  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.
