Ergodic Theory and Dynamical Systems

Research Article

Multiple polynomial correlation sequences and nilsequences

A. LEIBMANa1

a1 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 fxs2208C(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 fxs2208C(X); we show that the sequence ∫ g(n)Yf, nxs2208xs2124, 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,xs212C,μ,T), with μ(W)<, polynomials p1,…,pkxs2208xs2124[n], and sets A1,…,Akxs2208xs212C, the sequence μ(Tp1(n)A1xs2229xs22EFxs2229Tpk(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.

(Received May 13 2008)

(Revised February 23 2009)