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

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 $\mu (A\cap T^{-n}A\cap T^{-2n}\cap \cdots \cap T^{-kn}A)>0$. 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.

47 Pages; v.2 incorporates referee suggestions

References in corpus (3)

• Powers of sequences and recurrence
• Separating Bohr denseness from measurable recurrence
• Separating topological recurrence from measurable recurrence: exposition and extension of Kriz's example