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Article Dans Une Revue Annales de l'Institut Fourier Année : 2023

Combinatorial structure of Sturmian words and continued fraction expansions of Sturmian numbers

Yann Bugeaud
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Michel Laurent
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Résumé

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) k≥0 , 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. Let b ≥ 2 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.
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Dates et versions

hal-03571109 , version 1 (13-02-2022)

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Yann Bugeaud, Michel Laurent. Combinatorial structure of Sturmian words and continued fraction expansions of Sturmian numbers. Annales de l'Institut Fourier, 2023, 73 (5), pp.2029-2078. ⟨10.5082/aif.3561⟩. ⟨hal-03571109⟩
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