Recurrence, rigidity, and popular differences

TL;DR

This paper constructs a special set with properties that make every translate a recurrence and rigidity set for weak mixing systems, extending previous results and connecting to finite abelian groups.

Contribution

It introduces a novel set construction that generalizes prior work on recurrence and rigidity, and relates to popular differences in finite groups.

Findings

Every translate of the constructed set is a recurrence and rigidity set.

The construction generalizes and strengthens previous results by Katznelson, Saeki, Forrest, Fayad, and Kanigowski.

Provides a density analogue of Julia Wolf's results on popular differences.

Abstract

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.