## 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.

### MathOverflow Questions:

Extension of Khintchine’s recurrence in a simple case

### MSC:

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

Zbl 1077.37002
Full Text:

### References:

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