a1 Department of Mathematics, The Ohio State University, OH 43221, USA (email: email@example.com)
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, 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,,μ,T), with μ(W)<∞, polynomials p1,…,pk[n], and sets A1,…,Ak, the sequence μ(Tp1(n)A1Tpk(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)