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Hausdorff dimension and critical exponent of words

What is the Hausdorff dimension of the subset $S_c \subset [0,1]$ of points such that the critical exponent of their binary expansion is $c$? It's clear that $\dim_H S_{\infty}=1$, but what can be said for $c<\infty$?. Also, what is $\dim_H \cup_{c\in [2,\infty)} S_c$? Are there techniques for the evaluation of the Hausdorff dimension which are applicable to these cases? The text by Falconer 1 doesn't seem to cover this kind of questions.

1: Falconer, K. (2004). Fractal geometry: mathematical foundations and applications. John Wiley & Sons.