1.1 The Steinhaus lemma

H. Steinhaus [Reference Steinhaus34, Théorème VII] proved that if $A, B\subseteq \mathbb R$ both have positive Lebesgue measure, then $A+B$ contains an open interval. The relevant properties of Lebesgue measure are retained by Haar measure on locally compact abelian (LCA) groups, leading to the following generalization.

We will prove a special case of Theorem 1.1 in §2. Weil [Reference Weil37, p.50] proves the above for arbitrary locally compact (including nonabelian) groups. See [Reference Stromberg35] for a short proof of the special case where $B=A^{-1}$ .

Our main results are Theorem 1.5 and Theorem 3.16; these are analogues of Theorem 1.1 where Haar measure is replaced by a finitely additive translation-invariant measure. Such measures are usually discussed in terms of invariant means on the space of bounded functions on the underlying group.

1.2 Means on $\ell ^{\infty }(\Gamma )$

Let $\Gamma $ be a discrete abelian group. We write $\ell ^{\infty }(\Gamma )$ for the Banach space of bounded complex-valued functions on $\Gamma $ , equipped with the supremum norm $\|\cdot \|_{\infty }$ . For a function f defined on $\Gamma $ and $\gamma \in \Gamma $ , we write $f_\gamma $ for the translate given by $f_\gamma (x):=f(x-\gamma )$ . A mean on $\ell ^\infty (\Gamma )$ is a linear functional $m:\ell ^\infty (\Gamma )\to \mathbb C$ satisfying $m(1_\Gamma ) = 1$ and $m(f)\geq 0$ if $f(x)\geq 0$ for all $x\in \Gamma $ .

We write $f\sim _{m} g$ if $m(|f-g|)=0$ .

Note that each mean m induces a finitely additive probability measure $m^*$ on the algebra of subsets of $\Gamma $ , given by $m^*(A)=m(1_A)$ .

We write $A\subset _m B$ if $m^*(A\setminus B)=0$ , and we write $A\sim _{m} B$ if $m^{*}(A\triangle B)=0$ , or equivalently, if $1_{A} \sim _{m} 1_{B}$ .

We will abuse notation and write $m(A)$ for $m^{*}(A)$ .

If a mean m satisfies $m(f_\gamma )=m(f)$ for all $f\in \ell ^{\infty }(\Gamma )$ and $\gamma \in \Gamma $ , we say that m is an invariant mean; invariant means are sometimes called “Banach means” or “Banach mean values” in the literature, cf. [Reference Følner19], [Reference Følner20].

The set ${\mathcal {M}}(\Gamma )$ of means on $\ell ^{\infty }(\Gamma )$ is a weak $^{*}$ -compact convex subset of $\ell ^{\infty }(\Gamma )^{*}$ . Translation on $\Gamma $ induces an action $\tau $ of $\Gamma $ on ${\mathcal {M}}(\Gamma )$ by continuous linear operators: $(\tau _{\gamma } m) (f):=m(f_{\gamma })$ . The Markov-Kakutani fixed point theorem [Reference Conway14, p. 151] guarantees that there is at least one mean $m\in {\mathcal {M}}(\Gamma )$ satisfying $\tau _{\gamma }m=m$ for all $\gamma \in \Gamma $ . In other words, there is at least one invariant mean on $\ell ^{\infty }(\Gamma )$ .

We let ${\mathcal {M}}_{\tau }(\Gamma )$ denote the set of invariant means on $\ell ^\infty (\Gamma )$ . It is straightforward to verify that ${\mathcal {M}}_{\tau }(\Gamma )$ is a weak $^*$ -compact convex subset of $\ell ^\infty (\Gamma )^*$ . The Krein-Milman theorem [Reference Conway14, p. 142] then implies that the set ${\mathcal {M}}_{\tau }^{ext}(\Gamma )$ of extreme points of ${\mathcal {M}}_{\tau }(\Gamma )$ is nonempty. We call elements of ${\mathcal {M}}_{\tau }^{ext}(\Gamma )$ extreme invariant means.

Definition 1.2. Let $A\subseteq \Gamma $ . The upper Banach density of A is

$$\begin{align*}d^*(A):=\sup\{m(A):m \in {\mathcal{M}}_{\tau}(\Gamma)\}.\end{align*}$$

See Lemma 12.1 for conditions equivalent to A having positive upper Banach density.

1.3 Characters

We use standard definitions and background presented, for example, in Chapter 4 of [Reference Folland18] or Chapters 1 and 2 of [Reference Rudin33].

Write $\mathcal S^1$ for the group $\{z\in \mathbb C: |z|=1\}$ with the group operation of multiplication and the usual topology.

Let G be an LCA group. A character of G is a continuous homomorphism $\chi :G\to \mathcal S^{1}$ ; the set of all characters of G is denoted $\widehat {G}$ . The trivial character $\chi _{0}$ is constant: $\chi _{0}(\gamma )=1$ for all $\gamma \in \Gamma $ . With the group operation of pointwise multiplication and the topology of uniform convergence on compact subsets, $\widehat {G}$ is an LCA group. When G is compact, $\widehat {G}$ is discrete, and when G is discrete, $\widehat {G}$ is compact. A trigonometric polynomial is a finite linear combination of characters of G. A function $f:G\to \mathbb C$ is uniformly almost periodic if it is a uniform limit of trigonometric polynomials.

1.4 Bohr compactification

Let $\Gamma $ be a discrete abelian group. The Bohr compactification of $\Gamma $ is a compact Hausdorff abelian group $b\Gamma $ together with a one-to-one homomorphism $\iota :\Gamma \to b\Gamma $ such that

1. (i) $\iota (\Gamma )$ is topologically dense in $b\Gamma $ ;

2. (ii) for all $\chi \in \widehat {\Gamma }$ , there is a character $\tilde {\chi }\in \widehat {b\Gamma }$ such that $\tilde {\chi }\circ \iota = \chi $ .

We will identify $\Gamma $ with its image $\iota (\Gamma )$ , and consider $\Gamma $ as a topologically dense subgroup of $b\Gamma $ , so that each $\chi \in \widehat {\Gamma }$ is the restriction of a continuous $\tilde {\chi }\in \widehat {b\Gamma }$ : $\chi = \tilde {\chi }|_\Gamma $ .

The group $b\Gamma $ may be constructed as the Pontryagin dual of $\widehat {\Gamma }_d$ , where $\widehat {\Gamma }_d$ is the group $\widehat {\Gamma }$ with the discrete topology instead of the usual topology. In this construction, the embedding map $\iota $ is given by $\iota (\gamma ):=e_\gamma $ , where $e_\gamma (\chi )=\chi (\gamma )$ ; see Section 1.8 of [Reference Rudin33] or Section 4.7 of [Reference Folland18] for details.

The following is a special case of [Reference Folland18, Theorem 4.79] for discrete abelian groups.

We write $\mu _{b\Gamma }$ for Haar probability measure on $b\Gamma $ .

1.5 Main theorem

The next theorem is our main result; it relates sumsets in discrete abelian groups to sumsets in compact abelian groups. In §1.7 we show how it extends the results mentioned in the abstract. Here “ $F_{\sigma }$ ” means “countable union of compact sets.” Note that when A and B are $F_{\sigma }$ subsets of $b\Gamma $ , $A+B$ is also $F_{\sigma }$ , implying that $A+B$ is $\mu _{b\Gamma }$ -measurable.

Theorem 1.5. Let $\Gamma $ be a discrete abelian group. Let $\nu $ be an extreme invariant mean on $\ell ^{\infty }(\Gamma )$ , m an invariant mean on $\ell ^{\infty }(\Gamma )$ , and A, $B\subseteq \Gamma $ . There are $F_{\sigma }$ sets $\tilde {A}_{\nu }$ , $\tilde {B}_{m}\subseteq b\Gamma $ such that $\mu _{b\Gamma }(\tilde {A}_{\nu })\geq \nu (A)$ , $\mu _{b\Gamma }(\tilde {B}_{m})\geq m(B)$ , and

(1.1) $$ \begin{align} \nu(A+B)\geq \mu_{b\Gamma}(\tilde{A}_{\nu}+\tilde{B}_{m}).\\[-18pt] \nonumber\end{align} $$

Furthermore,

1. (i) if $V\subseteq b\Gamma $ is compact, then

   $$\begin{align*}\mu_{b\Gamma}(\tilde{A}_{\nu}\cap V)\geq \nu(A\cap V) \quad \text{and} \quad \mu_{b\Gamma}(\tilde{B}_{m}\cap V)\geq m(B\cap V);\end{align*}$$

2. (ii) if $V\subseteq b\Gamma $ is clopen, then

   $$\begin{align*}\mu_{b\Gamma}(\tilde{A}_{\nu}\cap V)=0\iff \nu(A\cap V)=0 \quad \text{and} \quad \mu_{b\Gamma}(\tilde{B}_{m}\cap V)=0\iff m(B\cap V)=0;\end{align*}$$

3. (iii) if $V\subseteq \tilde {A}_{\nu }+\tilde {B}_{m}$ is compact, then $V\cap \Gamma \subset _{\nu } A+B$ .

