Integer part polynomial correlation sequences

TL;DR

This paper proves that multiple correlation sequences involving integer part polynomial iterates can be decomposed into a structured nilsequence plus a small error, with convergence results for related ergodic averages.

Contribution

It extends the structure theorem for multiple correlation sequences to the setting of integer part polynomial iterates using a transference principle.

Findings

Multiple ergodic averages with integer part polynomial iterates converge in mean.

Correlation sequences decompose into nilsequences plus negligible error.

Under certain conditions, the limit of these averages is zero.

Abstract

Following an approach presented by N. Frantzikinakis, we prove that any multiple correlation sequence, defined by invertible measure preserving actions of commuting transformations with integer part polynomial iterates, is the sum of a nilsequence and an error term, small in uniform density. As an intermediate result, we show that multiple ergodic averages with iterates given by the integer part of real valued polynomials converge in the mean. Also, we show that under certain assumptions the limit is zero. An important role in our arguments plays a transference principle, communicated to us by M. Wierdl, that enables to deduce results for $\mathbb{Z}$-actions from results for flows.