Transcendence of Sturmian Numbers over an Algebraic Base

TL;DR

This paper proves that numbers generated by Sturmian sequences over an algebraic base are transcendental and explores their linear independence, with applications to dynamical systems and a conjecture by Bugeaud et al.

Contribution

It establishes the transcendence of Sturmian sequence sums over algebraic bases and characterizes their linear independence, advancing the understanding of these numbers in number theory.

Findings

Numbers of the form $S_\beta(\boldsymbol{u})$ are transcendental for algebraic $\beta$ with $|\beta|>1$.

Characterization of $\overline{\mathbb{Q}}$-linear independence for sets involving Sturmian sequences.

Application to dynamical systems confirms a conjecture about the nature of limit sets in contracted rotations.

Abstract

We consider numbers of the form $S_\beta(\boldsymbol{u}):=\sum_{n=0}^\infty \frac{u_n}{\beta^n}$ for $\boldsymbol{u}=\langle u_n \rangle_{n=0}^\infty$ a Sturmian sequence over a binary alphabet and $\beta$ an algebraic number with $|\beta|>1$. We show that every such number is transcendental. More generally, for a given base~$\beta$ and given irrational number~$\theta$ we characterise the $\overline{\mathbb{Q}}$-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 sequences having slope $\theta$. We give an application of our main result 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.