In this short post I show that in any countable amenable group the (left) upper Banach density of a set can be obtained by looking only at translations of a given Følner sequence. Definitions and the precise statement are given below.

This result is well know among experts but this fact doesn’t seem to be explicitly stated in the literature. The proof usually uses some functional analytic machinery, but the proof in this post is purely combinatorial in nature, which may give some additional information (as functional analytic tools are usually based on the axiom of choice, which prevents by principle the deduction of quantitative bounds).

Definition 1Let be a countable group. A sequence of finite sets is a (left)Følner sequenceif for all we have

A countable group with a Følner sequence is called *amenable* and we will only deal with such groups. The canonical example is with the Følner sequence formed by the sets . Every solvable group (and in particular every abelian group) is amenable.

The following proposition follows directly from the definition of Følner sequence:

Proposition 2Let be a countable amenable group, let be a Følner sequence and let be any sequence taking values on . The sequence is a Følner sequence.

The Følner will be called *a shift of* . In the example when we obtain the Følner sequences by shifts of .

Definition 3Let be an amenable group and a Følner sequence on . Let .

- The upper density of with respect to is:
- The upper Banach density of is:
- The upper Banach density of with respect to is:

The last definition is not standard, and the following theorem, whose proof is the main purpose of this post, explains why:

Theorem 4The upper Banach density with respect to a Følner sequence is the same as the upper Banach density. More precisely, let be an amenable group, let and let be any Følner sequence on . Then .

It follows directly from the definitions that, for any Følner sequence in we have . Thus it suffices to prove that, given any other Følner sequence in we have . The idea of the proof is to tile the each set (or, more precisely, an approximation for ) when is large, with shifts from .

From now on we fix the Følner sequences and in and a set . Also we define

The main point of this lemma is that does not depend on .

*Proof:* The proof goes by contradiction. Assume for each there is some such that . Then the upper density of with respect to shift of the Følner sequence would be larger than , which is a contradiction.

The following lemma gives us an asymptotically perfect tilling of by shifts of a finite set .

Lemma 6Let be a Følner sequence and let be a finite set. For each , defineDefine, for each , the number

(note that also depends on , we do not make this explicit to avoid notation even more cumbersome). Finally let .

Then as .

The set is constructed so that it can be tiled by shifts of . This lemma shows that occupies essentially all of .

*Proof:* Note that for we have

Let and note that is a finite set. Rephrasing we have that . Thus we have

Since is a Følner sequence we get that

for all . Thus, taking the union over the finite set we obtain that

From this we conclude that as desired.

As a Corollary we deduce that the density of with respect to can be calculated by looking at the intersections of with .

We now prove Theorem 4. Let and let be given by Lemma 5. Let be very large and let , and be all as in Lemma 6. We now have:

On the other hand:

Putting both together we get

By Corollary 7 we conclude that . Since was arbitrarily chosen we conclude the proof of Theorem 4.