The algebra and geometry of using continued fractions for approximating real and complex numbers
Date
2022
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Abstract
Through geometric analysis we forge a new interpretation of the link between nearest integer continued fractions and the Farey tessellation of hyperbolic space. Instead of just truncating the continued fraction to generate approximations, we focus on
• how to parse the product of maps derived from the nearest integer continued fraction into a product of parabolic and elliptic Mobius maps and ¨
• on the collection of points on which these maps act.
It turns out that we need to set apart the elliptic maps that permute a collection of six vertices, three values in R, namely ∞, 0 and 1, and three values in C, namely i, 1 + i and 1+i 2 . The action of the remaining parabolic maps on the same six vertices results in the creation of a sequence of nested Farey quadrilaterals, containing the target, whose boundaries are based in the Schmidt arrangement formed by the Farey sets and dual Farey sets of Schmidt.
Description
A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy to the Faculty of Science, University of the Witwatersrand, 2022
Keywords
Algebra and Geometry, Continued Fractions, Approximating real and complex numbers