A dichotomy for the stability of arithmetic progressions

TL;DR

This paper establishes a dichotomy for Borel subsets of [0,1], showing they either can be mapped to sets with no 3-term arithmetic progressions or always contain arbitrarily long progressions under any homeomorphism.

Contribution

It proves a precise characterization of when Borel sets can be transformed to avoid or contain long arithmetic progressions under homeomorphisms.

Findings

Meager sets can be mapped to sets with no 3-term arithmetic progressions.

Non-meager sets always contain arbitrarily long arithmetic progressions under any homeomorphism.

The dichotomy hinges on the meagerness of the set.

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).