We will prove Theorem 1.5 as a special case of Theorem 3.16 below; the latter obtains the same conclusion under a weaker hypothesis on B.

1.6 Large sets

Let $\Gamma $ be an arbitrary discrete abelian group (countable or uncountable). We list some common definitions for large subsets of $\Gamma $ . Equivalent formulations are provided in §12.

If $\gamma _{0}\in \Gamma $ we say U is a Bohr neighborhood of $\gamma _{0}$ if there is an open subset $V\subseteq b\Gamma $ such that $\gamma _{0}\in V$ and $V\cap \Gamma \subseteq U$ . By Lemma 12.6, U is a Bohr neighborhood of $\gamma _{0}$ if and only if there is a uniformly almost periodic function $\phi :\Gamma \to [0,\infty )$ supported on U such that $\phi (\gamma _{0})\neq 0$ .

A set $T\subseteq \Gamma $ is thick if for all finite $F\subseteq \Gamma $ , there is a $t\in \Gamma $ such that $F+t\subseteq T$ . By Lemma 12.2, T is thick if and only if there is an invariant mean m on $\ell ^{\infty }(\Gamma )$ such that $m(T)=1$ .

A set $S\subseteq \Gamma $ is syndetic if there is a finite set $F\subseteq \Gamma $ such that $S+F=\Gamma $ . By Lemma 12.3, S is syndetic if and only if S has nonempty intersection with every thick set. Every Bohr neighborhood is syndetic – see §12.3.

We say that $C\subseteq \Gamma $ is piecewise syndetic if there is an $m\in {\mathcal {M}}_{\tau }(\Gamma )$ and a syndetic set S such that $S\subset _{m} C$ . Equivalently (Lemma 12.4), C is piecewise syndetic if C contains a set of the form $S\cap T$ , where S is syndetic and T is thick.

We say that C is piecewise Bohr if there is an $m\in {\mathcal {M}}_{\tau }(\Gamma )$ and a Bohr neighborhood $U\subseteq \Gamma $ such that $U\subset _{m}C$ . Equivalently (Lemma 12.7), C is piecewise Bohr if there is a thick set T and a Bohr neighborhood U of some $\gamma \in \Gamma $ such that $T\cap U\subseteq \Gamma $ .

Every piecewise Bohr set is piecewise syndetic, but some syndetic sets are not piecewise Bohr; examples are given in [Reference Bergelson, Furstenberg and Weiss4, Theorem 4.3] and in Section 2.5 of [Reference Griesmer22]. In countable abelian groups there are many examples of sets which have positive upper Banach density and are not piecewise syndetic (and therefore not piecewise Bohr); see [Reference Bergelson, Huryn and Raghavan5] for a survey. In §12.4 we construct such examples in $\mathbb R$ and in some other uncountable abelian groups.

1.7 Sumset phenomena

Jin [Reference Jin27] used nonstandard analysis to give a new proof of the Steinhaus lemma in $\mathbb R$ , and to prove a new result about sumsets in $\mathbb Z$ : Corollary 3 of [Reference Jin27] says that if $A, B\subseteq \mathbb Z$ both have positive upper Banach density, then $A+B$ is piecewise syndetic. Under the same hypothesis, Bergelson, Furstenberg, and Weiss improved the conclusion to say that $A+B$ is piecewise Bohr; this is [Reference Bergelson, Furstenberg and Weiss4, Theorem I]. The latter result was extended to countable amenable groups by Beiglböck, Bergelson, and Fish in [Reference Beiglböck, Bergelson and Fish2], and (with a modified conclusion) to arbitrary countable groups by Björklund and Fish in [Reference Björklund and Fish7] and by Björklund in [Reference Björklund6]. As Corollary 1.6 shows, Theorem 1.5 recovers the abelian cases of these results and extends them to discrete abelian groups of arbitrary cardinality.

In [Reference Griesmer22, Theorem 1.4] we showed that when $d^{*}(A)>0$ and B satisfies a weaker positive density condition (cf. §3.2 below), we may still conclude that $A+B$ is piecewise Bohr. Corollary 4.2 extends this result to arbitrary discrete abelian groups.

1.8 Følner sequences and Følner nets

Let $\mathbf F:=(F_n)_{n\in \mathbb N}$ be a sequence of finite subsets of $\Gamma $ . For $A\subseteq \Gamma $ , we write

$$\begin{align*}\underline{d}_{\mathbf F}(A):= \liminf_{n\to\infty}\frac{|A\cap F_n|}{|F_n|}, \qquad \bar{d}_{\mathbf F}(A):= \limsup_{n\to\infty}\frac{|A\cap F_n|}{|F_n|} \end{align*}$$

for the lower density of A with respect to $\mathbf F$ and upper density of A with respect to $\mathbf F$ , respectively. If $\underline {d}_{\mathbf F}(A)=\bar {d}_{\mathbf F}(A)$ we write $d_{\mathbf F}(A)$ for the common value, which we call the density of A with respect to $\mathbf F$ .

If $\mathbf F:=(F_j)_{j\in I}$ is a net of finite subsets of $\Gamma $ indexed by a directed set I, we define the lower and upper density of A with respect to $\mathbf F$ to be $\underline {d}_{\mathbf F}(A):=\liminf _{j\in I} \frac {|A\cap F_j|}{|F_j|}$ , and $\bar {d}_{\mathbf F}(A):=\limsup _{j\in I} \frac {|A\cap F_j|}{|F_j|}$ , respectively. When $f\in \ell ^{\infty }(\Gamma )$ is real-valued, we write

$$\begin{align*}\underline{\mathbb E}_{\mathbf F}(f):=\liminf_{j\in I} \frac{1}{|F_{j}|}\sum_{\gamma\in F_{j}}f(\gamma) \end{align*}$$

and $\bar {\mathbb E}_{\mathbf F}(f)$ for the corresponding $\limsup $ . When $\underline {\mathbb E}_{\mathbf F}(f)=\bar {\mathbb E}_{\mathbf F}(f)$ , we write $\mathbb E_{\mathbf F}(f)$ for the common value.

We say that $(F_{n})_{n\in \mathbb N}$ is a Følner sequence if $\lim _{n\to \infty } \frac {|F_n\triangle (F_n+\gamma )|}{|F_n|}=0$ for all $\gamma \in \Gamma $ . Likewise, $(F_j)_{j\in I}$ is a Følner net if $\lim _{j\in I} \frac {|F_j\triangle (F_j+\gamma )|}{|F_j|}=0$ for all $\gamma \in \Gamma $ .

It is straightforward to verify that when $\mathbf F$ is Følner sequence or Følner net, we have

(1.2) $$ \begin{align} \underline{d}_{\mathbf F}(A+t)=\underline{d}_{\mathbf F}(A) \quad \text{ and }\quad \bar{d}_{\mathbf F}(A+t)=\bar{d}_{\mathbf F}(A) \text{ for all } A\subseteq \Gamma,\, t\in \Gamma. \end{align} $$

Lemma 1.9. Let $\mathbf F$ be a sequence (or net) of subsets of $\Gamma $ . Then there is a mean m on $\ell ^\infty (\Gamma )$ satisfying

(1.3) $$ \begin{align} \underline{\mathbb E}_{\mathbf F}(f)&\leq m(f)\leq \bar{\mathbb E}_{\mathbf F}(f) && \text{for all real valued } f\in \ell^{\infty}(\Gamma), \end{align} $$

(1.4) $$ \begin{align} \underline{d}_{\mathbf F}(A)& \leq m(A)\leq \bar{d}_{\mathbf F}(A) && \text{for all } A\subseteq \Gamma. \end{align} $$

If $\mathbf F$ is a Følner sequence or Følner net, there is an $m\in {\mathcal {M}}_{\tau }(\Gamma )$ satisfying (1.3) and (1.4).

