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

Let $\left(X,𝒳,\mu ,T\right)$ be a dynamical system (i.e., a measure preserving probability system, with $T$ invertible). The Khintchine recurrence theorem states:

If $A\in 𝒳$ with $\mu \left(A\right)>0$, then for every $\epsilon >0$, $\left\{n\in ℤ:\mu \left(A\cap {T}^{n}A\right)>\mu {\left(A\right)}^{2}-\epsilon \right\}$ is syndetic.

A multiple recurrence theorem due to Furstenberg states:

Let $\left(X,𝒳,\mu ,T\right)$ be a dynamical system, let $A\in 𝒳$ with $\mu \left(A\right)>0$ and let $k\ge 1$. Then

$\underset{N-M\to \infty }{lim inf}\phantom{\rule{0.166667em}{0ex}}\frac{1}{N-M}\sum _{n=M}^{N-1}\mu \left(\cap {T}^{n}A\cap {T}^{2n}A\cap \cdots \cap {T}^{kn}A\right)>0·$

(This $lim inf$ 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 $A\in 𝒳$, $\mu \left(A\right)>0$ and $\epsilon >0$, the set of $n\in ℤ$ such that $\mu \left(A\cap {T}^{n}A\cap {T}^{2n}A\cap \cdots \cap {T}^{kn}A\right)>\mu {\left(A\right)}^{k+1}-\epsilon$ 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 $k\ge 4$ (and also false in the non-ergodic cases of $k=2$ and $k=3$). The case of $k\ge 4$ 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 Measure-preserving transformations
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