A set of 2-recurrence whose perfect squares do not form a set of measurable recurrence. (English) Zbl 07862706
Summary: 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 N. Frantzikinakis et al. [Ann. Inst. Fourier 56, No. 6, 839–849 (2006; Zbl 1123.37001)], we construct a set of \(2\)-recurrence \(S\) with the property that \(\{n^2:n\in S\}\) is not a set of measurable recurrence.
MSC:
| 37A44 | Relations between ergodic theory and number theory |
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 37A05 | Dynamical aspects of measure-preserving transformations |
| 11B30 | Arithmetic combinatorics; higher degree uniformity |
Citations:
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