Single recurrence in abelian groups

TL;DR

This paper explores recurrence properties in countable abelian groups, constructing specific sets to distinguish between topological and measurable recurrence, and addressing open questions about sumsets and Bohr neighborhoods.

Contribution

It constructs examples that separate topological recurrence from measurable recurrence and addresses open problems in recurrence theory within abelian groups.

Findings

Constructed a set in $G_2$ where all translates are topologically recurrent but not measurably recurrent.

Provided a negative answer to a variant of a question about positive density sets and Bohr neighborhoods.

Created sets with high density where the sumset is not piecewise syndetic.

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 $G_{2}:=\bigoplus_{n=1}^{\infty} \mathbb Z/2\mathbb Z$, 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 $\varepsilon>0$, sets $S, A\subseteq G_{2}$ such that every translate of $S$ is a set of topological recurrence, $d^{*}(A)>1-\varepsilon$, and the sumset $S+A$ is not piecewise syndetic. Here $d^{*}$ denotes upper Banach density.