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Weighted multiple ergodic averages and correlation sequences

Published online by Cambridge University Press:  04 July 2016

NIKOS FRANTZIKINAKIS  and
BERNARD HOST
NIKOS FRANTZIKINAKIS
Affiliation:
University of Crete, Department of Mathematics, Voutes University Campus, Heraklion 71003, Greece email frantzikinakis@gmail.com
BERNARD HOST
Affiliation:
Université Paris-Est Marne-la-Vallée, Laboratoire d’analyse et de mathématiques appliquées, UMR CNRS 8050, 5 Bd Descartes, 77454 Marne la Vallée Cedex, France email bernard.host@u-pem.fr
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Abstract

We study mean convergence results for weighted multiple ergodic averages defined by commuting transformations with iterates given by integer polynomials in several variables. Roughly speaking, we prove that a bounded sequence is a good universal weight for mean convergence of such averages if and only if the average of this sequence times any nilsequence converges. Two decomposition results of independent interest play key roles in the proof. The first states that every bounded sequence in several variables satisfying some regularity conditions is a sum of a nilsequence and a sequence that has small uniformity norm (this generalizes a result of the second author and Kra); and the second states that every multiple correlation sequence in several variables is a sum of a nilsequence and a sequence that is small in uniform density (this generalizes a result of the first author). Furthermore, we use these results in order to establish mean convergence and recurrence results for a variety of sequences of dynamical and arithmetic origin and give some combinatorial implications.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 1 , February 2018 , pp. 81 - 142
Copyright
© Cambridge University Press, 2016 

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