On the Irrationality Exponents of Mahler Numbers

TL;DR

This paper develops a method to compute the irrationality exponents of Mahler numbers derived from specific functional equations, extending previous work and providing detailed background on continued fractions.

Contribution

It introduces a procedure to determine irrationality exponents of Mahler numbers from functional equations, generalizing prior results to cases where C(z) is non-zero.

Findings

Provides a formula for irrationality exponents of Mahler numbers.

Extends previous results to more general functional equations.

Includes detailed background on continued fractions for broader accessibility.

Abstract

We explore Mahler numbers originating from functions $f(z)$ that satisfy the functional equation $f(z) = (A(z)f(z^d) + C(z))/B(z)$. A procedure to compute the irrationality exponents of such numbers is developed using continued fractions for formal Laurent series, and the form of all such irrationality exponents is investigated. This serves to extend Dmitry Badziahin's paper, On the Spectrum of Irrationality Exponents of Mahler Numbers, where he does the same under the condition that $C(z) = 0$. Furthermore, we cover the required background of continued fractions in detail for unfamiliar readers. This essay was submitted as a thesis in the Pure Mathematics Honours program at the University of Sydney.