  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
$(\mathbb{R},<,+,0,r^{-\mathbb{N}})$ 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 $0$.
