Transcendence of Sturmian or morphic continued fractions
type: Item
instance of: scholarly article
title: Transcendence of Sturmian or morphic continued fractions
zbMATH Open document ID: 0998.11036
author: Jean-Paul Allouche
author: J. L. Davison
author: Martine Queffélec
author: Luca Q. Zamboni
published in: Journal of Number Theory
publication date: 2002-08-25T00:00:00Z
review text: The authors prove, using a theorem of W. Schmidt, that if the sequence of partial quotients of the continued fraction expansion of a positive irrational real number takes only two values and begins with arbitrary long blocks which are `almost squares', then this number is either quadratic or transcendental. This result applies in particular to real numbers whose partial quotients form a Sturmian (or quasi-Sturmian) sequence, or are given by the sequence \((1 + (\lfloor n\alpha \rfloor \bmod 2))_{n\geq 0}\), or are a `repetitive' fixed point of a binary morphism satisfying some technical conditions.
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 1787785
zbMATH Keywords: quasi-Sturmian sequence
zbMATH Keywords: Sturmian sequence
zbMATH Keywords: binary morphisms
reviewed by: Robert F. Tichy
MaRDI profile type: Publication
OpenAlex ID: W2079999463
cites work: A relative of the Thue-Morse sequence
cites work: Q3128793
cites work: Q3713496
cites work: Algebraic irrational binary numbers cannot be fixed points of non-trivial constant length or primitive morphisms
cites work: Représentation géométrique de suites de complexité $2n+1$
cites work: Quasiperiodic patterns and icosahedral symmetry
cites work: Continued fractions of transcendental numbers
cites work: Q4941168
cites work: Frequencies of factors of Sturmian sequences
cites work: Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions
cites work: Enumeration of factors in the Thue-Morse word
cites work: Limit values of the recurrence quotient of Sturmian sequences
cites work: Sequences with minimal block growth II
cites work: Sequences with minimal block growth
cites work: Uniform spectral properties of one-dimensional quasicrystals. III: \(\alpha\)-continuity.
cites work: The index of Sturmian sequences
cites work: Q3035382
cites work: Combinatoire des codages de rotations
cites work: Codings of rotations and continued fractions
cites work: The characterization of \(N\)-écritures and applications to the study of sequences of finally \(n+c^{st}\) complexity
cites work: Les transformations de Chacon : combinatoire, structure géométrique, lien avec les systèmes de complexité $2n+1$
cites work: Transcendence of numbers with a low complexity expansion
cites work: Ein Verteilungsproblem für arithmetische Folgen
cites work: Substitution Minimal Sets
cites work: A Note on The Metrical Theory of Continued Fractions
cites work: On low-complexity bi-infinite words and their factors
cites work: Fractional powers in Sturmian words
cites work: Fast Pattern Matching in Strings
cites work: Engel series and continued fractions
cites work: Q4529547
cites work: Q5182919
cites work: Q4097906
cites work: Quelques remarques sur l'approximation rationnelle des nombres algébriques.
cites work: Ergodic sets
cites work: Minimal symbolic flows having minimal block growth
cites work: Q3128798
cites work: Infinite 0-1 sequences without long adjacent identical blocks
cites work: Transcendence of Thue-Morse continued fractions
cites work: Q3160623
cites work: A generalization of Sturmian sequences: Combinatorial structure and transcendence
cites work: Sequences with subword complexity \(2n\)
cites work: On simultaneous approximations of two algebraic numbers by rationals
cites work: Real numbers with bounded partial quotients: A survey
cites work: Automaticity. IV: Sequences, sets, and diversity
cites work: Sturmian words and words with a critical exponent
cites work: Remarques sur la suite engendrée par des substitutions composées
DOI: 10.1006/JNTH.2001.2669
description: scientific article; zbMATH DE number 1787785