Nonstandard analysis and the sumset phenomenon in arbitrary amenable groups

Abstract

Beiglböck, Bergelson and Fish proved that if subsets $A$, $B$ of a countable discrete amenable group $G$ have positive Banach densities $\alpha$ and $\beta$ respectively, then the product set $AB$ is piecewise syndetic, that is, there exists $k$ such that the union of $k$-many left translates of $AB$ is thick. Using nonstandard analysis, we give a shorter alternative proof of this result that does not require $G$ to be countable and moreover yields the explicit bound $k\le1/\alpha\beta$. We also prove with similar methods that if $\{A_{i}\}_{i=1}^{n}$ are finitely many subsets of $G$ having positive Banach densities $\alpha_{i}$ and $G$ is countable, then there exists a subset $B$ whose Banach density is at least $\prod_{i=1}^{n}\alpha_{i}$ and such that $BB^{-1}\subseteq\bigcap_{i=1}^{n}A_{i}A_{i}^{-1}$. In particular, the latter set is piecewise Bohr.