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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    FRANTZIKINAKIS, NIKOS and HOST, BERNARD 2016. Weighted multiple ergodic averages and correlation sequences. Ergodic Theory and Dynamical Systems, p. 1.

    Frantzikinakis, Nikos 2015. Multiple correlation sequences and nilsequences. Inventiones mathematicae, Vol. 202, Issue. 2, p. 875.

    LEIBMAN, A. 2015. Nilsequences, null-sequences, and multiple correlation sequences. Ergodic Theory and Dynamical Systems, Vol. 35, Issue. 01, p. 176.

    Tu, Siming 2014. Nil Bohr0-sets and polynomial recurrence. Journal of Mathematical Analysis and Applications, Vol. 409, Issue. 2, p. 890.

    Eisner, Tanja 2013. Linear Sequences and Weighted Ergodic Theorems. Abstract and Applied Analysis, Vol. 2013, p. 1.


Multiple polynomial correlation sequences and nilsequences

  • A. LEIBMAN (a1)
  • DOI:
  • Published online: 01 June 2009

A basic nilsequence is a sequence of the form ψ(n)=f(Tnx), where x is a point of a compact nilmanifold X, T is a translation on X, and fC(X); a nilsequence is a uniform limit of basic nilsequences. Let X=G/Γ be a compact nilmanifold, Y be a subnilmanifold of X, g(n) be a polynomial sequence in G, and fC(X); we show that the sequence ∫ g(n)Yf, n∈ℤ, is the sum of a basic nilsequence and a sequence that converges to zero in uniform density. This implies that, given an ergodic invertible measure-preserving system (W,ℬ,μ,T), with μ(W)<, polynomials p1,…,pk∈ℤ[n], and sets A1,…,Ak∈ℬ, the sequence μ(Tp1(n)A1∩⋯∩Tpk(n)Ak) is the sum of a nilsequence and a sequence that converges to zero in uniform density. We also obtain a version of this result for the case where pi are polynomials in several variables.

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[1]V. Bergelson , B. Host and B. Kra . Multiple recurrence and nilsequences. Invent. Math. 160(2) (2005), 261303.

[2]H. Furstenberg . Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981.

[4]B. Host and B. Kra . Non-conventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161(1) (2005), 397488.

[5]B. Host and B. Kra . Convergence of polynomial ergodic averages. Israel J. Math. 149 (2005), 119.

[8]A. Leibman . Convergence of multiple ergodic averages along polynomials of several variables. Israel J. Math. 146 (2005), 303315.

[13]T. Ziegler . Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20(1) (2007), 5397.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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