Transcendence of Sturmian Numbers over an Algebraic Base

We consider numbers of the form $S_\beta(\boldsymbol{u}):=\sum_{n=0}^\infty \frac{u_n}{\beta^n}$ for $\boldsymbol{u}=\langle u_n \rangle_{n=0}^\infty$ a Sturmian sequence over a binary alphabet and $\beta$ an algebraic number with $|\beta|>1$. We show that every such number is transcendental. More generally, for a given base~$\beta$ and given irrational number~$\theta$ we characterise the $\overline{\mathbb{Q}}$-linear independence of sets of the form $\left\{ 1, S_\beta(\boldsymbol{u}^{(1)}),\ldots,S_\beta(\boldsymbol{u}^{(k)}) \right\}$, where $\boldsymbol{u}^{(1)},\ldots,\boldsymbol{u}^{(k)}$ are Sturmian sequences having slope $\theta$. We give an application of our main result to the theory of dynamical systems, showing that for a contracted rotation on the unit circle with algebraic slope, its limit set is either finite or consists exclusively of transcendental elements other than its endpoints $0$ and $1$. This confirms a conjecture of Bugeaud, Kim, Laurent, and Nogueira.