Multiple correlation sequences and nilsequences

TL;DR

This paper proves that multiple correlation sequences from commuting transformations with polynomial iterates can be decomposed into a structured nilsequence plus a small error, leading to new convergence criteria for multiple ergodic averages.

Contribution

It extends the decomposition of correlation sequences to multiple commuting transformations with polynomial iterates without relying on characteristic factors theory.

Findings

Correlation sequences decompose into nilsequences plus small error.

Convergence of multiple ergodic averages follows from single transformation cases.

New proof technique using orthogonality and inverse theorems.

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 involving iterates that grow linearly, and prove the rather surprising fact that for such sequences, convergence results for actions of commuting transformations follow automatically from the special case of actions of a single transformation. Our proof of the decomposition result differs from previous works of V. Bergelson, B. Host, B. Kra, and A. 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 B. Host and B. Kra for general bounded sequences.