What is the Hausdorff dimension of the subset $S\_c \subset [0,1]$ of points such that the [critical exponent](https://en.wikipedia.org/wiki/Critical_exponent_of_a_word) 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.



Hausdorff dimension and critical exponent of words
Here are a few observations: (the last one is maybe the main point of interest)


$\bullet$ Your $S\_c$ is a closed subset of $[0,1]$ invariant under $x \mapsto 2x \pmod 1$ (usually there's a subtlety about points with multiple expansions, but none of those are in $S\_c$ for $c < \infty$), and so I think a result of Furstenberg ([1]) shows that it has equal Hausdorff and box dimension. They should both be equal to the topological entropy $h(S\_c)$ of $S\_c$ viewed as a subshift on $\{0,1\}$ (using $\log\_2$ in definition of entropy).


$\bullet$ In [1], Kolpakov proves that $dim(S\_3) = h(S\_3) \geq 0.5435$ (I've converted to $\log\_2$, he used $\ln$)


$\bullet$ for all $c < \infty$, $dim(S\_c) = h(S\_c) < 1$, since at the least you forbid the word $0^{\lceil c \rceil}$, and all proper subshifts of $\{0,1\}^{\mathbb{N}}$ have entropy less than $1$.


$\bullet$ (this is quite long and probably hard to read, sorry) There are probably better methods, but you can use the techniques of my paper [3] to bound the entropy (and therefore dimension) of $S\_c$ from below. Let's restrict to integer $c$ for now. Then $S\_c$ is a subshift defined by forbidden list $\{w^c \ : \ w \in \{0,1\}^\*\}$, where $w^c$ is the $c$th power of $w$.


The technique from that paper to bound entropy from below is to rescale at some length $j$, and then view your subshift $X$ (here $S\_c$) as a new subshift whose alphabet is $A^{(j)} = L\_j(X)$ (the set of $j$-letter words in $X$) and whose forbidden list is $F^{(j)}$, a set of forbidden concatenations of `letters' $A^{(j)}$ which induces the same forbidden words as those for $X$. More formally,
$a\_1 \ldots a\_i$ is in $F^{(j)}$ if the $ij$-letter concatenation $a\_1 \ldots a\_i$ contains a forbidden word for $X$, but neither $a\_1 \ldots a\_i-1$ nor $a\_2 \ldots a\_i$ contains a forbidden word for $X$. This forbidden list $F^{(j)}$ on $A^{(j)}$ induces a subshift $X^{(j)}$ where each point is a sequence of $c$-letter blocks whose concatenation is in $X$. (Formally, $X^{(j)}$ is isomorphic to $X$ under the $c$th power of the shift.)


Here, if we use $X = S\_c$ and set $j = c$, then it's easy to bound from above the number of $i$-'letter' words in $F^{(c)}$. There are none for $i = 1$ by definition. For larger $i$, if $a\_1 \ldots a\_i \in F^{(c)}$, then either $a\_1 \ldots a\_i$ is a $c$th power of an $i$-letter word (less than $2^i$ possibilities) or contains a $c$th power of an $(i-1)$-letter word ($c$ choices for where in $a\_1$ the power starts, less than $2^{i-1}$ possibilities for the base of the power, less than $2^c$ choices for the remaining letters). If only shorter $c$th powers were in $a\_1 \ldots a\_i$, then one of $a\_1 \ldots a\_{i-1}$ or $a\_2 \ldots a\_i$ would contain a $c$th power, disallowed by definition. So, the number of $i$-letter forbidden words in $F^{(c)}$ is less than $2^i + c2^c 2^{i-1} \leq c 2^{c+i}$.


Now, Theorem 4.1 from [3] gives a lower bound for entropy of a subshift. It states that if $X$ is a subshift with alphabet $A$ and an infinite forbidden list $F$ with $F\_n$ words of length $n$ for every $n$, and if
$\sum F\_i x^i < x(|A| - k + 1) - 1$ for some $x > 1/|A|$ and $k < |A|$, then
$h(X) \geq \log k$. This is hideous, but we can make it easier.


We'll apply to the $c$-letter rescaling $S\_c^{(c)}$ of $S\_c$ defined above, and assume that $c \leq 10$. Then $|A^{(c)}| = 2^c - 2$ (the only illegal $c$-letter words in $S\_c$ are $0^c$ and $1^c$). Let's take $k = 2^{c-1} - 1$ and $x = 4/2^c$. Then $x(|A^{(c)}| - k + 1) = 1$, so we need only verify that
$\sum (F^{(c)})\_i x^i < 1$; recall that $(F^{(c)})\_i \leq c 2^{c+i}$. Finally,


$\sum\_{i = 2}^\infty (F^{(c)})\_i x^i \leq \sum\_{i=2}^\infty c 2^{c+i} (4/2^c)^i = 
c 2^c \sum\_{i=2}^\infty (8/2^c)^i = c 2^c \frac{64/2^{2c}}{1 - (8/2^c)}
= \frac{64c}{2^c-8}$.


This is less than $1$ for $c \leq 10$, so Theorem 4.1 shows that in that case,
$h(S\_c^{(c)}) \geq \log k = \log(2^{c-1} - 1)$. But $h(S\_c^{(c)}) = c h(S\_c)$ (since the former is isomorphic to the latter under the $c$th power of the shift). So, at long last, we get $h(S\_c) \geq \frac{\log(2^{c-1} - 1)}{c}$.


In particular, this approaches $1$ as $c \rightarrow \infty$, so this should answer one of your questions: $dim(\bigcup S\_c) = 1$.


I should say that I found several papers estimating entropies of $S\_c$ from below, and would not at all be surprised if one gives a much easier proof that $dim(\bigcup S\_c) = 1$. But all I found were for specific values of $c$.


[1] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory, 1 (1967)


[2] R. Kolpakov, Efficient lower bounds on the number of repetition-free words. J. Integer Seq. 10 (2007), no. 3


[3] R. Pavlov, Ronnie, On subshifts with slow forbidden word growth.
Ergodic Theory Dynam. Systems 42 (2022), no. 4


