1},\dots,g_{k}\in G$ such that

$\mu\left(\bigcup_{i=1}^{k}(T_{g_{i}})^{-1}A\right)>1-\varepsilon.$

###### Proof.

Let $A\in\mathcal{B}$ with $\mu(A)>0$. We claim that

(6.3) $\sup\left\\{\mu\left(\bigcup_{g\in B}(T_{g})^{-1}A\right):B\subseteq
G\textrm{ and }|B|\leq|\mathbb{N}|\right\\}=1.$

We proceed by contradiction. Let us assume that this is not the case. Then
there exists $0<\delta<1$ such that

(6.4) $\sup\left\\{\mu\left(\bigcup_{g\in B}(T_{g})^{-1}A\right):B\subseteq
G\textrm{ and }|B|\leq|\mathbb{N}|\right\\}=1-\delta.$

Let $0<\varepsilon^{\prime}<\frac{\mu(A)\delta}{2}$, and choose $B\subseteq G$
with $|B|\leq|\mathbb{N}|$ such that

(6.5) $\mu\left(\bigcup_{g\in B}(T_{g})^{-1}A\right)\geq
1-\delta-\varepsilon^{\prime}.$

By assumption, there is some $C\subseteq X\setminus\bigcup_{g\in
B}(T_{g})^{-1}A$ with
$\mu(C)\geq\delta$. Now, by Lemma 6.6, we can find $g_{0}\in G$ such that
$\mu(C\cap(T_{g_{0}})^{-1}A)\geq\frac{\mu(A)\mu(C)}{2}$. (Notice that, in
particular, it follows that $g_{0}\notin B$). We have

(6.6) $\mu\left(\bigcup_{g\in B\cup\\{g_{0}\\}}(T_{g})^{-1}A\right)\geq
1-\delta-\varepsilon^{\prime}+\mu(C\cap(T_{g_{0}})^{-1}A)\geq
1-\delta-\varepsilon^{\prime}+\frac{\mu(A)\delta}{2}>1-\delta,$

by our choice of $\varepsilon^{\prime}$ and $g_{0}$. This contradicts the
definition of supremum, since clearly $B\cup\\{g_{0}\\}$ is still a countable
subset of $G$. Thus, (6.3) holds.

To complete the proof, let $\varepsilon>0$ and choose $B\subseteq G$ with
$|B|\leq|\mathbb{N}|$ such that

$\mu\left(\bigcup_{g\in B}(T_{g})^{-1}A\right)>1-\frac{\varepsilon}{2}.$

Since $\mu$ is continuous and $B$ is countable, there exist
$g_{1},\dots,g_{k}\in B$ such that

$\mu\left(\bigcup_{i=1}^{k}(T_{g_{i}})^{-1}A\right)>1-\frac{\varepsilon}{2}-\frac{\varepsilon}{2},$

so we are done. ∎

Before proving an ergodic version of Furstenberg’s correspondence principle
for means, we formulate two additional results. We start with a version of
Theorem 2.3 for means, which we will juxtapose with its ergodic counterpart,
Theorem 6.11, below.

######  Theorem 6.8 (Furstenberg correspondence principle for means (cf.
[BLei] and [BMc])).

Let $G$ be a discrete amenable semigroup. Let $m\in\mathfrak{M}(G)$,
$E\subseteq G$ with $m(\mathbb{1}_{E})>0$. Then there exists a probability
measure preserving system $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ such that $X$
is compact and Hausdorff, $\mathcal{B}=\textrm{Borel}(X)$ and $(T_{g})_{g\in
G}$ is a $G$-action on $X$ by continuous self-maps of $X$. Finally, $A$ is a
set in $\mathcal{B}$ for which $\mu(A)=m(\mathbb{1}_{B})$, and $\mu$ is such
that for all $k\in\mathbb{N}$, $g_{1},\dots,g_{k}\in G$ we have

(6.7)
$m\left(\mathbb{1}_{E}\prod_{i=1}^{k}\mathbb{1}_{g_{i}^{-1}E}\right)=\mu(A\cap
T_{g_{1}}^{-1}A\cap\dots\cap T_{g_{k}}^{-1}A).$

###### Proof.

The proof for discrete amenable groups provided in [BLei] extends verbatim to
our context without any major modification. ∎

######  Remark 6.9.

The proof of Theorem 6.8 in [BLei] can be easily adjusted to include unions
and complements as in Theorem 2.3.

