Recurrence, rigidity, and popular differences

We construct a set of integers $S$ such that every translate of $S$ is a set of recurrence and a set of rigidity for a weak mixing measure preserving system. Here ‘set of rigidity’ means that enumerating $S$ as $(s_{n})_{n\in \mathbb{N}}$ produces a rigidity sequence. This construction generalizes or strengthens results of Katznelson, Saeki (on equidistribution and the Bohr topology), Forrest (on sets of recurrence and strong recurrence), and Fayad and Kanigowski (on rigidity sequences). The construction also provides a density analogue of Julia Wolf’s results on popular differences in finite abelian groups.