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


[1] Baker, A., Continued fractions of transcendental numbers, Mathematika, 9, 1-8 (1962) · Zbl 0105.03903
[2] Berstel, J.; Seebold, P., A characterization of overlap-free morphisms, Discrete Appl. Math., 46, 275-281 (1993) · Zbl 0824.68093
[3] Christol, G.; Kamae, T.; Mendès France, M.; Rauzy, G., Suites algébriques, automates et substitutions, Bull. Soc. Math. France, 108, 401-419 (1980) · Zbl 0472.10035
[4] Davenport, H.; Roth, K. F., Rational approximations to algebraic numbers, Mathematika, 2, 160-167 (1955) · Zbl 0066.29302
[5] Davison, J. L., A class of transcendental numbers with bounded partial quotients, (Mollin, R. A., Number Theory and Applications (1989)), 365-371 · Zbl 0693.10028
[6] Hardy, G. H.; Wright, E. M., An Introduction to the Theory of Numbers (1975), Clarendon: Clarendon Oxford · Zbl 0020.29201
[7] Mahler, K., Lectures on Transcendental Numbers. Lectures on Transcendental Numbers, Lecture Notes in Math, 546 (1972), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0213.32703
[8] Maillet, E., Introduction à la théorie des nombres transcendants (1906), Chapitre VII: Chapitre VII Paris · JFM 37.0237.02
[9] Schmidt, W., On simultaneous approximations of two algebraic numbers by rationals, Acta Math., 119, 27-50 (1967) · Zbl 0173.04801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.