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.