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 1 Let be a countable group. A sequence of finite sets is a (left) Følner sequence if 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 2 Let 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 3 Let 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:
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 .
Define, 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 .
On the other hand:
Putting both together we get