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

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