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Multiple recurrence and nilsequences (with an appendix by Imre Ruzsa). (English) Zbl 1087.28007

Let \((X,{\mathcal X},\mu, T)\) be a dynamical system (i.e., a measure preserving probability system, with \(T\) invertible). The Khintchine recurrence theorem states:
If \(A\in{\mathcal X}\) with \(\mu(A)> 0\), then for every \(\varepsilon> 0\), \(\{n\in\mathbb{Z}: \mu(A\cap T^nA)>\mu(A)^2- \varepsilon\}\) is syndetic.
A multiple recurrence theorem due to Furstenberg states:
Let \((X,{\mathcal X},\mu, T)\) be a dynamical system, let \(A\in{\mathcal X}\) with \(\mu(A)> 0\) and let \(k\geq 1\). Then \[ \liminf_{N-M\to\infty}\,{1\over N- M} \sum^{N- 1}_{n= M} \mu(\cap T^n A\cap T^{2n} A\cap\cdots\cap T^{kn} A)> 0. \] (This \(\liminf\) was subsequently shown by the latter two authors [Ann. Math. (2) 161, No. 1, 397–488 (2005; Zbl 1077.37002)] to be a limit.)
Both of these theorems can be regarded a generalizations of the Poincaré recurrence theorem, and the original aim of the authors was to give a simultaneous generalization of both theorems, to show that for such a dynamical system and for \(A\in{\mathcal X}\), \(\mu(A)> 0\) and \(\varepsilon> 0\), the set of \(n\in\mathbb Z\) such that \(\mu(A\cap T^nA\cap T^{2n}A\cap\dots\cap T^{kn}A)> \mu(A)^{k+1}- \varepsilon\) is syndetic. However, surprisingly, the authors are able to show that if \(T\) is ergodic, then this is true for \(k= 2\) and \(k= 3\), but false for \(k\geq 4\) (and also false in the non-ergodic cases of \(k= 2\) and \(k= 3\)). The case of \(k\geq 4\) uses a combinatorial result due to Ruzsa, which appears as an appendix.

MSC:

28D05 Measure-preserving transformations
37A30 Ergodic theorems, spectral theory, Markov operators
37A05 Dynamical aspects of measure-preserving transformations

Citations:

Zbl 1077.37002
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References:

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