Bohr neighborhoods in generalized difference sets

TL;DR

This paper generalizes a theorem about the structure of sumsets with positive density, showing they contain Bohr neighborhoods in broader algebraic settings, extending previous results from integers to countable abelian groups.

Contribution

It extends the Bergelson-Ruzsa theorem to subsets of countable abelian groups and sums with more than three terms, broadening the scope of Bohr neighborhood results.

Findings

Sumsets with positive density contain Bohr neighborhoods in countable abelian groups.

Generalization from integers to broader algebraic structures.

Applicable to sums with more than three summands.

Abstract

If $A$ is a set of integers having positive upper Banach density and $r,s,t$ are nonzero integers whose sum is zero, a theorem of Bergelson and Ruzsa says that the set $rA+sA+tA:=\{ra_1+sa_2+ta_3:a_i\in A\}$ contains a Bohr neighborhood of zero. We prove the natural generalization of this result for subsets of countable abelian groups and more summands.