1.9 Invariant means and almost periodicity

It is well known that for every uniformly almost periodic function $\phi $ and any two invariant means m, $m'$ on $\ell ^{\infty }(\Gamma )$ , $m(\phi )=m'(\phi )$ . This is a consequence of the fact that $m(\chi )=0$ for every invariant mean m and every nontrivial $\chi \in \widehat {\Gamma }$ (see Lemma 3.5 below). Linearity of $m-m'$ then implies $m(\phi )=m'(\phi )$ for every trigonometric polynomial $\phi $ , and boundedness of $m-m'$ implies $m(\phi )=m'(\phi )$ every uniform limit of trigonometric polynomials.

We use the following consequence in the next example: if $\phi :\mathbb Z\to \mathbb C$ is uniformly almost periodic, $(F_{n})_{n\in \mathbb N}$ is a Følner sequence for $\mathbb Z$ , and m is any invariant mean on $\ell ^{\infty }(\mathbb Z)$ , then

(1.5) $$ \begin{align} \lim_{n\to\infty} \frac{1}{|F_{n}|}\sum_{x\in F_{n}} \phi(x)=m(\phi). \end{align} $$

To construct the example, fix an increasing sequence of even natural numbers $b_{n}$ with $b_{n+1}/b_{n}\to \infty $ . Let $F_{n}=\{1,\dots ,b_{n}\}$ and $E_{n}=\{b_{n}/2,\dots , b_{n}\}$ , so that $F_{n}$ and $E_{n}$ are Følner sequences. Let m be a weak $^{*}$ limit of $m_{n}$ , given by $m_{n}(f)=\frac {1}{|F_{n}|}\sum _{j\in F_{n}}f(j)$ . Let $A=B=\bigcup _{n=1}^{\infty } E_{n}$ . Then $m(A)=1/2$ , and $A+A\sim _{m} A$ .

Assume, to get a contradiction, that there is a Bohr neighborhood U such that $U\subset _{m} A$ . By Lemma 12.6 (B.3), there is a uniformly almost periodic function $\phi :\mathbb Z\to [0,1]$ with $m(\phi )>0$ supported on U. This implies $m(\phi 1_{A})=m(\phi )$ . But (1.5) implies $m(\phi 1_{A})=\lim _{n\to \infty } \frac {1}{|F_{n}|}\sum _{j\in E_{n}} \phi (j)=\lim _{n\to \infty } \frac {|E_{n}|}{|F_{n}|}\frac {1}{|E_{n}|}\sum _{j\in E_{n}}\phi (j)=\frac {1}{2}m(\phi )$ . Thus $0<m(\phi )=m(\phi 1_{A})< m(\phi )$ , which is the desired contradiction. Since $A\sim _{m} A+A$ , we have shown that no Bohr neighborhood satisfies $U\subset _{m} A+A$ .

1.10 Small sumsets and subgroups

Corollary 1.12 demonstrates the utility of Theorem 1.5: statements about sumsets in compact abelian groups can be transferred to statements about sumsets in discrete abelian groups. In [Reference Griesmer23] we extend this analysis to prove new inverse theorems for sumsets in discrete abelian groups.

Proof. Let $\Gamma $ , $\nu $ , $\eta $ , A, and B be as in the hypothesis, and let $\tilde {A} =\tilde {A}_{\nu }$ , $ \tilde {B}=\tilde {B}_{\eta }\subseteq b\Gamma $ be the sets provided by Theorem 1.5. We will prove that $(\tilde {A}+\tilde {B})\cap \Gamma \subset _{\nu } A+B$ and that $(\tilde {A}+\tilde {B})\cap \Gamma $ is a union of cosets of a finite index subgroup of $\Gamma $ . First we claim

(1.6) $$ \begin{align} \mu_{b\Gamma}(\tilde{A}+\tilde{B})<\mu_{b\Gamma}(\tilde{A})+\mu_{b\Gamma}(\tilde{B}). \end{align} $$

To see this, apply Theorem 1.5 to find

$$ \begin{align*}\mu_{b\Gamma}(\tilde{A}+\tilde{B})&\leq \nu(A+B) && \\ &< \nu(A)+\eta(B) && \text{by hypothesis} \\ &\leq \mu_{b\Gamma}(\tilde{A})+\mu_{b\Gamma}(\tilde{B}). \end{align*} $$

Because of (1.6), Satz 1 of [Reference Kneser29] provides a compact open (hence finite index) subgroup $\tilde {H}\leq b\Gamma $ satisfying $\tilde {A}+\tilde {B}=\tilde {A}+\tilde {B}+\tilde {H}$ . Thus $\tilde {A}+\tilde {B}$ is compact and is a union of cosets of $\tilde {H}$ . We may take $V=\tilde {A}+\tilde {B}$ in Theorem 1.5 and conclude that $(\tilde {A}+\tilde {B})\cap \Gamma \subset _{\nu } A+B$ . Setting $H=\tilde {H}\cap \Gamma $ , Lemma 11.1 says that H has finite index in $\Gamma $ . The same lemma implies $(\tilde {A}+\tilde {B})\cap \Gamma $ is a union of cosets of H, so $t+H\subset _{\nu } A+B$ for some $t\in \Gamma $ .

2.1 Fourier coefficients and convolutions

As above we draw from Chapter 4 of [Reference Folland18] and Chapters 1 and 2 of [Reference Rudin33] for background.

Let K be a compact abelian group with Haar probability measure $\mu $ . For $f\in L^{2}(\mu )$ , the Fourier transform $\hat {f}:\widehat {K}\to \mathbb C$ is defined as $\hat {f}(\chi ):=\int f \overline {\chi } \, d\mu $ ; individual values $\hat {f}(\chi )$ are called Fourier coefficients of f. The characters form an orthonormal subset of $L^2(\mu )$ spanning a dense subspace of $L^2(\mu )$ . The Fourier transform maps $L^{2}(\mu )$ onto $\ell ^{2}(\widehat {K})$ isometrically: when f, $g\in L^{2}(\mu )$ , we have the Parseval identity

(2.1) $$ \begin{align} \sum_{\chi\in\widehat{K}} \hat{f}(\chi)\overline{\hat{g}(\chi)} = \int f\bar{g}\, d\mu. \end{align} $$

Specializing with $g=f$ , we have

(2.2) $$ \begin{align}\sum_{\chi\in \widehat{K}}|\hat{f}(\chi)|^2=\int |f|^2\, d\mu, \end{align} $$

and the Fourier series $\sum _{\chi \in \widehat {K}} \hat {f}(\chi )\chi $ converges to f in the norm topology of $L^2(\mu )$ . Here when we say a series $\sum _{j\in I} a_j$ converges to a in some topology, we mean convergence in the sense of unordered sums: for every neighborhood O of a, there is a finite set $I_O\subseteq I$ such that $\sum _{j\in J} a_j\in O$ whenever J is a finite set containing $I_O$ .

Definition 2.1. Let $f, g \in L^2(\mu )$ . The convolution $f*g:K\to \mathbb C$ is defined by

$$\begin{align*}f*g(x):=\int f(t)g(x-t)\, d\mu(t). \end{align*}$$

We have the standard identity [Reference Rudin33, Theorem 1.2.4 (b)]

(2.3) $$ \begin{align} \widehat{f*g}(\chi)=\hat{f}(\chi)\hat{g}(\chi) \qquad \text{for all } \chi\in\widehat{K}. \end{align} $$

The usual proof of (2.3) uses Fubini’s theorem to interchange the order of an iterated integral involving Haar measure $\mu $ on K. In our proof of Theorem 1.5, we will have a finitely additive measure in place of Haar measure, so Fubini’s theorem will not be available. In §5 we explain how using an extreme invariant mean recovers what we need.

2.2 Notation

If $f:D\to \mathbb R$ is a function and $c\in \mathbb R$ , $\{f>c\}$ denotes the level set $\{x\in D: f(x)>c\}$ .

