Single recurrence in abelian groups

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.