Sumset Phenomenon in Countable Amenable Groups

TL;DR

This paper extends Jin's sumset phenomenon from integers and certain abelian groups to all countable amenable groups, showing such sumsets are piecewise Bohr and characterizing them in abelian groups.

Contribution

It generalizes Jin's theorem to all countable amenable groups and proves that sumsets are piecewise Bohr, providing a comprehensive understanding of sumset structure in these groups.

Findings

Sumsets in countable amenable groups are piecewise Bohr.

Sumsets contain sums of sets with positive upper Banach density in abelian groups.

Extension of Jin's theorem to a broader class of groups.

Abstract

Jin proved that whenever $A$ and $B$ are sets of positive upper density in $\Z$, $A+B$ is piecewise syndetic. Jin's theorem was subsequently generalized by Jin and Keisler to a certain family of abelian groups, which in particular contains $\Z^d$. Answering a question of Jin and Keisler, we show that this result can be extended to countable amenable groups. Moreover we establish that such sumsets (or -- depending on the notation -- "productsets") are piecewise Bohr, a result which for $G=\Z$ was proved by Bergelson, Furstenberg and Weiss. In the case of an abelian group $G$, we show that a set is piecewise Bohr if and only if it contains a sumset of two sets of positive upper Banach density.