The other result we need can be found in [P]:

######  Theorem 6.10 (Proposition 0.1 [P]).

The space of left invariant means $\mathfrak{M}(G)$ is a weak*-compact, convex
spanning subset of $\ell^{\infty}(G)^{*}$.

We are now in position to formulate and prove a general version of the ergodic
Furstenberg correspondence principle.

######  Theorem 6.11 (Ergodic Furstenberg correspondence principle for means).

Let $E\subseteq G$ be such that $m(\mathbb{1}_{E})>0$ for some mean
$m\in\mathfrak{M}(G)$. Then there exists a mean $\tilde{m}\in\mathfrak{M}(G)$,
and an ergodic measure preserving system $(X,\mathcal{B},\mu,(T_{g})_{g\in
G})$ such that for all $k\in\mathbb{N}$ we have

(6.8) $\tilde{m}(\mathbb{1}_{E^{w_{0}}\star
g_{1}^{-1}E^{w_{1}}\star\dotso\star
g_{k}^{-1}E^{w_{k}}})=\mu(A^{w_{0}}\star(T_{g_{1}})^{-1}A^{w_{1}}\star\dots\star(T_{g_{k}})^{-1}A^{w_{k}}),$

where $A\in\mathcal{B}$ is such that
$0<\mu(A)=m(\mathbb{1}_{E})\leq\tilde{m}(\mathbb{1}_{E})$, and each of the
stars denotes either union or intersection with the understanding that

  * (i)

for all $1\leq i\leq k-1$, the operation represented by $\star$ which stands
between $E^{w_{i}}$ and $E^{w_{i+1}}$ is the same as the operation appearing
between $A^{w_{i}}$ and $A^{w_{i+1}}$.

  * (ii)

the choices of parentheses which are needed to make the expressions on both
sides of formula (2.1) well defined also match.

###### Proof.

First, we remark that, as in the proof of Theorem 6.8, any invariant mean on
$G$ is given by a $G$-invariant probability measure on $\beta G$, the Stone-
Čech compactification of $G$, via the isomorphism $(\ell^{\infty}G)^{*}\cong
C(\beta G)^{*}$121212See [GJ] for the background on Stone-Čech
compactifications.. It is easy to see that this isomorphism is behind the
formula (6.8).

By Choquet’s theorem (which we can apply in view of Theorem 6.10), we can
write

(6.9) $m=\int_{\textrm{Ext}(\mathfrak{M}(G))}m_{t}\ d\lambda(t),$

for some probability measure $\lambda$ supported on
$\textrm{Ext}(\mathfrak{M}(G))$, the set of extreme points of
$\mathfrak{M}(G)$.

Notice that extreme points of $\mathfrak{M}(G)$ get mapped to extreme points
of the set of probability measures on $\beta G$ via the isomorphism
$(\ell^{\infty}(G))^{*}\cong C(\beta G)^{*}$. It is well known that the
measures that are extreme points in the set of $G$-invariant probability
measures on $\beta G$ are in fact ergodic.

It follows from formula (6.9), that since $m(\mathbb{1}_{E})>0$, we have that
$m_{t}(\mathbb{1}_{E})>0$ for a set of $t$ of positive $\lambda$-measure.