To prove Theorem 2.2, let $A, B\subseteq K$ be $\mu $ -measurable sets. We average translates of $1_B$ by elements of A, where “average” is with respect to Haar measure:

$$\begin{align*}F(x):=\int 1_{t+B}(x)1_A(t)\, d\mu(t). \end{align*}$$

We see that $1_{t+B}(x)1_{A}(t)>0$ only if $t\in A$ and $x\in t+B$ , meaning $x\in A+B$ . Thus F is supported on $A+B$ . Simplifying, we get $F(x)=\int 1_{A}(t)1_B(x-t)\, d\mu (t)=1_{A}*1_{B}(x)$ . Theorem 2.2 is now a consequence of the following lemma. From it we conclude that $1_A*1_B$ is continuous and not identically $0$ ; the level set $\{1_{A}*1_{B}>0\}$ will then be the desired open subset of $A+B$ .

The proof uses the identity $\int h(-t)\, d\mu (t)=\int h(t)\, d\mu (t)$ , which follows from uniqueness of Haar measure.

Proof. Note that when $f,g\in L^2(\mu _K)$ , (2.2) implies $\hat {f},\hat {g}\in \ell ^{2}(\widehat {K})$ . Then (2.3) and Cauchy-Schwarz imply $\sum _{\chi \in \widehat {K}}|\widehat {f*g}(\chi )|$ converges. Hence the Fourier series $\sum _{\chi \in \widehat {K}} \hat {f}(\chi )\hat {g}(\chi )\chi $ converges uniformly, and its sum is continuous. To prove $f*g$ is continuous, we show that $f*g$ is identically equal to the sum of its Fourier series. This follows from (2.1): writing $\tilde {g}_x(t)$ for $\overline {g(x-t)}$ , we have

$$ \begin{align*} \widehat{\tilde{g}_x}(\chi)&= \int \overline{g(x-t)}\overline{\chi(t)}\, d\mu(t)\\ &= \int \overline{g(-t)} \overline{\chi(x+t)}\, d\mu(t) && \text{replacing } t \text{ with } t+x\\ &= \int \overline{g(-t)}\overline{\chi(t)} \overline{\chi(x)}\, d\mu(t) \\ &= \int \overline{g(t)}\overline{\chi(-t)} \, d\mu(t)\, \overline{\chi(x)} && \text{replacing } t \text{ with } -t, \text{ factoring } \overline{\chi(x)}\\ &= \overline{\hat{g}(\chi)} \overline{\chi(x)}, \end{align*} $$

so $\widehat {\tilde {g}_x}(\chi )=\overline {\hat {g}(\chi )} \overline {\chi (x)}$ for all $x\in K$ , $\chi \in \widehat {K}$ . Then

(2.4) $$ \begin{align} f*g(x) = \int f(t) g(x-t)\, d\mu(t) = \int f \overline{\tilde{g}_{x}}\, d\mu = \sum_{\chi\in\widehat{K}} \hat{f}(\chi)\overline{\widehat{\tilde{g}_x}(\chi)}=\sum_{\chi\in\widehat{K}} \hat{f}(\chi)\hat{g}(\chi)\chi(x) \end{align} $$

for all $x\in K$ . The identity $\int f*g\, d\mu = \int f\, d\mu \int g\, d\mu $ is the special case of (2.3) where $\chi $ is the trivial character.

3.1 The mean ergodic theorem

For a Hilbert space $\mathcal H$ and a unitary operator $U:\mathcal H\to \mathcal H$ , we write $\mathcal H_{U\text {-inv}}$ for the closed subspace $\{w\in \mathcal H: Uw=w\}$ and $P_{U\text {-inv}}$ for orthogonal projection onto $\mathcal H_{U\text {-inv}}$ . The mean ergodic theorem is often stated as follows.

Theorem 3.1. Let $U:\mathcal H\to \mathcal H$ be a unitary operator and $w\in \mathcal H$ . Then

$$\begin{align*}\lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^{N} U^{n} w = P_{U\text{-inv}}w\end{align*}$$

in the norm topology of $\mathcal H$ .

Weak convergence is often all that is required for applications: we have

$$\begin{align*}\lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^{N} \langle v,U^{n}w\rangle=\langle v, P_{U\text{-inv}}w\rangle \qquad \text{for all } v, w\in \mathcal H. \end{align*}$$

There are many sequences of integers $(a_{n})_{n\in \mathbb N}$ that can be used in place of $a_{n}=n$ while retaining convergence to $P_{U\text {-inv}}w$ . We say that $(a_{n})_{n\in \mathbb N}$ is an ergodic sequence Footnote ¹ if for every unitary operator on a Hilbert space $\mathcal H$ and all $v, w\in \mathcal H$

$$\begin{align*}\lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^{N} \langle v ,U^{a_{n}}w\rangle = \langle v, P_{U\text{-inv}} w\rangle. \end{align*}$$

Many examples of ergodic sequences are known – see §10 for discussion and references.

3.2 Matrix coefficients and FS-means

The mean ergodic theorem remains true when generalized to arbitrary (possibly uncountable) abelian groups, with the minor caveat that in uncountable groups, one must average over Følner nets, rather than sequences. We will need only weak convergence, so we use means instead of averaging over nets of finite subsets of $\Gamma $ .

Definition 3.2. A unitary action of $\Gamma $ on a Hilbert space $\mathcal H$ is an action of $\Gamma $ on $\mathcal H$ by unitary operators $U_{\gamma }$ , $\gamma \in \Gamma .$ For $v, w\in \mathcal H$ , the matrix coefficient determined by U, v, and w is the function $\phi _{v,w}:\Gamma \to \mathbb C$ ,

$$\begin{align*}\phi_{v,w}(\gamma):=\langle v, U_{\gamma }w\rangle.\end{align*}$$

We write $\mathcal {H}_{U\text {-inv}}$ for the closed subspace $\{x\in \mathcal H: U_{\gamma } x = x \text { for all } \gamma \in \Gamma \}$ , and write $P_{U\text {-inv}}w$ for the orthogonal projection of w onto $\mathcal H_{U\text {-inv}}$ .

The following is a weak ergodic theorem for abelian groups.

Theorem 3.3. Let $\Gamma $ be a discrete abelian group. If $m\in {\mathcal {M}}_{\tau }(\Gamma )$ , U is a unitary action of $\Gamma $ on a Hilbert space $\mathcal H$ , and $v, w\in \mathcal H$ , then

(3.1) $$ \begin{align} m(\phi_{v,w})=\langle v, P_{U\text{-inv}} w\rangle. \end{align} $$

Although Theorem 3.3 is classical (and in fact a generalization to arbitrary groups goes back at least to [Reference Godement21]), we give a proof in §9. The conclusion of Theorem 3.3 provides all properties of m required for Theorem 1.5, motivating the following definition.

Every invariant mean is an FS-mean, by Theorem 3.3, and FS-means can be thought of as generalizations of ergodic sequences. We write ${\mathcal {M}}^{FS}(\Gamma )$ for the set of $FS$ -means on $\Gamma $ . So

$$\begin{align*}{\mathcal{M}}_{\tau}^{ext}(\Gamma)\subseteq {\mathcal{M}}_{\tau}(\Gamma) \subseteq {\mathcal{M}}^{FS}(\Gamma). \end{align*}$$

In countable abelian groups there are many non-invariant FS-means. See §10 for details.

3.3 Eigenspaces

For a unitary action U of $\Gamma $ on a Hilbert space $\mathcal H$ and $\chi \in \widehat {\Gamma }$ , the $\chi $ -eigenspace of U is

$$\begin{align*}\mathcal H_{\chi}:=\{w\in \mathcal H: U_{\gamma} w=\chi(\gamma) w \text{ for all } \gamma \in \Gamma\}. \end{align*}$$

We write $\mathcal H_{c}$ to denote the smallest closed subspace of $\mathcal H$ containing every eigenspace of $\mathcal U$ . In other words, $\mathcal H_{c}$ is the closure of the span of $\bigcup _{\chi \in \widehat {\Gamma }} \mathcal H_{\chi }$ .

The next lemma states the properties of FS-means we use in the proof of Theorem 1.5.

Lemma 3.5 is proved as part of Lemma 9.5 in §9.

Lemma 3.5 is often derived as a consequence of Wiener’s lemma for Fourier transforms of measures – see [Reference Eberlein16] for such a statement in the generality of our setting. We present essentially this argument in our proof of Lemma 9.5. Lemma 3.5 also follows from Corollary 3.9 of [Reference Badea and Grivaux1], an explicit computation of $m_{\mathcal B}(\phi _{v,w})$ . The same computation is done in Lemma 6.4 and Proposition 6.7 of [Reference Björklund and Fish7]. Although the latter article makes the standing assumption that the acting group is countable, that assumption is not used for these results.

