I have asked the following question on [Math.SE](https://math.stackexchange.com/questions/3150310/is-the-prouhet-thue-morse-constant-transcendental-in-any-integer-base-b2) some time ago and offered a bounty, yet received no answers nor comments, so I'm posting it here.




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The Prouhet-Thue-Morse constant, defined as 


$$
\tau =\sum \_{{i=0}}^{{\infty }}{\frac {t\_{i}}{2^{{i+1}}}}=0.412454033640\ldots 
$$


where the $t\_i$ are elements of the Thue-Morse sequence, is transcendental. But is 


$$
\tau\_b =\sum \_{{i=0}}^{{\infty }}{\frac {t\_{i}}{b^{{i+1}}}}
$$


also transcendental, for $b>2$?



Is the Prouhet-Thue-Morse constant transcendental in any integer base $b>2$?
Michel Waldschmidt, [Words and transcendence](https://hal.archives-ouvertes.fr/hal-00407221/file/WordsTranscendence.pdf), writes in Section 3.1, on page 461, "Mahler also proved in 1929 that the so-called Prouhet–Thue–Morse–Mahler number in base $g\ge2$, given by $$\xi\_g=\sum\_{n\ge0}{a\_n\over g^n}$$
 where $(a\_n)\_{n\ge0}$ is the Prouhet–Thue–Morse sequence, is transcendental; see [52] and [15, Section 13.4]." [15] is Allouche and Shallit, Automatic sequences: Theory, Applications, Generalizations (Cambridge
University Press, 2003). [52] is K. Nishioka, Mahler Functions and Transcendence (Lecture Notes in Mathematics 1631, Springer- Verlag, 1996).


Waldschmidt gives "the idea of proof," and refers to [15, Section 13.4] and [52, Example 1.3.1] for the full proof. 


