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Multiple polynomial correlation sequences and nilsequences

  • A. LEIBMAN (a1)

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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