Like Theorem 3.3, the natural generalization of Lemma 3.5 to arbitrary (including uncountable and nonamenable) groups goes back to [Reference Godement21].

3.4 Fourier coefficients, $b\Gamma $ , and the Besicovitch-Radon-Nikodym density

Let $\Gamma $ be a discrete abelian group.

Recall from §1.4 that we identify $\Gamma $ with a topologically dense subgroup of its Bohr compactification $b\Gamma $ , and that every character $\chi \in \widehat {\Gamma }$ is the restriction of a unique $\tilde {\chi }\in \widehat {b\Gamma }$ .

Lemma 3.10. If $\eta $ is an FS-mean on $\ell ^\infty (\Gamma )$ and $f\in \ell ^{\infty }(\Gamma )$ , then $\hat {f}^{\eta }\in \ell ^{2}(\widehat {\Gamma })$ , and

$$\begin{align*}\tilde{f}^{(\eta)}:=\sum_{\chi\in \widehat{\Gamma}} \hat{f}^\eta(\chi)\tilde{\chi} \end{align*}$$

defines an element of $L^{2}(\mu _{b\Gamma })$ with the following properties:

1. (i) $\widehat {\tilde {f}^{(\eta )}}(\tilde {\chi })=\hat {f}^{\eta }(\chi )$ for all $\chi \in \widehat {\Gamma }$ ;

2. (ii) $\int \tilde {\phi }\tilde {f}^{(\eta )}\, d\mu _{b\Gamma } = \eta (\tilde {\phi }|_{\Gamma } f)$ for all $\tilde {\phi }\in C(b\Gamma )$ ;

3. (iii) $\int \tilde {f}^{(\eta )}\, d\mu _{b\Gamma }=\eta (f)$ ;

4. (iv) $\|\tilde {f}^{(\eta )}\|_{L^{\infty }(\mu _{b\Gamma })}\leq \|f\|_{\infty }$ . Furthermore, if f is real-valued, then $\tilde {f}^{(\eta )}$ is real-valued and $\inf f \leq \tilde {f}^{(\eta )}(x) \leq \sup f$ for $\mu _{b\Gamma }$ -almost every x.

   In particular, if $f:\Gamma \to [0,1]$ , then $0\leq \tilde {f} \leq 1\ \mu _{b\Gamma }$ -a.e.

We call $\tilde {f}^{(\eta )}$ the Besicovitch-Radon-Nikodym density of f with respect to $\eta $ ; cf. Section 4 of [Reference Björklund and Griesmer9].

Proof. Let $\eta $ and f be as in the hypothesis. By Corollary 3.6, the series $\sum _{\chi \in \widehat {\Gamma }}|\hat {f}^{\eta }(\chi )|^{2}$ converges, so $\tilde {f}^{(\eta )}$ is well-defined in $L^{2}(\mu _{b\Gamma })$ .

Property (i) follows immediately from the definition of $\tilde {f}^{(\eta )}$ . Note that (i) is equivalent to the special case of (ii) where $\tilde {\phi }\in \widehat {b\Gamma }$ . The general case of (ii) now follows by linearity and the fact that the set of trigonometric polynomials is uniformly dense in $C(b\Gamma )$ . (iii) is the special case of (i) where $\chi \in \widehat {\Gamma }$ is trivial.

We prove the special case of (iv) where $f:\Gamma \to [0,1]$ ; the general case follows by linearity. We first prove that

(3.2) $$ \begin{align} 0\leq \int \tilde{\phi} \tilde{f}^{(\eta)} \, d\mu_{b\Gamma}\leq \|\tilde{\phi}\|_{L^{1}(\mu_{b\Gamma})} \end{align} $$

whenever $\tilde {\phi }\in L^{1}(\mu )$ is real-valued and nonnegative. To see this, note that (3.2) holds for every continuous $\tilde {\phi }:b\Gamma \to [0,1]$ , since

$$\begin{align*}\int \tilde{\phi} \tilde{f}^{(\eta)} \, d\mu_{b\Gamma}=\eta(\tilde{\phi}|_{\Gamma} f)\leq \eta(\tilde{\phi}|_{\Gamma})=\int \tilde{\phi}\, d\mu_{b\Gamma} = \|\tilde{\phi}\|_{L^{1}(\mu_{b\Gamma})}. \end{align*}$$

The general case of (3.2) follows, as every nonnegative real-valued $\tilde {\phi }\in L^{1}(\mu _{b\Gamma })$ can be approximated in the $L^{1}$ -norm by continuous functions taking nonnegative values. Thus the map $\tilde {\phi }\mapsto \int \tilde {\phi } \tilde {f}^{(\eta )} \mu _{b\Gamma }$ is a linear functional on $L^{1}(\mu _{b\Gamma })$ with norm at most $1$ , mapping nonnegative real-valued $\tilde {\phi }$ to nonnegative real numbers. The usual identification of $L^{\infty }(\mu _{b\Gamma })$ with the dual of $L^{1}(\mu _{b\Gamma })$ then implies $\tilde {f}^{(\eta )}$ is equal, $\mu _{b\Gamma }$ -almost everywhere, to a function taking values in $[0,1]$ .

Proof. To prove (i), assume $V\subseteq b\Gamma $ is compact. By assumption, $\tilde {A}_{\eta }\sim _{\mu _{b\Gamma }} \{\tilde {f}^{(\eta )}>0\}$ . Since $0\leq \tilde {f}^{(\eta )}\leq 1\ \mu _{b\Gamma }$ -a.e., it suffices to prove that

(3.3) $$ \begin{align} \int 1_{V} \tilde{f}^{(\eta)}\, d\mu_{b\Gamma}\geq \eta(A\cap V). \end{align} $$

Lemma 3.10 says that $\int \tilde {\phi } \tilde {f}^{(\eta )}\, d\mu _{b\Gamma }=\eta (\tilde {\phi }|_{\Gamma } 1_A)$ for every continuous $\tilde {\phi }:b\Gamma \to [0,1]$ . Thus, for every continuous $\tilde {\phi }:b\Gamma \to [0,1]$ with $\tilde {\phi }|_{V}\equiv 1$ , we have $\int \tilde {\phi }\, \tilde {f}^{(\eta )}\, d\mu _{b\Gamma }\geq \eta (A\cap V)$ . By outer regularity of Haar measure, this implies (3.3).

To prove (ii), assume $V\subseteq b\Gamma $ is clopen. Part (i) provides the implication $\mu _{b\Gamma }(\tilde {A}_\eta \cap V)=0 \implies \eta (A\cap V)=0$ . To prove the reverse implication, assume $\eta (A\cap V)=0$ . Since V is clopen, $1_V$ is continuous, so Lemma 3.9 implies $\int 1_V \tilde {f}^{(\eta )}\, d\mu = \eta (1_V|_\Gamma 1_A)=\eta (A\cap V)=0$ . Since $\tilde {A}_\eta \sim _{\mu _{b\Gamma }} \{\tilde {f}^{(\eta )}>0\}$ , this means $\mu _{b\Gamma }(\tilde {A}_\eta \cap V)=0$ .

Lemma 3.14. Let $\nu \in {\mathcal {M}}_{\tau }^{ext}(\Gamma )$ and $\eta \in {\mathcal {M}}^{FS}(\Gamma )$ . Let A, $B\subseteq \Gamma $ , let $f=1_A$ , $g=1_B$ , and let $\tilde {f}^{(\nu )}$ , $\tilde {g}^{(\eta )}$ be as in Lemma 3.10. Let $\tilde {h}=\tilde {f}^{(\nu )}*\tilde {g}^{(\eta )}$ and let $h=\tilde {h}|_{\Gamma }$ . Then the series

(3.4) $$ \begin{align} \sum_{\chi\in\widehat{\Gamma}} \hat{f}^{\nu}(\chi)\hat{g}^{\eta}(\chi)\chi \end{align} $$

converges uniformly to h, and

1. (i) $\nu (h1_{A+B})=\nu (h)$ ;

2. (ii) for all $\delta>0$ , $\{h\geq \delta \}\subset _\nu A+B$ ;

3. (iii) if $V\subseteq \{\tilde {h}>0\}$ is compact, then $V\cap \Gamma \subset _{\nu } A+B$ ;

4. (iv) $\nu (A+B)\geq \mu _{b\Gamma }(\{\tilde {h}>0\})$ .

