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

Let $\theta =[0;a_1,a_2,\dots ]$ be the continued fraction expansion of an irrational real number $\theta \in (0,1)$. It is well-known that the characteristic Sturmian word of slope $\theta$ is the limit of a sequence of finite words $(M_k{)}_{k\ge 0}$, with $M_k$ of length $q_k$ (the denominator of the $k$-th convergent to $\theta$) 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 $b\ge 2$ be an integer. Our second result gives the continued fraction expansion of any real number $\xi$ whose $b$-ary expansion is a Sturmian word $\mathbf{s}$ over the alphabet $\{0,b-1\}$. This extends a classical result of B"ohmer who considered only the case where $\mathbf{s}$ is characteristic. As a consequence, we obtain a formula for the irrationality exponent of $\xi$ in terms of the slope and the intercept of $\mathbf{s}$.