# 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.

Update: it occurred to me that even the estimate of the (upper) box dimension of $$S_c$$ is not trivial. Already for integer $$c$$, it leads to a tricky combinatorial problem, that is, how many distinct binary words of length $$n$$ are there having no factor of the form $$w^c$$ ($$w \in \{0,1\}^*$$)? It's also not easy for me to guess whether the Hausdorff and box dimensions could disagree.

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

• In the previous revision you had "1: Falconer, K. (2004)." and most likely you intended to write d"[1]: Falconer, K. (2004)." I have tried to fix this - you can check the source and the revision history to see what I changed. You can learn about various ways how links can be formatted in the editing help (the section "Links"). – Martin Sleziak Apr 18 at 12:14