Transcendence of Thue-Morse continued fractions
type: Item
instance of: scholarly article
title: Transcendence of Thue-Morse continued fractions
zbMATH Open document ID: 0920.11045
published in: Journal of Number Theory
publication date: 1999-09-10T00:00:00Z
review text: 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 \textit{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.
reviewed by: Günter Köhler
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 1247919
zbMATH Keywords: Thue-Morse sequence
zbMATH Keywords: continued fraction expansion
zbMATH Keywords: transcendence
author: Martine Queffélec
MaRDI profile type: Publication
full work available at URL: jnth.1998.2305
OpenAlex ID: W2014572193
cites work: Continued fractions of transcendental numbers
cites work: A characterization of overlap-free morphisms
cites work: Suites algébriques, automates et substitutions
cites work: Rational approximations to algebraic numbers
cites work: Q3035382
cites work: Q5772619
cites work: Q5615451
cites work: On simultaneous approximations of two algebraic numbers by rationals
DOI: 10.1006/JNTH.1998.2305
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description: scientific article; zbMATH DE number 1247919