Thus we can choose $\tilde{m}$, an extreme point of $\mathfrak{M}(G)$, for
which $\tilde{m}(\mathbb{1}_{E})\geq m(\mathbb{1}_{E})>0$. Using the
aforementioned isomorphism, we obtain an ergodic measure $\mu$ for which (6.8)
holds. ∎

######  Remark 6.12.

It is worth noticing that the ergodicity of the system
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ in Theorem 6.11 stems from the fact
that passed from the mean $m$ to $\tilde{m}$. If one restricts oneself to only
using a given mean $m$, as in Theorem 6.8, ergodicity cannot be guaranteed. We
also see in the statement of Theorem 6.11 that moving form $m$ to $\tilde{m}$
does not affect combinatorial applications, as we can require
$\tilde{m}(\mathbb{1}_{E})\geq m(\mathbb{1}_{E})$. This situation is similar
to the one discussed in the comments at the end of Section 2.

We are now ready for a proof of Theorem 6.2:

###### Proof of Theorem 6.2.

Let $E\subseteq G$ with $d^{*}(E)>0$. Then, there is some
$m_{0}\in\mathfrak{M}(G)$ such that $m_{0}(\mathbb{1}_{E})>0$. Let
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ be an ergodic measure preserving system
satisfying the equality (6.8) for some set $A\in\mathcal{B}$ with $\mu(A)>0$.
(See Theorem 6.11 above). Let $E\subseteq G$ with $d^{*}(E)>0$. Then, there is
some $m_{0}\in\mathfrak{M}(G)$ such that $m_{0}(\mathbb{1}_{E})>0$. Let
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ be an ergodic measure preserving system
satisfying the equality (6.8) for some set $A\in\mathcal{B}$ with $\mu(A)>0$.
(See Theorem 6.11 above).

Let $E\subseteq G$ with $d^{*}(E)>0$. Then, there is some
$m_{0}\in\mathfrak{M}(G)$ such that $m_{0}(\mathbb{1}_{E})>0$. Let
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ be an ergodic measure preserving system
satisfying the equality (6.8) for some set $A\in\mathcal{B}$ with $\mu(A)>0$.
(See Theorem 6.11 above).

Let $\varepsilon>0$. By Lemma 6.7, we can find $g_{1},\dots,g_{k}\in G$ such
that

$\mu\left(\bigcup_{i=1}^{k}(T_{g_{i}})^{-1}A\right)>1-\varepsilon,$

since $\mu(A)>0$. Equality (6.8) then implies that for some
$m\in\mathfrak{M}(G)$ we have

$m(\mathbb{1}_{\bigcup_{i=1}^{k}g_{i}^{-1}E})>1-\varepsilon,$

whence the result follows from the definition of $d^{*}$. ∎

We conclude this section with brief remarks on yet another generalization of
Hindman’s covering theorem. Assume that $G$ is a locally compact amenable
group (i.e. that $G$ has a left invariant _topological_ mean, see [Gre] for
more details). The Furstenberg’s correspondence principle which was proved in
[BCRZ] (see also [BF]) can be "upgraded" to an ergodic Furstenberg
correspondence principle similar to Theorem 2.8 and Theorem 6.8. Based on this
enhancement one can prove a version of Hindman’s Theorem for locally compact
groups. Before providing the formulation we need two definitions.

######  Definition 6.13.

Let $G$ be a locally compact amenable group and let $E\subseteq G$. We define
the upper Banach density of $E$ as follows:

(6.10) $d^{*}(E)=\sup\\{m(\mathbb{1}_{E}):m\textrm{ is a left-invariant
topological mean on }L^{1}(G,\mu)\\},$

where $\mu$ is a Haar measure on $G$.

######  Definition 6.14.

Let $G$ be a locally compact amenable group. We say that a set $E\subseteq G$
is _substantial_ if $E\supseteq UW$, where $U$ is a non-empty open subset in
$G$ containing $\textrm{id}_{G}$ and $W$ is a measurable set with
$d^{*}(W)>0$.

