Diophantine approximation of Mahler numbers

TL;DR

This paper investigates the approximation properties of Mahler numbers, showing they cannot be Liouville numbers and classifying their nature as rational, transcendental, or specific types of transcendental numbers.

Contribution

It establishes new results on the Diophantine approximation of Mahler numbers, including their non-Liouville nature and classification when regular.

Findings

Mahler numbers cannot be Liouville numbers.

Regular Mahler functions at 1/b produce either rational or transcendental values.

Transcendental Mahler numbers are classified as S-numbers or T-numbers.

Abstract

Suppose that $F(x)\in\mathbb{Z}[[x]]$ is a Mahler function and that $1/b$ is in the radius of convergence of $F(x)$. In this paper, we consider the approximation of $F(1/b)$ by algebraic numbers. In particular, we prove that $F(1/b)$ cannot be a Liouville number. If $F(x)$ is also regular, we show that $F(1/b)$ is either rational or transcendental, and in the latter case that $F(1/b)$ is an $S$-number or a $T$-number.