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arXiv · 1303.4684

A dichotomy for the stability of arithmetic progressions

Michael Boshernitzan·Jon Chaika
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Abstract

Let H stand for the set of homeomorphisms on [0,1]. We prove the following dichotomy for Borel subsets A of [0,1]: either there exists a homeomorphism f in H such that the image f(A) contains no 3-term arithmetic progressions; or, for every f in H, the image f(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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Michael Boshernitzan, Jon Chaika. 2013-03-19. A dichotomy for the stability of arithmetic progressions. https://arxiv.org/abs/1303.4684

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