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

Queffélec, M.

J. Number Theory 73, No. 2, 201-211 (1998).

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.

Reviewer: G.Köhler (Würzburg)

MSC:

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

Keywords:

Thue-Morse sequence; continued fraction expansion; transcendence

Citations:

Zbl 0173.04801

References:

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