Quantitative expansivity for ergodic Z d $\mathbb {Z}^d$ -actions

Abstract

We study expansiveness properties of positive measure subsets of ergodic $Z^d$-actions along two different types of structured subsets of $Z^d$, namely, cyclic subgroups and images of integer polynomials. We prove quantitative expansiveness properties in both cases, strengthening combinatorial results from two distinct works—one by Björklund and Fish, the other by Bulinski and Fish. Our methods unify and strengthen earlier approaches used in Björklund and Fish and Bulinski and Fish and to our surprise, also yield a counterexample to a certain pinned variant of the polynomial Bogolyubov theorem.