Lemma 3.14 is proved in §7.

The next lemma connects the level set $\{\tilde {h}>0\}$ to a sumset in $b\Gamma $ .

Lemma 3.15 is proved in §8.

The next lemma deduces property (ii) in Lemma 3.14 from (i).

Combining Lemmas 3.14 and 3.15 will yield our main result, the promised generalization of Theorem 1.5.

Theorem 3.16. Let $\Gamma $ be a discrete abelian group. Let $\nu \in {\mathcal {M}}_{\tau }^{ext}(\Gamma )$ , $\eta \in {\mathcal {M}}^{FS}(\Gamma )$ , and A, $B\subseteq \Gamma $ . Then there are $F_{\sigma }$ sets $\tilde {A}_{\nu }$ , $\tilde {B}_{\eta }\subseteq b\Gamma $ such that $\mu _{b\Gamma }(\tilde {A}_{\nu })\geq \nu (A)$ , $\mu _{b\Gamma }(\tilde {B}_{\eta })\geq \eta (B)$ , and

(3.5) $$ \begin{align} \nu(A+B)\geq \mu_{b\Gamma}(\tilde{A}_{\nu}+\tilde{B}_{\eta}). \end{align} $$

Furthermore,

1. (i) if $V\subseteq b\Gamma $ is compact, then

   $$\begin{align*}\mu_{b\Gamma}(\tilde{A}_{\nu}\cap V)\geq \nu(A\cap V) \quad \text{and} \quad \mu_{b\Gamma}(\tilde{B}_{\eta}\cap V)\geq \eta(B\cap V);\end{align*}$$

2. (ii) if $V\subseteq b\Gamma $ is clopen, then

   $$\begin{align*}\mu_{b\Gamma}(\tilde{A}_{\nu}\cap V)=0\iff \nu(A\cap V)=0 \quad \text{and} \quad \mu_{b\Gamma}(\tilde{B}_{\eta}\cap V)=0\iff \eta(B\cap V)=0;\end{align*}$$

3. (iii) if $V\subseteq \tilde {A}_{\nu }+\tilde {B}_{\eta }$ is compact, then $V\cap \Gamma \subset _{\nu } A+B$ .

4.1 Connection to previous work

Lemma 3.14 and Theorem 3.16 improve [Reference Bergelson, Furstenberg and Weiss4, Theorem I], which says that if $A, B\subseteq \mathbb Z$ have positive upper Banach density, then $A+B$ is piecewise Bohr. The improvement is in three aspects:

1. (i) The containments $\{h\geq \delta \} \subset _\nu A+B$ and $(\tilde {A}_{\nu }+\tilde {B}_{\eta })\cap V \subset _{\nu } A+B $ in Lemma 3.14 and Theorem 3.16 say that the piecewise Bohr structure of $A+B$ can be found where the largeness of A itself is realized; [Reference Bergelson, Furstenberg and Weiss4, Theorem I] does not specify the location of the thick set T where the piecewise Bohr structure of $A+B$ is realized.

2. (ii) The hypothesis on B is weakened from “positive upper Banach density” to $\eta (B)>0$ for some FS-mean $\eta $ .

3. (iii) Relating $A+B$ to $\tilde {A}_{\nu }+\tilde {B}_{\eta }$ allows one to transfer results about sumsets in compact abelian groups directly to discrete abelian groups.

Theorem 3.16 is analogous to a special case of [Reference Björklund and Fish7, Proposition 1.10]. The latter does specify the structure of $A+B$ in terms of a convolution $\tilde {f}*\tilde {g}$ on $b\Gamma $ , and the results there apply to general countable groups (not necessarily abelian, or even amenable). Our Theorem 3.16 locates the piecewise Bohr structure of $A+B$ in terms of $\nu $ , whereas [Reference Björklund and Fish7, Proposition 1.10] locates the structure of $A+B$ in terms of a thick set T whose relation to $\nu $ is unspecified. The proofs in [Reference Björklund and Fish7] rely on detailed analysis of orbits in measure preserving dynamical systems, which our proofs avoid.

Results in [Reference Björklund and Fish8] do locate structure in $A+B$ in terms of a $\nu \in {\mathcal {M}}_{\tau }^{ext}(\Gamma)$ with $\nu (A)>0$ .

Our results apply to discrete abelian groups of arbitrary cardinality, whereas [Reference Björklund and Fish7] and [Reference Björklund and Fish8] consider only countable discrete groups.

5.1 Integral notation for means

While means are only finitely additive, we follow [Reference Björklund and Fish7] and [Reference Björklund and Fish8], using integral notation to denote some evaluations of means. So $m(f)$ may be denoted $\int f(t)\, dm(t)$ .

5.2 Pointwise convolution versus weak convolution

To prove Lemma 3.14, one would naturally set $f=1_A$ , $g=1_B$ , and define a convolution $f*_\eta g\in \ell ^\infty (\Gamma )$ by $f*_\eta g(x):= \int f(x-t)g(t)\, d\eta (t)$ . One easily verifies that $f*_\eta g(x)=0$ unless $x\in A+B$ , so we would like to show that

(5.1) $$ \begin{align} f*_{\eta} g \sim_{\nu} \sum_{\chi\in{\widehat{\Gamma}}} {\hat{f}}^{\nu}(\chi) {\hat{g}}^{\eta}(\chi) \chi, \end{align} $$

cf. equation (2.4) above. This is difficult, due the fact that $\nu $ is not countably additive: the inability to exchange the order of integration à la Fubini is an obstruction to computing Fourier coefficients of $f*_{\eta } g$ . To circumvent this issue, we use the alternative Definition 5.2, denoted . This can be thought of as a weak $L^{2}(\nu )$ limit of averages of translates of f, weighted by g and $\eta $ . When $\nu $ is only assumed to be invariant, we still cannot prove an identity like (5.1) for , due to examples like Example 1.11. Thanks to Lemma 5.4, the desired identity is available if we assume $\nu \in {\mathcal {M}}_{\tau }^{ext}(\Gamma )$ and $\eta \in {\mathcal {M}}^{FS}(\Gamma )$ .

5.3 Averaging unitary actions

We generalize the mean ergodic theorem (Theorem 3.1 above) in two ways: (i) instead of averaging over $n\in \{1,\dots ,N\}$ , we average with respect to a mean on $\ell ^\infty (\Gamma )$ , and (ii) we introduce some coefficients $g(n)$ , so our averages will be generalizations of $\frac {1}{N}\sum _{n=1}^N g(n)U^n w$ . We will see that limits of such averages can often be computed in terms of Fourier coefficients and eigenspaces; cf. [Reference Jones, Rosenblatt and Tempelman28].

Let $\eta $ be any mean on $\ell ^{\infty }(\Gamma )$ and U an action of $\Gamma $ by unitary operators $U_t$ on a Hilbert space $\mathcal H$ . For each $w\in \mathcal H$ and $g\in \ell ^{\infty }(\Gamma )$ we define

$$\begin{align*}\int (U_{t} w)g(t)\, d\eta(t)\end{align*}$$

to be the unique $z\in \mathcal H$ satisfying $\langle v, z\rangle = \int \langle v, U_t w\rangle \, g(t) d\eta (t)$ for all $v\in \mathcal H$ . The Reisz representation theorem for Hilbert spaces provides this z, as the map $v\mapsto \int \langle v, U_t w\rangle \, g(t) d\eta (t)$ is a continuous linear functional on $\mathcal H$ .

Recall from §3.3 that for a given $\chi \in \widehat {\Gamma }$ , $\mathcal H_\chi :=\{w\in \mathcal H:U_{\gamma }w= \chi (\gamma )w \text { for all } \gamma \in \Gamma \}$ is the $\chi $ -eigenspace of U. For each $g\in \ell ^{\infty }(\Gamma )$ and each $w\in \mathcal H_{\chi }$ , we have $\int (U_{t} w) g(t)\, d\eta (t)=\int \chi (t)g(t)w \, d\eta (t)$ , so

(5.2) $$ \begin{align}\int (U_{t}w)g(t)\,d\eta(t) = \eta(g\chi)w \qquad \text{ for each } w\in \mathcal H_{\chi}, g\in \ell^{\infty}(\Gamma). \end{align} $$

5.4 Hilbert space associated to a mean

Let m be a mean on $\ell ^{\infty }(\Gamma )$ . As above we write $f\sim _m g$ if $m(|f-g|)=0$ .

