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Multiple polynomial correlation sequences and nilsequences

Published online by Cambridge University Press:  23 June 2009

A. LEIBMAN*
Affiliation:
Department of Mathematics, The Ohio State University, OH 43221, USA (email: leibman@math.ohio-state.edu)

Abstract

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.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Bergelson, V., Host, B. and Kra, B.. Multiple recurrence and nilsequences. Invent. Math. 160(2) (2005), 261303.CrossRefGoogle Scholar
[2]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981.CrossRefGoogle Scholar
[3]Green, B. and Tao, T.. The quantitative behaviour of polynomial orbits on nilmanifolds. Preprint, 2007, arXiv:0709.3562v2.Google Scholar
[4]Host, B. and Kra, B.. Non-conventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161(1) (2005), 397488.CrossRefGoogle Scholar
[5]Host, B. and Kra, B.. Convergence of polynomial ergodic averages. Israel J. Math. 149 (2005), 119.CrossRefGoogle Scholar
[6]Leibman, A.. Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergod. Th. & Dynam. Sys. 25 (2005), 201213.CrossRefGoogle Scholar
[7]Leibman, A.. Pointwise convergence of ergodic averages for polynomial actions of ℤd by translations on a nilmanifold. Ergod. Th. & Dynam. Sys. 25 (2005), 215225.CrossRefGoogle Scholar
[8]Leibman, A.. Convergence of multiple ergodic averages along polynomials of several variables. Israel J. Math. 146 (2005), 303315.CrossRefGoogle Scholar
[9]Leibman, A.. Rational sub-nilmanifolds of a compact nilmanifold. Ergod. Th. & Dynam. Sys. 26 (2006), 787798.CrossRefGoogle Scholar
[10]Leibman, A.. Orbits on a nilmanifold under the action of a polynomial sequences of translations. Ergod. Th. & Dynam. Sys. 27 (2007), 12391252.CrossRefGoogle Scholar
[11]Leibman, A.. Orbit of the diagonal in the power of a nilmanifold. Trans. Amer. Math. Soc. to appear.Google Scholar
[12]Malcev, A.. On a class of homogeneous spaces. Amer. Math. Soc. Transl. 9 (1962), 276307.Google Scholar
[13]Ziegler, T.. Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20(1) (2007), 5397.CrossRefGoogle Scholar