Continued Fractions and Hyperbolic Geometry

Walker, Mairi (2016). Continued Fractions and Hyperbolic Geometry. PhD thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.0000ef81

Abstract

This thesis uses hyperbolic geometry to study various classes of both real and complex continued fractions. This intuitive approach gives insight into the theory of continued fractions that is not so easy to obtain from traditional algebraic methods. Using it, we provide a more extensive study of both Rosen continued fractions and even-integer continued fractions than in any previous works, yielding new results, and revisiting classical theorems. We also study two types of complex continued fractions, namely Gaussian integer continued fractions and Bianchi continued fractions. As well as providing a more elegant and simple theory of continued fractions, our approach leads to a natural generalisation of continued fractions that has not been explored before.

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