On the transcendence of a series related to Sturmian words

TL;DR

This paper proves that certain infinite series, generated by Sturmian sequences and involving algebraic numbers, are transcendental, and provides conditions for their linear independence over algebraic numbers.

Contribution

It establishes the transcendence of series related to Sturmian words and offers criteria for their linear independence over algebraic numbers.

Findings

All such series are transcendental.

Provides sufficient conditions for linear independence.

Applicable to sequences generated by irrational rotations.

Abstract

Let $b$ be an algebraic number with $|b|>1$ and $\mathcal{H}$ a finite set of algebraic numbers. We study the transcendence of numbers of the form $\sum_{n=0}^\infty \frac{a_n}{b^n}$ where $a_n \in \mathcal{H}$ for all $n\in\mathbb{N}$. We assume that the sequence $(a_n)_{n=0}^\infty$ is generated by coding the orbit of a point under an irrational rotation of the unit circle. In particular, this assumption holds whenever the sequence is Sturmian. Our main result shows that all numbers of the above form are transcendental. We moreover give sufficient conditions for a finite set of such numbers to be linearly independent over~$\bar{\mathbb{Q}}$.