A set of 2-recurrence whose perfect squares do not form a set of measurable recurrence
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916237925154816 |
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| author | Griesmer, John T. |
| author_facet | Griesmer, John T. |
| contents | 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 $(X,μ)$ and every $A\subseteq X$ with $μ(A)>0$, there is an $n\in S$ such that $μ(A\cap T^{-n} A\cap T^{-2n}\cap \dots \cap T^{-kn}A)>0$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_11851 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A set of 2-recurrence whose perfect squares do not form a set of measurable recurrence Griesmer, John T. Dynamical Systems Combinatorics 37A44, 11B30 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 $(X,μ)$ and every $A\subseteq X$ with $μ(A)>0$, there is an $n\in S$ such that $μ(A\cap T^{-n} A\cap T^{-2n}\cap \dots \cap T^{-kn}A)>0$. 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. |
| title | A set of 2-recurrence whose perfect squares do not form a set of measurable recurrence |
| topic | Dynamical Systems Combinatorics 37A44, 11B30 |
| url | https://arxiv.org/abs/2207.11851 |