Transcendence of Hecke-Mahler Series

Florian Luca, Joel Ouaknine, James Worrell

Published 2024-12-10, updated 2024-12-17Version 2

We prove transcendence of the Hecke-Mahler series $\sum_{n=0}^\infty f(\lfloor n\theta+\alpha \rfloor) \beta^{-n}$, where $f(x) \in \mathbb{Z}[x]$ is a non-constant polynomial $\alpha$ is a real number, $\theta$ is an irrational real number, and $\beta$ is an algebraic number such that $|\beta|>1$.