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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.

11J81 Transcendence (general theory)
11J70 Continued fractions and generalizations
11B83 Special sequences and polynomials
Zbl 0173.04801
Full Text: DOI
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