A dichotomy for the stability of arithmetic progressions
Dynamical Systems
2013-03-20 v1
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).
Keywords
Cite
@article{arxiv.1303.4684,
title = {A dichotomy for the stability of arithmetic progressions},
author = {Michael Boshernitzan and Jon Chaika},
journal= {arXiv preprint arXiv:1303.4684},
year = {2013}
}