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

We consider numbers of the form S_β(u):=∑_n=0^∞u_n/β^n, where u=⟨ u_n ⟩_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 ℚ-linear independence of sets of the form { 1, S_β(u_1),…,S_β(u_k) }, where u_1,…,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.