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Dichotomy for arithmetic progressions in subsets of reals


Authors: Michael Boshernitzan and Jon Chaika
Journal: Proc. Amer. Math. Soc. 144 (2016), 5029-5034
MSC (2010): Primary 11B25, 26A21
DOI: https://doi.org/10.1090/proc/13273
Published electronically: August 18, 2016
MathSciNet review: 3556249
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Abstract: Let $ \mathcal {H}$ stand for the set of homeomorphisms $ \phi \colon \![0,1]\to [0,1]$. We prove the following dichotomy for Borel subsets $ A\subset [0,1]$:

  • either there exists a homeomorphism $ \phi \in \mathcal {H}$ such that the image $ \phi (A)$ contains no $ 3$-term arithmetic progressions;
  • or, for every $ \phi \in \mathcal {H}$, the image $ \phi (A)$ contains arithmetic progressions of arbitrary finite length.
In fact, we show that the first alternative holds if and only if the set $ A$ is meager (a countable union of nowhere dense sets).

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Additional Information

Michael Boshernitzan
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005
Email: michael@rice.edu

Jon Chaika
Affiliation: Department of Mathematics, University of Utah, 155 S. 1400 E Room 233, Salt Lake City, Utah 84112
Email: chaika@math.utah.edu

DOI: https://doi.org/10.1090/proc/13273
Keywords: Meager subset, arithmetic progressions, homeomorphism
Received by editor(s): December 4, 2013
Received by editor(s) in revised form: March 2, 2015, and June 30, 2015
Published electronically: August 18, 2016
Additional Notes: The first author was supported in part by DMS-1102298
The second author was supported in part by DMS-1300550
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2016 American Mathematical Society