Upper density of subsets of an amenable group

Question

Let $G$ be an amenable group (so locally compact Hausdorff) and also assume it is second countable if needed. My question is that what are the standard ways (across literature) of defining the upper density for a subset of $G$?

If we are talking about $\mathbb{N}$, then given a Folner sequence $\mathcal{F}=\{F_n: n\in \omega\}$ (basically each $F_n$ is finite and for any $m\in \mathbb{N}$, $\lim_{n\to \infty} \frac{|(m+F_n)\Delta F_n|}{|F_n|}=0$), then the upper density associated with $\mathcal{F}$ can be $\bar{d}_\mathcal{F}(A)=\limsup_{n\to \infty} \frac{|A\cap F_n|}{|F_n|}$.

One way to define the density is to replace the counting measure by the Haar measure $\mu$ on $G$ and the Folner sequence in a more general sense ($F_n$ are compact sets now) and we can say $\bar{d}_\mathcal{F}(A)=\limsup_{n\to \infty} \frac{\mu^*(A\cap F_n)}{\mu(F_n)}$, where $\mu^*$ is the outer measure (I want to define density for any subset $A$).

Another way is to say $\bar{d}(A)=\sup\{\alpha: \text{for every finite }H\subset G \text{ and $\varepsilon>0$, there is a finite $K$ with }\frac{|hK\Delta K|}{|K|}<\varepsilon\; \forall h\in H \text{ and }\frac{|A\cap K|}{|K|}\geq \alpha\}$.

I'm not a group theorist nor topologist and hopefully this question is okay here. Thanks.

Answer 1

In the case that $G$ is discrete, what you've defined last as $\bar{d}(A)$ is usually considered the upper (Banach) density of $A$. Moreover, in this case $\bar{d}(A)$ is the supremum of $\bar{d}_{\mathcal{F}}(A)$ over all Folner sequences $\mathcal{F}$ (or nets in the uncountable case).

Recall also that a discrete group $G$ is amenable if and only if it admits at least one Folner sequence (or net) if and only if it admits a left-invariant finitely additive probability measure on its subsets. So another equivalent formulation of $\bar{d}(A)$ in this case is: $$ \bar{d}(A)=\sup\{\mu(A):\text{$\mu$ is a left-invariant finitely additive probability measure on $G$}\}. $$

In the non-discrete case, all of the same facts are true if we use compact sets instead of finite sets. In particular, a locally compact group $G$ (with Haar measure $\eta$) is amenable if and only if for any compact $H\subseteq G$ and $\epsilon>0$, there is a Borel set $K\subseteq G$, with $0<\eta(K)<\infty$, such that $\eta(hK{\vartriangle} K)/\eta(K)<\epsilon$ for all $h\in H$.

In this case, if one defines $\bar{d}(A)$ analogously (for Borel $A$), then \begin{align*} \bar{d}(A) &= \sup\{\bar{d}_{\mathcal{F}}(A):\text{$\mathcal{F}$ is a Folner net for $G$}\}.\\ &= \sup\{\mu(A):\text{$\mu$ is a left-invariant finitely additive Borel probability measure on $G$}\}. \end{align*}

A final comment is that in groups like $\mathbb{Z}$, there are other "canonical" upper densities, such as the upper asymptotic density: $$ \bar{\delta}(A)=\limsup_n|A\cap [\text{-}n,n]|/(2n+1). $$ But notice that this is just $\bar{d}_{\mathcal{F}}(A)$ for a particular choice of Folner sequence in $\mathbb{Z}$. So in more general groups $G$, one can work with upper densities defined by "special" Folner sequences too.

Edit: The discussion above I think motivates some further remarks about combinatorics. For simplicity, let $G$ be a countable discrete amenable group. Then the notion of "density $0$ with respect to every Folner sequence", i.e. $\bar{d}(A)=0$, is a useful notion of "sparse". For example Szemeredi's Theorem can be rephrased as saying that (in $\mathbb{Z}$) if $\bar{d}(A)>0$ then $A$ contains arbitrarily large finite arithmetic progressions).

For some easier facts, define the lower Banach density with respect to a Folner sequence $\mathcal{F}$ to be $\underline{d}_{\mathcal{F}}(A)=\liminf|A\cap F_n|/|F_n|$. The lower Banach density is then $$ \underline{d}(A)=\inf\{\underline{d}_{\mathcal{F}}(A):\text{$\mathcal{F}$ a Folner sequence}\} $$

Then some basic combinatorial facts are:

1. $\bar{d}(A)=1-\underline{d}(G\backslash A)$ for any $A$ (this works at the level of Folner sequences.

2. $\underline{d}(A)>0$ if and only if $G=FA$ for some finite $F\subseteq G$ (i.e., $A$ is syndetic).

3. $\bar{d}(A)=1$ if and only if for any finite $F\subseteq G$, there is some $g\in G$ such that $Fg\subseteq A$ (i.e., $A$ is thick).

A final combinatorial notion is that of a piecewise syndetic set, which is a set $A$ such that $FA$ is thick for some finite $F\subseteq G$. Then another fact that is used a lot is that if $A$ is piecewise syndetic then $\bar{d}(A)>0$ (but the converse fails in general).

We also have the following facts.

1. $\underline{d}(A)=\inf\{\mu(A):\text{$\mu$ is a left-invariant finitely additive probability measure on $G$}\}$.

2. For any set $A$ there are Folner sequences $\mathcal{E}$ and $\mathcal{F}$ such that $\bar{d}(A)=\bar{d}_{\mathcal{F}}(A)$ and $\underline{d}(A)=\underline{d}_{\mathcal{F}}(A)$.

In additional to Szemeredi's Theorem, other well-known results are:

1. (Jin) If $A,B\subseteq\mathbb{Z}$ are such that $\bar{d}(A),\bar{d}(B)>0$, then $A-B$ is piecewise syndetic. This is a more difficult variation of the easier fact that if $\bar{d}(A)>0$ then $A-A$ is syndetic.

2. (Moreira, Richter, Robertson) If $A\subseteq\mathbb{Z}$ is such that $\bar{d}(A)>0$ then $A$ contains $B+C$ for some infinite sets $B$ and $C$. This was originally conjectured by Erdos as a replacement for the failure of a density version of Hindman's Theorem.

Answer 2

There is a dual description of $\overline d$ which goes back to Banach and Følner (in the general countable case), see Theorem 6 of Granirer. For locally compact groups it should be more or less the same if you deal with the means defined on $L^\infty$ with respect to the Haar measure.