A multidimensional Szemerédi theorem for Hardy sequences of different growth
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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.References
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Additional Information
- Nikos Frantzikinakis
- Affiliation: Department of Mathematics, University of Crete, Voutes University Campus, Heraklion 71003, Greece
- MR Author ID: 712393
- ORCID: 0000-0001-7392-5387
- Email: frantzikinakis@gmail.com
- 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.
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3347186