On uniform asymptotic upper density in locally compact abelian groups

TL;DR

This paper introduces a new concept of asymptotic uniform upper density for locally compact abelian groups, extending classical results from Z^d and R^d to a broader setting, and proves that sets with positive density have syndetic difference sets.

Contribution

The paper defines a novel notion of asymptotic uniform upper density for locally compact abelian groups and extends classical density results to this general setting.

Findings

The new density notion aligns with classical cases like Z^d and R^d.

Sets with positive density have syndetic difference sets in general locally compact abelian groups.

Extension of Furstenberg's result to broader group contexts.

Abstract

Starting out from results known for the most classical cases of N, Z^d, R^d or for sigma-finite abelian groups, here we define the notion of asymptotic uniform upper density in general locally compact abelian groups. Even if a bit surprising, the new notion proves to be the right extension of the classical cases of Z^d, R^d. The new notion is used to extend some analogous results previously obtained only for classical cases or sigma-finite abelian groups. In particular, we show the following extension of a well-known result for Z of Furstenberg: if in a general locally compact Abelian group G a subset S of G has positive uniform asymptotic upper density, then S-S is syndetic.