Michigan Math. J. 69 (2), 297-322, (May 2020) DOI: 10.1307/mmj/1576033219
KEYWORDS: 52C22, 20G20, 51M10, 11J70
This paper sets out to extend the results in the paper Geodesic Continued Fractions to continued fractions with Gaussian integer coefficients. The Farey graph , whose vertices are reduced Gaussian rationals in and whose edges join Farey neighbors, is introduced. The graph is modeled by the concrete realization in where Farey neighbors are joined by hyperbolic geodesics (Farey geodesics) as seen in the Farey tessellation of by Farey octahedrons. A natural distance on is also recalled, where is the least number of edges in from to , where is called the generation of w and a relevant path in is called a geodesic expansion for . The Farey neighborhood of a reduced Gaussian rational is introduced and partitioned into neighbors of generation , , and . Subsequently, it is seen that there can be at most four Farey neighbors of generation in the neighborhood. An ancestral path is introduced, and a bound on the number of geodesic paths to any 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.