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.

  • $\begingroup$ 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"). $\endgroup$ – Martin Sleziak Apr 18 at 12:14

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