Consider the Hermitian form $\langle \cdot , \cdot \rangle _m$ on $\ell ^\infty (\Gamma )$ given by $\langle f,g\rangle _m:=m(f\overline {g})$ and the associated seminorm $\|f\|_{m}:=\langle f,f\rangle _m$ . Observe that $\langle \cdot , \cdot \rangle _m$ is not an inner product, as $\|f\|_m$ may be zero for nonzero f. Nevertheless, $\langle \cdot , \cdot \rangle _m$ is positive semidefinite and sesquilinear; in particular Cauchy-Schwarz holds:

(5.3) $$ \begin{align} m(f\overline{g})\leq \|f\|_{m}\|g\|_{m}. \end{align} $$

Let $\ell ^{\infty }(\Gamma )/{\sim }_{m}$ denote the quotient $\ell ^\infty (\Gamma )/\{f:\|f\|_m=0\}$ . We write $L^2(m)$ for the metric completion of $\ell ^\infty (\Gamma )/{\sim }_{m}$ with respect to the metric $d_m([f],[g]):=\|f-g\|_m$ , where $[f]$ and $[g]$ denote the respective $\sim _{m}$ equivalence classes of f and g.

Elements of $L^2(m)$ are represented by Cauchy sequences $(f_n)_{n\in \mathbb N}$ of elements of $\ell ^\infty (\Gamma )$ , where “Cauchy” means $\lim _{n\to \infty } \sup _{j,k\geq n} \|f_j-f_k\|_m=0$ . When appropriate, we identify $f\in \ell ^{\infty }(\Gamma )$ with the element of $L^{2}(m)$ determined by the constant Cauchy sequence $f_{n}=f$ . In particular, each $\chi \in \widehat {\Gamma }$ determines an element of $L^{2}(m)$ .

Now $L^{2}(m)$ is a Hilbert space; the inner product of two elements v, w represented by Cauchy sequences $(f_{n})_{n\in \mathbb N}$ , $(g_{n})_{n\in \mathbb N}$ is $\langle v,w \rangle _{m}=\lim _{n\to \infty } \langle f_{n},g_{n}\rangle _{m}.$

If $w\in L^{2}(m)$ and $g\in \ell ^{\infty }(\Gamma )$ , we write $w\sim _{m} g$ to mean that w is represented by the constant Cauchy sequence $g_{n}=g$ .

5.5 Translation action

Now assume m is an invariant mean on $\ell ^{\infty }(\Gamma )$ . For $f\in \ell ^\infty (\Gamma )$ , we write $\gamma \cdot f$ for $f_\gamma $ , where $f_\gamma (x)=f(x-\gamma )$ . Since m is invariant, translation preserves the seminorm: $\|\gamma \cdot f\|_{m}=\|f\|_{m}$ for all $\gamma \in \Gamma $ and $f\in \ell ^{\infty }(m)$ . Translation on $\ell ^{\infty }(\Gamma )$ therefore induces an action of $\Gamma $ by unitary operators $U_{\gamma }$ on $L^2(m)$ : if $w\in L^2(m)$ is represented by the Cauchy sequence $(f_n)_{n\in \mathbb N}$ , we define $U_\gamma w$ to be the element of $L^2(m)$ represented by the Cauchy sequence $(\gamma \cdot f_n)_{n\in \mathbb N}$ . It is routine to check that the element of $L^{2}(m)$ determined by $(\gamma \cdot f_n)_{n\in \mathbb N}$ depends only on w and not on the specific choice of Cauchy sequence, so $U_{\gamma }w$ is well-defined.

Definition 5.2. Let m be an invariant mean and let $\eta $ be any mean on $\ell ^{\infty }(\Gamma )$ . Let $f,g\in \ell ^\infty (\Gamma )$ , and write U for the translation action of $\Gamma $ on $L^{2}(m)$ . We define (the convolution of f and g with respect to m and $\eta $ ) to be the element of $L^{2}(m)$ given by

(5.4)

The “f” on the right-hand side of (5.4) is the element of $L^{2}(m)$ represented by the constant Cauchy sequence $f_{n}=f$ , while “g” denotes the actual function g. Thus is the unique element of $L^{2}(m)$ satisfying for all $\phi \in L^{2}(m)$ .

Lemma 5.3. Let m be an invariant mean on $\ell ^\infty (\Gamma )$ and let $\eta $ be any mean on $\ell ^\infty (\Gamma )$ . If f, $g:\Gamma \to [0,1]$ are supported on A and B, respectively, then is nonnegative and is supported on $A+B$ . More precisely: if $\phi :\Gamma \to [0,1]$ , then

(5.5)

(5.6)

In light of Equation (5.2), simplifying in terms of characters requires us to identify the $\chi $ -eigenspace $\mathcal H_{\chi }$ of the translation action U on $L^{2}(m)$ . Clearly we have $\overline {\chi }\in \mathcal H_{\chi }$ , but in general $\mathcal H_{\chi }$ may be infinite-dimensional. Lemma 5.4 says that for extreme invariant means $\nu $ , each eigenspace of the translation action on $L^{2}(\nu )$ is one-dimensional.

Lemma 5.4 follows from the well-known correspondence between extreme invariant means on $\ell ^{\infty }(\Gamma )$ and Radon probability measures $\mu $ on $\beta \Gamma $ ( $=$ the Stone-Cech compactification of the discrete group $\Gamma $ ) for which left translation by elements of $\Gamma $ induces an ergodic measure-preserving action on $(\beta \Gamma ,\mu )$ . We present an elementary proof in §6.

Lemma 5.5 is proved at the end of §5.6.

5.6 Simplifying convolutions

Let m be an invariant mean on $\ell ^{\infty }(\Gamma )$ and let $\eta $ be an arbitrary mean on $\ell ^{\infty }(\Gamma )$ . Let U be a unitary action of $\Gamma $ on a Hilbert space $\mathcal H$ .

Recall that $\mathcal H_c$ is the smallest closed subspace of $\mathcal H$ containing all the eigenspaces $\mathcal H_{\chi }$ . Since each $U_\gamma $ is unitary, the eigenspaces are closed and mutually orthogonal, and each $w\in \mathcal H$ may be expanded uniquely as

$$\begin{align*}w=w_0+\sum_{\chi\in\widehat{\Gamma}} w_\chi,\end{align*}$$

where $w_\chi \in \mathcal H_\chi $ and $w_0\in \mathcal H_c^\perp $ . With this expansion, equation (5.2) implies

(5.7) $$ \begin{align} \int (U_{t}w) g(t)\, d\eta(t) = \int (U_{t}w_{0}) g(t)\, d\eta(t)+ \sum_{\chi\in\widehat{\Gamma}} \eta(g\chi) w_{\chi}. \end{align} $$

We will simplify $\int (U_{t}w_{0}) g(t)\, d\eta (t)$ under the assumption that $\eta $ is an FS-mean, and we will simplify $\eta (g\chi )w_{\chi }$ under the assumption that U is the translation action associated to an extreme invariant mean. While these assumptions are independent of one another, our proof of Lemma 5.5 uses both.

Proof. Let $v\in \mathcal H$ and $w\in \mathcal H_{c}^{\perp }$ . Since $\eta $ is an FS-mean, Lemma 3.5 (ii) says that $\int |\langle v, U_{t}w\rangle |^{2}\, d\eta (t)=0$ . We apply Cauchy-Schwarz to find

$$\begin{align*}\Bigl|\int \langle v,U_{t}w \rangle g(t) \, d\eta(t)\Bigr|^{2} \leq \int |\langle v, U_{t}w\rangle|^{2}\, d\eta(t)\, \eta(|g|^{2}) =0, \end{align*}$$

so $\int \langle v,U_{t}w \rangle g(t) \, d\eta (t)=0$ . Since this holds for all $v\in \mathcal H$ , we have $\int (U_{t}w)\, g(t)\, d\eta (t)=0$ .

Proof of Lemma 5.5.

