## Transcendence of Thue-Morse continued fractions. (Transcendance des fractions continues de Thue-Morse.)(French)Zbl 0920.11045

For distinct positive integers $$a$$ and $$b$$, the Thue-Morse sequence $$a_1a_2a_3\dots$$ is defined to be the fixed point of the substitution $$\zeta(a)=ab$$, $$\zeta(b)=ba$$ on $$\{a,b\}$$; it begins with $abbabaabbaababbabaa babbaabba \dots$ Theorem: The real number $$x$$ between 0 and 1 with simple continued fraction expansion $$x=[0,a_1,a_2,a_3, \dots]$$ is transcendent. The author deduces this result from a theorem due to W. Schmidt [Acta Math. 119, 27-50 (1967; Zbl 0173.04801)] which tells us that a real number is transcendent if it is well approximable, in a specified sense, by quadratic irrationals. It is shown that various other criteria for transcendence cannot be used to establish the result.

### MSC:

 11J81 Transcendence (general theory) 11J70 Continued fractions and generalizations 11B83 Special sequences and polynomials

Zbl 0173.04801
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### References:

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