  In a recent paper [Adv. Math. 305:165--196, 2017], Komornik et al.~proved a
long-conjectured formula for the Hausdorff dimension of the set $\mathcal{U}_q$
of numbers having a unique expansion in the (non-integer) base $q$, and showed
that this Hausdorff dimension is continuous in $q$. Unfortunately, their proof
contained a gap which appears difficult to fix. This article gives a completely
different proof of these results, using a more direct combinatorial approach.
