Transcendence of Hecke-Mahler Series

Abstract

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 $|\beta |>1$.