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A multidimensional Szemerédi theorem for Hardy sequences of different growth


Author: Nikos Frantzikinakis
Journal: Trans. Amer. Math. Soc. 367 (2015), 5653-5692
MSC (2010): Primary 37A45; Secondary 28D05, 05D10, 11B25
DOI: https://doi.org/10.1090/S0002-9947-2014-06275-2
Published electronically: December 5, 2014
MathSciNet review: 3347186
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Abstract: We prove a variant of the multidimensional polynomial Szemerédi theorem of Bergelson and Leibman where one replaces polynomial sequences with other sparse sequences defined by functions that belong to some Hardy field and satisfy certain growth conditions. We do this by studying the limiting behavior of the corresponding multiple ergodic averages and obtaining a simple limit formula. A consequence of this formula in topological dynamics shows denseness of certain orbits when the iterates are restricted to suitably chosen sparse subsequences. Another consequence is that every syndetic set of integers contains certain non-shift invariant patterns, and every finite coloring of $ \mathbb{N}$, with each color class a syndetic set, contains certain polychromatic patterns, results very particular to our non-polynomial setup.


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Additional Information

Nikos Frantzikinakis
Affiliation: Department of Mathematics, University of Crete, Voutes University Campus, Heraklion 71003, Greece
Email: frantzikinakis@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2014-06275-2
Keywords: Ergodic theory, recurrence, Hardy field, Ramsey theory, nilmanifolds.
Received by editor(s): April 24, 2012
Received by editor(s) in revised form: June 20, 2013
Published electronically: December 5, 2014
Additional Notes: The author was partially supported by Marie Curie IRG 248008.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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