Sparse regular subsets of the reals

Abstract

This paper concerns the expansion of the real ordered additive group by a predicate for a subset of $[0,1]$ whose base-$r$ representations are recognized by a Büchi automaton. In the case that this predicate is closed, a dichotomy is established for when this expansion is interdefinable with the structure $(\mathbb{R},<,+,0,r^{-\mathbb{N}})$ for some $r\in {\mathbb{N}}_{>1}$. In the case that the closure of the predicate has Hausdorff dimension less than $1$, the dichotomy further characterizes these expansions of $(\mathbb{R},<,+,0,1)$ by when they have NIP and NTP${}_2$, which is precisely when the closure of the predicate has Hausdorff dimension $0$.

Accepted by Israel Journal of Mathematics, 2025