######  Theorem 6.15.

Let $G$ be a locally compact amenable group. Let $E\subseteq G$ be a
substantial set. Let $\varepsilon>0$. Then there exist $g_{1},\dots,g_{k}\in
G$ such that

$d^{*}\left(\bigcup_{i=1}^{k}g_{i}^{-1}E\right)>1-\varepsilon.$

## References

  * [BBF] M. Beiglböck, V. Bergelson and A. Fish, Sumset phenomenon in countable amenable groups, Adv. Math. 223 (2010), no. 2, 416–432.
  * [B1] V. Bergelson, Ergodic Ramsey theory, in Logic and combinatorics (Arcata, Calif., 1985), 63–87, Contemp. Math., 65, Amer. Math. Soc., Providence, RI.
  * [B2] V. Bergelson. Ergodic theory and Diophantine problems. Topics in symbolic dynamics and applications (Temuco, 1997), 167–205, London Math. Soc. Lecture Note Ser., 279, Cambridge Univ. Press, Cambridge, 2000.
  * [BCRZ] V. Bergelson, J. C. Christopherson, D. Robertson, P. Zorin-Kranich. Finite products sets and minimally almost periodic groups, J. Funct. Anal. 270 (2016), no. 6, 2126–2167.
  * [BF] V. Bergelson and H. Furstenberg, WM groups and Ramsey theory, Topology Appl. 156 (2009), no. 16, 2572–2580.
  * [BG] V. Bergelson and D. Glasscock, On the interplay between additive and multiplicative largeness and its combinatorial applications, J. Combin. Theory Ser. A 172 (2020), 105203, 60 pp.
  * [BHK] V. Bergelson, B. Host and B. Kra, Multiple recurrence and nilsequences, Invent. Math. 160 (2005), no. 2, 261–303.
  * [BK] V. Bergelson and I. J. Håland Knutson, Weak mixing implies weak mixing of higher orders along tempered functions, Ergodic Theory Dynam. Systems 29 (2009), no. 5, 1375–1416.
  * [BKMST] V. Bergelson, G. Kolesnik, M. Madritsch, Y. Son and R. Tichy. Uniform distribution of prime powers and sets of recurrence and van der Corput sets in $\mathbb{Z}^{k}$, Israel J. Math. 201 (2014), no. 2, 729–760.
  * [BKS] V. Bergelson, G. Kolesnik and Y. Son, Uniform distribution of subpolynomial functions along primes and applications, J. Anal. Math. 137 (2019), no. 1, 135–187.
  * [BLei] V. Bergelson and A. Leibman, Cubic averages and large intersections, in Recent trends in ergodic theory and dynamical systems, 5–19, Contemp. Math., 631, Amer. Math. Soc., Providence, RI.
  * [BLes] V. Bergelson and E. Lesigne, Van der Corput sets in ${Z}^{d}$, Colloq. Math. 110 (2008), no. 1, 1–49.
  * [BMc] V. Bergelson and R. McCutcheon, Recurrence for semigroup actions and a non-commutative Schur theorem, in Topological dynamics and applications (Minneapolis, MN, 1995), 205–222, Contemp. Math., 215, Amer. Math. Soc., Providence, RI.
  * [BMo] V. Bergelson and J. Moreira, Van der Corput’s difference theorem: some modern developments, Indag. Math. (N.S.) 27 (2016), no. 2, 437–479.
  * [BMoR] V. Bergelson, J. Moreira and F. K. Richter. Single and multiple recurrence along non-polynomial sequences. arXiv:1171.05729, 2017.
  * [Bi] G. Birkhoff, A note on topological groups, Compositio Math. 3 (1936), 427–430.
  * [BKQW] M. Boshernitzan, G. Kolesnik, A. Quas and M. Wierdl. Ergodic averaging sequences, J. Anal. Math. 95 (2005), 63–103.
  * [Bu] R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach Science Publishers, New York, 1970.
  * [C1] G. Choquet, Existence et unicité des représentations intégrales au moyen des points extrémaux dans les cônes convexes, in Séminaire Bourbaki, Vol. 4, Exp. 139, 33–47, Soc. Math. France, Paris.
  * [C2] G. Choquet, Les cônes convexes faiblement complets dans l’analyse, in Proc. Internat. Congr. Mathematicians (Stockholm, 1962), 317–330, Inst. Mittag-Leffler, Djursholm.
  * [DHZ] T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups, J. Reine Angew. Math. 747 (2019), 277–298.
