2024/11/16 by Andrew Rajchert, Rajchert, Andrew
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Algebraic and Geometric Analysis #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2411.10733
openalex publication_date 2024/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We explore Mahler numbers originating from functions f(z) that satisfy the functional equation f(z) = (A(z)f(zd) + 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.