On the transcendence of a series related to Sturmian words
Florian Luca, Jo\"el Ouaknine, James Worrell

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 be an algebraic number with and a finite set of algebraic numbers. We study the transcendence of numbers of the form where for all . We assume that the sequence 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~.
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Taxonomy
Topicssemigroups and automata theory · Mathematical Dynamics and Fractals · Geometric and Algebraic Topology
