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

TL;DR

This paper investigates quantitative expansiveness in ergodic ^d-actions along structured subsets, strengthening previous combinatorial results and providing new insights including a counterexample to a polynomial Bogolyubov variant.

Contribution

It introduces new quantitative expansiveness results for ergodic ^d-actions along cyclic and polynomial subsets, unifying and extending prior approaches.

Findings

Strengthens previous combinatorial results on expansiveness.

Provides a counterexample to a pinned polynomial Bogolyubov theorem.

Unifies methods for analyzing structured subsets in ergodic 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 Bj\"orklund 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.