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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Multiple ergodic theorems for arithmetic sets
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by Nikos Frantzikinakis and Bernard Host PDF
Trans. Amer. Math. Soc. 369 (2017), 7085-7105 Request permission

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
  • MR Author ID: 712393
  • ORCID: 0000-0001-7392-5387
  • 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
  • MR Author ID: 88605
  • Email: bernard.host@u-pem.fr
  • 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
  • © Copyright 2017 American Mathematical Society
  • 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
  • MathSciNet review: 3683104