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arXiv · 2207.11851

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

John T. Griesmer
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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.

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John T. Griesmer. 2022-07-24. A set of 2-recurrence whose perfect squares do not form a set of measurable recurrence. https://doi.org/10.1017/etds.2023.51

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