On the Irrationality Exponents of Mahler Numbers

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.