High density piecewise syndeticity of product sets in amenable groups

Abstract

M. Beiglböck, V. Bergelson, and A. Fish proved that if $G$ is a countable amenable group and $A$ and $B$ are subsets of $G$ with positive Banach density, then the product set $AB$ is piecewise syndetic. This means that there is a finite subset $E$ of $G$ such that $EAB$ is thick, that is, $EAB$ contains translates of any finite subset of $G$. When $G=\mathbb{Z}$, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a Følner sequence) of the set of witnesses to the thickness of . When $G={\mathbb{Z}}^d$, this result was first proven by the current set of authors using completely different techniques.

7 pages; the proof of the main result has been simplified and sharpened by removing a technical assumption; also, a lower density version has been added