Note on polynomial recurrence

TL;DR

This paper proves that in measure-preserving systems with commuting transformations, the set of times where multiple polynomial iterates of a set intersect with nearly maximal measure has bounded gaps, under certain degree conditions.

Contribution

It establishes bounded gaps for polynomial multiple recurrence sets in measure-preserving systems with commuting transformations, assuming polynomials have distinct degrees.

Findings

The recurrence set has bounded gaps.

The result applies to systems with commuting transformations.

It generalizes polynomial recurrence phenomena.

Abstract

Let $(X,\mu,T_1,...,T_l)$ be a measure-preserving system with those $T_i$ are commuting. Suppose that the polynomials $p_1(t),...,p_{l}(t)\in\Z[t]$ with $p_j(0)=0$ have distinct degrees. Then for any $\epsilon>0$ and $A\subseteq X$ with $\mu(A)>0$, the set $$ \{n:\,\mu(A\cap T_1^{-p_1(n)}A\cap...\cap T_l^{-p_l(n)}A)\geq\mu(A)^{l+1}-\epsilon\} $$ has bounded gaps.