On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words
<|reference_start|>On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words: We consider numbers of the form $S_\beta(\boldsymbol{u}):=\sum_{n=0}^\infty \frac{u_n}{\beta^n}$, where $\boldsymbol{u}=\langle u_n \rangle_{n=0}^\infty$ 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.<|reference_end|>
@article{kebis2024on,
    title={On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words},
    author={Pavol Kebis, Florian Luca, Joel Ouaknine, Andrew Scoones, James
  Worrell},
    journal={arXiv preprint arXiv:2405.05279},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.05279},
    primaryClass={math.NT cs.FL}
}