A set of 2-recurrence whose perfect squares do not form a set of measurable recurrence

TL;DR

This paper constructs a specific 2-recurrence set whose perfect squares do not form a set of measurable recurrence, answering a question in ergodic theory about recurrence properties of squares.

Contribution

It introduces a novel 2-recurrence set with the property that its squares are not a measurable recurrence set, addressing a question posed by Frantzikinakis, Lesigne, and Wierdl.

Findings

Constructed a 2-recurrence set with non-measurable recurrence squares

Provided a counterexample to a conjecture about squares and recurrence

Advances understanding of recurrence properties in ergodic theory

Abstract

We say that $S\subset\mathbb Z$ is a set of $k$-recurrence if for every measure preserving transformation $T$ of a probability measure space $(X,\mu)$ and every $A\subseteq X$ with $\mu(A)>0$, there is an $n\in S$ such that $\mu(A\cap T^{-n} A\cap T^{-2n}\cap \dots \cap T^{-kn}A)>0$. A set of $1$-recurrence is called a set of measurable recurrence. Answering a question of Frantzikinakis, Lesigne, and Wierdl, we construct a set of $2$-recurrence $S$ with the property that $\{n^2:n\in S\}$ is not a set of measurable recurrence.