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

Abstract

Abstract We say that $S\subseteq \mathbb Z$ 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 [Sets of k-recurrence but not (k+1)-recurrence. Ann. Inst. Fourier (Grenoble)56(4) (2006), 839–849], we construct a set of $2$ -recurrence S with the property that $\{n^2:n\in S\}$ is not a set of measurable recurrence.