Almost all dynamically syndetic sets are multiplicatively thick

If a set of integers is syndetic (finitely many translates cover the integers), must it contain two integers whose ratio is a square?  No one knows.  In the broader context of the disjointness between additive and multiplicative configurations and actions in ergodic Ramsey theory, it makes sense to ask similar questions about dynamically syndetic sets, those sets that contain the visit times of a point to an open set in a minimal topological dynamical system.  The main result of the talk is that almost every dynamically syndetic set is multiplicatively very rich: it is “thick” in some coset of a multiplicative subsemigroup.  We will discuss some applications: a “thick-starters” van der Waerden theorem; the existence of multiplicative structure in sets of the form A – A + t; and the topological disjointness of minimal niltranslations and minimal, aperiodic multiplicative actions.  Time permitting, we will discuss three tools that proved useful in the topic: the prolongation relation (the closure of the orbit-closure relation) developed by Auslander, Akin, and Glasner; the theory of rational points and polynomials on nilmanifolds developed by Leibman, Green, Tao; and the machinery of topological characteristic factors developed recently by Glasner, Huang, Shao, Weiss, and Ye.  This talk is based on work in https://arxiv.org/abs/2207.00098.