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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.