Transcendence of Hecke-Mahler Series

TL;DR

The paper proves the transcendence of the Hecke-Mahler series $\sum_{n=0}^ fty f(\lfloor n\theta+\alpha\rfloor)\beta^{-n}$ for a non-constant $f\in\mathbb{Z}[x]$, irrational $\theta$, and algebraic $\beta$ with $|\beta|>1$. It develops a general transcendence criterion (Condition, denoted (*)) for sequences, using the p-adic Subspace Theorem to show that $\sum_{m\ge0} u_m\beta^{-m}$ is transcendental whenever $\mathbf{u}$ satisfies (*) and grows at most polynomially. The key technical contribution is defining Expanding-Gaps and Polynomial-Variation within a linear-recurrence–like framework and proving these properties hold for $u_n=f(\lfloor n\theta+\alpha\rfloor)$ via continued-fraction analysis of $\theta$. Consequently, the Hecke-Mahler series is transcendental for any non-constant $f\in\mathbb{Z}[x]$, extending prior results beyond linear $f$ and fixed $\alpha$ and illustrating a powerful method combining Diophantine approximation with combinatorial structure.

Abstract

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

Theorems & Definitions (15)

• Theorem 1
• Proposition 2
• Example 3
• Definition 4
• Theorem 5
• proof
• Claim 6
• proof
• Theorem 7
• proof