  * [E] W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217–240.
  * [EW] M. Einsiedler and T. Ward, Ergodic theory with a view towards number theory, Graduate Texts in Mathematics, 259, Springer-Verlag London, Ltd., London, 2011.
  * [FK] R. Feres and A. Katok, Ergodic theory and dynamics of $G$-spaces (with special emphasis on rigidity phenomena), in Handbook of dynamical systems, Vol. 1A, 665–763, North-Holland, Amsterdam.
  * [Fra] N. Frantzikinakis, Ergodicity of the Liouville system implies the Chowla conjecture, Discrete Anal. 2017, Paper No. 19, 41 pp.
  * [F1] H. Furstenberg, Prediction Theory, ProQuest LLC, Ann Arbor, MI, 1958.
  * [F2] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49.
  * [F3] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256.
  * [F4] H. Furstenberg, Poincaré recurrence and number theory, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 3, 211–234.
  * [F5] H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, NJ, 1981.
  * [FKO] H. Furstenberg, Y. Katznelson and D. Ornstein, The ergodic theoretical proof of Szemerédi’s theorem, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 527–552.
  * [Gre] F. P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York, 1969.
  * [GJ] L. Gillman and M. Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, NJ, 1960.
  * [GLR] A. Gomilko, M. Lemanczyk, T. de la Rue, On Furstenberg systems of aperiodic multiplicative functions of Matomaki, Radziwill and Tao. arXiv preprint: https://arxiv.org/abs/2006.09958.
  * [GrS] G. Greschonig and K. Schmidt, Ergodic decomposition of quasi-invariant probability measures, Colloq. Math. 84/85 (2000), part 2, 495–514.
  * [HK] B. Hasselblatt and A. Katok, Principal structures, in Handbook of dynamical systems, Vol. 1A, 1–203, North-Holland, Amsterdam.
  * [H1] N. Hindman, Finite sums from sequences within cells of a partition of $N$, J. Combinatorial Theory Ser. A 17 (1974), 1–11.
  * [H2] N. Hindman, On density, translates, and pairwise sums of integers, J. Combin. Theory Ser. A 33 (1982), no. 2, 147–157.
  * [HF1] N. Frantzikinakis and B. Host, The logarithmic Sarnak conjecture for ergodic weights, Ann. of Math. (2) 187 (2018), no. 3, 869–931.
  * [HF2] N. Frantzikinakis and B. Host, Furstenberg systems of bounded multiplicative functions and applications, arXiv preprint: https://arxiv.org/abs/1804.08556.
  * [J] R. I. Jewett, The prevalence of uniquely ergodic systems, J. Math. Mech. 19 (1969/1970), 717–729.
  * [K] S. Kakutani, Selected papers. Vol. 2, Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA, 1986.
  * [Kr] W. Krieger, On unique ergodicity, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, 327–346, Univ. California Press, Berkeley, CA.
  * [N] I. Namioka, Følner’s conditions for amenable semi-groups, Math. Scand. 15 (1964), 18–28.
  * [P] A. L. T. Paterson, Amenability, Mathematical Surveys and Monographs, 29, American Mathematical Society, Providence, RI, 1988.
  * [Ph] R. R. Phelps, Lectures on Choquet’s theorem, second edition, Lecture Notes in Mathematics, 1757, Springer-Verlag, Berlin, 2001.
  * [R] A. Rosenthal, Strictly ergodic models and amenable group action. PhD Thesis. Paris 6, 1986.
  * [S] K. Schmidt, Asymptotic properties of unitary representations and mixing, Proc. London Math. Soc. (3) 48 (1984), no. 3, 445–460.
  * [Sz] E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arith. 27 (1975), 199–245.
  * [V] V. S. Varadarajan, Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc. 109 (1963), 191–220.
  * [W] B. Weiss, Strictly ergodic models for dynamical systems, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 143–146.