Dichotomy for arithmetic progressions in subsets of reals

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).