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

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.