Single recurrence in abelian groups

Abstract

We collect problems on recurrence for measure preserving and topological actions of a countable abelian group, considering combinatorial versions of these problems as well. We solve one of these problems by constructing, in $G_2:={⨁}_{n=1}^{\infty }\mathbb{Z}/2\mathbb{Z}$, a set $S$ such that every translate of $S$ is a set of topological recurrence, while $S$ is not a set of measurable recurrence. This construction answers negatively a variant of the following question asked by several authors: if $A\subset \mathbb{Z}$ has positive upper Banach density, must $A-A$ contain a Bohr neighborhood of some $n\in \mathbb{Z}$? We also solve a variant of a problem posed by the author by constructing, for all $\epsilon >0$, sets $S,A\subseteq G_2$ such that every translate of $S$ is a set of topological recurrence, $d^{\ast }(A)>1-\epsilon$, and the sumset $S+A$ is not piecewise syndetic. Here $d^{\ast }$ denotes upper Banach density.

Revision 2: Typos corrected, minor changes to exposition, including definition of cylinder sets and restrictions in Section 3. Intentional text overlap with arxiv:1608.01014 in order to keep both articles self contained. 39 pages, comments welcome!