Recurrence, rigidity, and popular differences

We construct a set $S$ such that every translate of $S$ is a set of recurrence and a set of rigidity for a weak mixing measure preserving system. This construction generalizes or strengthens results of Katznelson, Saeki, Forrest, and Fayad and Kanigowski. The construction provides a density analogue of Julia Wolf’s results on popular differences in finite abelian groups.