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

We consider numbers of the form S_β(u):=∑ₙ₌₀^∞ uₙ/βⁿ, where u= uₙ ₙ₌₀^∞ is an infinite word over a finite alphabet and β∈ C satisfies |β|>1. Our main contribution is to present a combinatorial criterion on u, called echoing, that implies that S_β(u) is transcendental whenever β 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 Q̄-linear independence of sets of the form \ 1, S_β(u₁),…,S_β(uₖ) \, where u₁,…,uₖ 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.