Combinatorial structure of Sturmian words and continued fraction expansion of Sturmian numbers
Annales de l'Institut Fourier, Volume 73 (2023) no. 5, pp. 2029-2078.

Let θ=[0;a 1 ,a 2 ,] be the continued fraction expansion of an irrational real number θ(0,1). It is well-known that the characteristic Sturmian word of slope θ is the limit of a sequence of finite words (M k ) k0 , with M k of length q k (the denominator of the k-th convergent to θ) being a suitable concatenation of a k copies of M k-1 and one copy of M k-2 . Our first result extends this to any Sturmian word s. Let b2 be an integer. Our second result gives the continued fraction expansion of any real number ξ whose b-ary expansion is a Sturmian word s over the alphabet {0,b-1}. This extends a classical result of Böhmer who considered only the case where s is characteristic. As a consequence, we obtain a formula for the irrationality exponent of ξ in terms of the slope and the intercept of s.

Soit θ=[0;a 1 ,a 2 ,] le développement en fraction continue d’un nombre irrationnel θ(0,1) et soit q k le dénominateur de la k-ième réduite de θ. On sait que les préfixes M k de longueur q k du mot sturmien caractéristique de pente θ vérifient la relation de récurrence M k =M k-1 a k M k-2 pour tout k2. Nous établissons une relation de concaténation analogue pour les préfixes d’un mot sturmien quelconque s. Soit b un entier 2. Nous obtenons en deuxième lieu une formule explicite pour le développement en fraction continue de tout nombre réel ξ(0,1) dont la suite des chiffres en base b forme une suite sturmienne s sur l’alphabet {0,b-1}. On généralise ainsi un résultat classique de Böhmer qui traitait le cas particulier où s est une suite sturmienne caractéristique. Nous en déduisons une formule donnant l’exposant d’irrationalité de ξ en fonction de la pente et de l’intercept de s.

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DOI: 10.5802/aif.3561
Classification: 11J04, 11J70, 11J81, 68R15
Keywords: Rational approximation, continued fraction, transcendence, Sturmian sequence, combinatorics on words.
Mot clés : Approximation rationnelle, fraction continue, transcendance, suite sturmienne, combinatoire des mots.
Bugeaud, Yann 1, 2; Laurent, Michel 3

1 IRMA, UMR7501 Université de Strasbourg et CNRS 7, rue René Descartes 67084 Strasbourg (France)
2 Institut universitaire de France
3 Aix-Marseille Université CNRS Institut de Mathématiques de Marseille 163 avenue de Luminy, Case 907 13288 Marseille Cedex 9 (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Bugeaud, Yann; Laurent, Michel. Combinatorial structure of Sturmian words and continued fraction expansion of Sturmian numbers. Annales de l'Institut Fourier, Volume 73 (2023) no. 5, pp. 2029-2078. doi : 10.5802/aif.3561. https://aif.centre-mersenne.org/articles/10.5802/aif.3561/

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