Dichotomy for arithmetic progressions in subsets of reals
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- by Michael Boshernitzan and Jon Chaika PDF
- Proc. Amer. Math. Soc. 144 (2016), 5029-5034 Request permission
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
- MR Author ID: 39965
- Email: michael@rice.edu
- Jon Chaika
- Affiliation: Department of Mathematics, University of Utah, 155 S. 1400 E Room 233, Salt Lake City, Utah 84112
- MR Author ID: 808329
- Email: chaika@math.utah.edu
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 5029-5034
- MSC (2010): Primary 11B25, 26A21
- DOI: https://doi.org/10.1090/proc/13273
- MathSciNet review: 3556249