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Multiple ergodic theorems for arithmetic sets


Authors: Nikos Frantzikinakis and Bernard Host
Journal: Trans. Amer. Math. Soc. 369 (2017), 7085-7105
MSC (2010): Primary 37A45; Secondary 05D10, 11B30, 11N37, 28D05
DOI: https://doi.org/10.1090/tran/6870
Published electronically: March 30, 2017
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Abstract: We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemerédi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements we can restrict the implicit parameter $ n$ to those integers that have an even number of distinct prime factors or satisfy any other congruence condition. In order to obtain these refinements we study the limiting behavior of some closely related multiple ergodic averages with weights given by appropriately chosen multiplicative functions. These averages are then analyzed using a recent structural result for bounded multiplicative functions proved by the authors.


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Additional Information

Nikos Frantzikinakis
Affiliation: Department of Mathematics, Voutes University Campus, University of Crete, 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

DOI: https://doi.org/10.1090/tran/6870
Keywords: Multiple ergodic averages, multiple recurrence, multiplicative functions, higher degree uniformity.
Received by editor(s): March 27, 2015
Received by editor(s) in revised form: March 30, 2015, and November 7, 2015
Published electronically: March 30, 2017
Article copyright: © Copyright 2017 American Mathematical Society