Multiple correlation sequences and nilsequences

Abstract

We study the structure of multiple correlation sequences defined by measure preserving actions of commuting transformations. When the iterates of the transformations are integer polynomials we prove that any such correlation sequence is the sum of a nilsequence and an error term that is small in uniform density; this was previously known only for measure preserving actions of a single transformation. We then use this decomposition result to give convergence criteria for multiple ergodic averages and deduce some rather surprising results, for instance we infer convergence for actions of commuting transformations from the special case of actions of a single transformation. Our proof of the decomposition result differs from previous works of Bergelson, Host, Kra, and Leibman, as it does not rely on the theory of characteristic factors. It consists of a simple orthogonality argument and the main tool is an inverse theorem of Host and Kra for general bounded sequences.