In a comment made by Gjergji Zaimi to [this](https://mathoverflow.net/q/61615/167834) older question, it is conjectured that $\frac 73$ is the threshold separating countability and uncountability of the sets of infinite binary words having a given [critical exponent](https://en.wikipedia.org/wiki/Critical_exponent_of_a_word). In a later comment, James Currie cites the (remarkable!) paper:


"J. Karhumaki and J. Shallit. Polynomial versus exponential growth in repetition-free binary words. J. Combin. Theory. Ser. A 105 (2004), 335–347."


as the one proving the conjecture. Is it so, though?


In general, it is not true that, if every element of $S\subset\{0,1\}^\omega$ has at most polynomially many finite subwords of length $n$, then $|S|\le \aleph\_0$. In fact, it is sufficient to allow $n+1$ subwords of length $n$ to get an uncountable set, as exemplified by the well-known [Sturmian words](https://en.wikipedia.org/wiki/Sturmian_word). So the conjecture ie either false or there must be a more specific argument involving critical exponents to establish the conjecture. What is it?



Words with critical exponent $< \frac 73$
[From user Ben's comment]


In the paper "The ubiquitous Prouhet-Thue-Morse sequence" ([DOI link behind paywall](https://doi.org/10.1007/978-1-4471-0551-0_1), [preprint version](https://cs.uwaterloo.ca/%7Eshallit/Papers/ubiq15.pdf)), Allouche and Shallit claim that there are uncountably many overlap-free binary words, which implies that the threshold is 2. I believe this follows from Fife's theorem.


