On spectrum of irrationality exponents of Mahler numbers

TL;DR

This paper investigates the irrationality exponents of Mahler numbers, proving they are rational or rationally bounded, and computes exact values when continued fractions are known, advancing understanding of Mahler function approximations.

Contribution

It establishes that Mahler numbers at integer points are either rational or have rational irrationality exponents, and provides a method to compute these exponents exactly.

Findings

Irrationality exponents of Mahler numbers are always rational or rationally bounded.

Exact irrationality exponents can be computed when the Mahler function's continued fraction is known.

Improves previous results by providing precise values rather than upper bounds.

Abstract

We consider Mahler functions $f(z)$ which solve the functional equation $f(z) = \frac{A(z)}{B(z)} f(z^d)$ where $\frac{A(z)}{B(z)}\in \mathbb{Q}(z)$ and $d\ge 2$ is integer. We prove that for any integer $b$ with $|b|\ge 2$ either $f(b)$ is rational or its irrationality exponent is rational. We also compute the exact value of the irrationality exponent for $f(b)$ as soon as the continued fraction for the corresponding Mahler function is known. This improves the result of Bugeaud, Han, Wei and Yao where only an upper bound for the irrationality exponent was provided.