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May 2020 Geodesic Gaussian Integer Continued Fractions
Meira Hockman
Michigan Math. J. 69(2): 297-322 (May 2020). DOI: 10.1307/mmj/1576033219

Abstract

This paper sets out to extend the results in the paper Geodesic Continued Fractions to continued fractions with Gaussian integer coefficients. The Farey graph F, whose vertices are reduced Gaussian rationals in Q(i) and whose edges join Farey neighbors, is introduced. The graph is modeled by the concrete realization in H3 where Farey neighbors are joined by hyperbolic geodesics (Farey geodesics) as seen in the Farey tessellation of H3 by Farey octahedrons. A natural distance ϱ on Q(i) is also recalled, where ϱ(,w)=n is the least number of edges in F from to wQ(i), where n is called the generation of w and a relevant path in F is called a geodesic expansion for w. The Farey neighborhood of a reduced Gaussian rational is introduced and partitioned into neighbors of generation n1, n, and n+1. Subsequently, it is seen that there can be at most four Farey neighbors of generation n1 in the neighborhood. An ancestral path is introduced, and a bound on the number of geodesic paths to any w is established. Central to the paper are conditions for a path to be a geodesic path. The paper also addresses conditions for the existence of an infinite geodesic Gaussian integer continued fraction and suggestions of extending the paper to continued fraction with integer quaternion entries.

Citation

Download Citation

Meira Hockman. "Geodesic Gaussian Integer Continued Fractions." Michigan Math. J. 69 (2) 297 - 322, May 2020. https://doi.org/10.1307/mmj/1576033219

Information

Received: 8 May 2018; Revised: 6 August 2019; Published: May 2020
First available in Project Euclid: 11 December 2019

zbMATH: 07244374
MathSciNet: MR4104375
Digital Object Identifier: 10.1307/mmj/1576033219

Subjects:
Primary: 11J70, 20G20, 51M10, 52C22

Rights: Copyright © 2020 The University of Michigan

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Vol.69 • No. 2 • May 2020
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