  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.
