On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words

We consider numbers of the form $S\beta(\boldsymbol{u}):=\sum{n=0}^\infty \frac{un}{\beta^n}$, where $\boldsymbol{u}=\langle un \rangle{n=0}^\infty$ is an infinite word over a finite alphabet and $\beta\in \mathbb{C}$ satisfies $|\beta|>1$. Our main contribution is to present a combinatorial criterion on $\boldsymbol u$, called echoing, that implies that $S\beta(\boldsymbol{u})$ is transcendental whenever $\beta$ is algebraic. We show that every Sturmian word is echoing, as is the Tribonacci word, a leading example of an Arnoux-Rauzy word. We furthermore characterise $\overline{\mathbb{Q}}$-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 words having the same slope. Finally, we give an application of the above linear independence criterion 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.