We consider numbers of the form Sβ(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 Sβ(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, Sβ(u1), …, Sβ(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.
Proceedings of ICALP 24, LIPIcs 297, 2024. 15 pages.
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© 2024 Florian Luca, Joël Ouaknine, and James Worrell.