Let $\nu \in {\mathcal {M}}_{\tau }^{ext}(\Gamma )$ and $\eta \in {\mathcal {M}}^{FS}(\Gamma )$ . Let U be the translation action on $L^{2}(\nu )$ , and let $f, g\in \ell ^{\infty }(\Gamma )$ . Specializing equation (5.7) with f in place of w, the left-hand side becomes , so we have

(5.8)

where $f_{\chi }$ is the orthogonal projection of f onto the $\chi $ -eigenspace of U, and $f_{0}$ is orthogonal to every eigenspace of U in $L^{2}(\nu )$ . Lemma 5.6 implies $\int (U_{t}f_{0}) g(t)\, d\eta (t)=0$ . Part (ii) of Lemma 5.4 implies $f_{\chi }\sim _{\nu }\langle f,\overline {\chi }\rangle _{\nu }\overline {\chi }$ , which we can write as $f_{\chi }\sim _{\nu }\nu (f\chi )\overline {\chi }$ . The right-hand side of (5.8) then simplifies as $\sum _{\chi \in \widehat {\Gamma }} \eta (g\chi )\nu (f\chi )\overline {\chi }$ . Replacing $\chi $ with $\overline {\chi }$ and permuting the order of multiplication, we have . By definition, this is equal to $\sum _{\chi \in \widehat {\Gamma }} \hat {f}^\nu (\chi )\hat {g}^\eta (\chi )\chi $ , as desired.

To prove uniform convergence, first apply Corollary 3.6 to get $\sum _{\chi \in \widehat {\Gamma }}|\eta (g\overline {\chi })|^{2}\leq \eta (|g|^{2})$ , and likewise $\sum _{\chi \in \widehat {\Gamma }} |\nu (f\overline {\chi })|^{2}\leq \nu (|f|^{2})$ . Cauchy-Schwarz then implies $\sum _{\chi \in \widehat {\Gamma }} |\nu (f\overline {\chi })\eta (g\overline {\chi })|$ converges, so $\sum _{\chi \in \widehat {\Gamma }} \eta (g\overline {\chi })\nu (f\overline {\chi })\chi $ converges uniformly.

6.1 Truncations

Let $\alpha>0$ , and define $\psi _{\alpha }:\mathbb C\to \mathbb C$ by

$$\begin{align*}\psi_{\alpha}(z) := \begin{cases} z &\text{ if } |z|<\alpha\\ \alpha z/|z| &\text{ if } |z|\geq \alpha \end{cases} \end{align*}$$

Note that $\psi _{\alpha }$ is Lipschitz, with constant 1: $|\psi _{\alpha }(z)-\psi _{\alpha }(z')|\leq |z-z'|$ for all $z, z'\in \mathbb C$ .

For a function f with codomain $\mathbb C$ , define the truncation $f^{(\alpha )}$ by $f^{(\alpha )}:=\psi _{\alpha } \circ f$ . Then $f^{(\alpha )}(x)=f(x)$ when $|f(x)|\leq \alpha $ , and $|f^{(\alpha )}(x)|=\alpha $ when $|f(x)|\geq \alpha $ .

Proof. Let $\alpha>0$ and $f, g\in \ell ^{\infty }(\Gamma )$ . Pointwise we have $|f^{(\alpha )}-g^{(\alpha )}|= |\psi _{\alpha }(f)-\psi _{\alpha }(g)|\leq |f-g|$ , so

(6.1) $$ \begin{align} \|f^{(\alpha)} -g^{(\alpha)}\|_{m}\leq \|f-g\|_{m}. \end{align} $$

It follows that if $(f_{n})_{n\in \mathbb N}$ is Cauchy, so is $(f_{n}^{(\alpha )})_{n\in \mathbb N}$ . Thus $w^{(\alpha )}$ is well-defined.

Now let $(f_{n})_{n\in \mathbb N}$ be a Cauchy sequence of elements of $\ell ^{\infty }(\Gamma )$ representing w. We will prove that $\lim _{\alpha \to \infty }w^{(\alpha )}=w$ . Let $\varepsilon>0$ .

Fix $n\in \mathbb N$ such that $\|f_{j}-f_{\ell }\|_{m}<\varepsilon $ for all $j,\ell \geq n$ . Then for all $\alpha>0$ and all $j, \ell \geq n$ , (6.1) implies $\|f_{j}^{(\alpha )}-f_{\ell }^{(\alpha )}\|_{m}<\varepsilon $ . Choose $\alpha _{0}>0$ so that $f_{n}^{(\alpha _{0})}=f_{n}$ . For every $\alpha \geq \alpha _{0}$ and all $j,\ell \geq n$ , we have

$$\begin{align*}\|f_{j}^{(\alpha)}-f_{\ell}\|_{m}\leq \|f_{j}^{(\alpha)}-f_{n}^{(\alpha)}\|_{m}+\|f_{n}^{(\alpha)}-f_{n}\|_{m}+\|f_{n}-f_{\ell}\|_{m}<2\varepsilon,\end{align*}$$

so $\|w^{(\alpha )}-w\|_{m}<2\varepsilon $ . Since $\varepsilon>0$ was arbitrary, this shows that $\lim _{\alpha \to \infty } w^{(\alpha )}=w$ .

9.1 The weak ergodic theorem

Recall Theorem 3.3: if $m\in {\mathcal {M}}_{\tau }(\Gamma )$ , U is a unitary action of $\Gamma $ on a Hilbert space $\mathcal H$ , $v, w\in \mathcal H$ , and $\phi _{v,w}(\gamma )=\langle v, U_{\gamma } w\rangle $ is the corresponding matrix coefficient, then $ m(\phi _{v,w})=\langle v, P_{U\text {-inv}} w\rangle .$

Proof of Theorem 3.3.

Writing $w=w_{0}+w_{1}$ , where $w_{1}=P_{U\text {-inv}}w$ and $w_{0}\perp \mathcal H_{U\text {-inv}}$ , we have $\phi _{v,w}=\langle v, w_{1}\rangle +\phi _{v,w_{0}}$ . It suffices to prove that $m(\phi _{v,w_{0}})=0$ . Consider the linear functional $\Lambda :\mathcal H\to \mathbb C$ given by

$$\begin{align*}\Lambda(x) := \int \langle x, U_{\gamma }w_{0}\rangle \, dm(\gamma). \end{align*}$$

By the Reisz representation theorem for Hilbert spaces, there is a unique $z\in \mathcal H$ such that $\Lambda (x)=\langle x,z\rangle $ for all $x\in \mathcal H$ . We will show that $z=0$ , by showing that $z\in \mathcal H_{U\text {-inv}}$ and $z\in \mathcal H_{U\text {-inv}}^{\perp }$ . To see that $z\in \mathcal {H}_{U\text {-inv}}$ , fix $x\in \mathcal H$ and let $t \in \Gamma $ . Then

$$ \begin{align*} \langle x,U_{t} z\rangle &=\langle U_{t}^{-1} x, z\rangle && \text{since } U_{t} \text{ is unitary}\\ &=\int \langle U_{t}^{-1} x, U_{\gamma} w_{0}\rangle\, dm(\gamma)\\ &=\int \langle x, U_{t+\gamma} w_{0}\rangle\, dm(\gamma)\\ &= \int \langle x, U_{\gamma} w_{0}\rangle\, dm(\gamma) && \text{since } m \text{ is invariant}\\ &=\langle x, z\rangle. \end{align*} $$

Since $\langle x,z\rangle = \langle x, U_{t} z\rangle $ for all $x\in \mathcal H$ and all $t\in \Gamma $ , we have $U_{t}z=z$ for all $t\in \Gamma $ .

To see that $z\in \mathcal {H}_{U\text {-inv}}^{\perp }$ , let $x\in \mathcal {H}_{U\text {-inv}}$ . Then $\langle x, U_{\gamma } w_{0}\rangle = 0$ for every $\gamma \in \Gamma $ , so $\langle x,z\rangle =\int \langle x, U_{\gamma } w_{0}\rangle \, dm(\gamma )=0$ . This shows that $z=0$ , hence $\Lambda (x)=0$ for all $x\in \mathcal H$ , meaning $m(\phi _{v,w_{0}})=0$ .

9.2 Multiplication action

In the setting where $\Gamma $ is a countable abelian group, $\widehat {\Gamma }$ is compact and metrizable, hence every Borel measure on $\widehat {\Gamma }$ is a Radon measure. In general (when $\Gamma $ may be uncountable), we restrict our attention to Radon measures on $\widehat {\Gamma }$ , as they retain the approximation properties we require. See Chapter 7 of [Reference Folland17] for details.

The next lemma is one of the building blocks of spectral theory for unitary actions of discrete abelian groups.