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

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

If A𝒳 with μ(A)>0, then for every ε>0, {n:μ(AT n A)>μ(A) 2 -ε} is syndetic.

A multiple recurrence theorem due to Furstenberg states:

Let (X,𝒳,μ,T) be a dynamical system, let A𝒳 with μ(A)>0 and let k1. Then

lim inf N-M 1 N-M n=M N-1 μ(T n AT 2n AT kn A)>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 A𝒳, μ(A)>0 and ε>0, the set of n such that μ(AT n AT 2n AT kn A)>μ(A) k+1 -ε 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 k4 (and also false in the non-ergodic cases of k=2 and k=3). The case of k4 uses a combinatorial result due to Ruzsa, which appears as an appendix.

MSC:
28D05Measure-preserving transformations
37A30Ergodic theorems, spectral theory, Markov operators
37A05Measure-preserving transformations
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