Separating measurable recurrence from strong recurrence via rigidity sequences

TL;DR

This paper explores the relationship between measure expanding sets, recurrence, and rigidity sequences in countable abelian groups, proving that measure expanding sets contain sequences that are rigidity sequences for certain systems, and distinguishing measurable recurrence from strong recurrence.

Contribution

It establishes that measure expanding sets in countable abelian groups contain sequences that serve as rigidity sequences, and it proves a conjecture of Ackelsberg regarding recurrence properties.

Findings

Existence of measure expanding sequences that are rigidity sequences.

Proof that measure expanding sets can contain sequences not exhibiting strong recurrence.

Confirmation of Ackelsberg's conjecture on recurrence properties.

Abstract

If $G$ is an abelian group, we say $S\subset G$ is a set of recurrence if for every probability measure preserving $G$-system $(X,\mu,T)$ and every $D\subset X$ having $\mu(D)>0$, there is a $g\in S$ such that $\mu(D\cap T^{g}D)>0$. We say $S$ is a set of strong recurrence if for every set $D$ having $\mu(D)>0$ there is a $c>0$ such that $\mu(D\cap T^{g}D)>c$ for infinitely many $g\in S$. We call $S$ measure expanding if for all $g\in G$, the translate $S+g$ is a set of recurrence. A rigidity sequence for $(X,\mu,T)$ is a sequence of elements $s_n\in G$ satisfying $\lim_{n\to\infty} \mu(D\triangle T^{s_n}D)=0$ for all measurable $D\subset X$. For all but countably many countable abelian groups $G$, we prove that if $S$ is measure expanding, there is a sequence of elements $s_n\in S$ such that $\{s_n:n\in \mathbb N\}$ is also measure expanding and every translate of $(s_n)$ is a rigidity sequence for some free weak mixing measure preserving $G$-system. The special case where $S=G$ proves a conjecture of Ackelsberg. As a consequence, we prove that for every countably infinite abelian group $G$ and every measure expanding set $S\subset G$ there is a subset $S'\subset S$ such that $S'$ is measure expanding and no translate of $S'$ is a set of strong recurrence.