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An odd Furstenberg-Szemeredi theorem and quasi-affine systems

Abstract : We prove a version of Furstenberg's ergodic theorem with restrictions on return times. More specifically, for a measure preserving system (X, B, mu, T), integers 0 less than or equal to j < k, and E subset of X with mu(E) > 0, we show that there exists n equivalent to j (mod k) with mu(E boolean AND T(-n) E boolean AND T(-2n) E boolean AND T(-3n)E) > 0, so long as T(k) is ergodic. This result requires a deeper understanding of the limit of some nonconventional ergodic averages and the introduction of a new class of systems, the 'Quasi-Affine Systems'.
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https://hal-upec-upem.archives-ouvertes.fr/hal-00693660
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Submitted on : Wednesday, May 2, 2012 - 10:01:22 PM
Last modification on : Thursday, March 19, 2020 - 12:26:02 PM

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Bernard Host, B Kra. An odd Furstenberg-Szemeredi theorem and quasi-affine systems. Journal d'analyse mathématique, Springer, 2002, 86 (?), pp.183--220. ⟨10.1007/BF02786648⟩. ⟨hal-00693660⟩

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