Boshernitzan, MichaelChaika, Jon2017-05-222017-05-222016Boshernitzan, Michael and Chaika, Jon. "Dichotomy for arithmetic progressions in subsets of reals." <i>Proceedings of the American Mathematical Society,</i> 144, (2016) American Mathematical Society: 5029-5034. http://dx.doi.org/10.1090/proc/13273.https://hdl.handle.net/1911/94346Let H stand for the set of homeomorphisms φ:[0, 1] → [0, 1]. We prove the following dichotomy for Borel subsets A ⊂ [0, 1]: • either there exists a homeomorphism φ ∈ Hsuch that the image φ(A) contains no 3-term arithmetic progressions; • or, for every φ ∈ H, the image φ(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).engArticle is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.Journal articlehttp://dx.doi.org/10.1090/proc/13273