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

Abstract

We study expansiveness properties of positive measure subsets of ergodic ${\mathbb{Z}}^d$-actions along two different types of structured subsets of ${\mathbb{Z}}^d$, namely, cyclic subgroups and images of integer polynomials. We prove quantitative expansiveness properties in both cases and strengthen combinatorial results obtained by Björklund and Fish in arXiv:2401.03724, and Bulinski and Fish in arXiv:2102.05862. Our methods unify and strengthen earlier approaches used in arXiv:2401.03724 and arXiv:2102.05862 and to our surprise, also yield a counterexample to a certain pinned variant of the polynomial Bogolyubov theorem.