Transcendence of Sturmian Numbers over an Algebraic Base

Abstract

We consider numbers of the form $S_{\beta }(\mathbf{u}):={\sum }_{n=0}^{\infty }\frac{u_n}{{\beta }^n}$ for $\mathbf{u}=⟨u_n{⟩}_{n=0}^{\infty }$ 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 $\left\{1,S_{\beta }({\mathbf{u}}^{(1)}),\dots ,S_{\beta }({\mathbf{u}}^{(k)})\right\}$, where ${\mathbf{u}}^{(1)},\dots ,{\mathbf{u}}^{(k)}$ 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.