Sparse regular subsets of the reals

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 B\"uchi automaton. In the case that this predicate is closed, a dichotomy is established for when this expansion is interdefinable with the structure \((\mathbb{R},1}\). In the case that the closure of the predicate has Hausdorff dimension less than \(1\), the dichotomy further characterizes these expansions of \((\mathbb{R},