Geodesic Gaussian Integer Continued Fractions

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 $\mathcal{F}$, whose vertices are reduced Gaussian rationals in ${\mathbb{Q}}_{\infty }(i)$ and whose edges join Farey neighbors, is introduced. The graph is modeled by the concrete realization in ${\mathbb{H}}^3$ where Farey neighbors are joined by hyperbolic geodesics (Farey geodesics) as seen in the Farey tessellation of ${\mathbb{H}}^3$ by Farey octahedrons. A natural distance $\varrho$ on ${\mathbb{Q}}_{\infty }(i)$ is also recalled, where $\varrho (\infty ,w)=n$ is the least number of edges in $\mathcal{F}$ from $\infty$ to $w\in \mathbb{Q}(i)$, where $n$ is called the generation of w and a relevant path in $\mathcal{F}$ is called a geodesic expansion for $w$. The Farey neighborhood of a reduced Gaussian rational is introduced and partitioned into neighbors of generation $n-1$, $n$, and $n+1$. Subsequently, it is seen that there can be at most four Farey neighbors of generation $n-1$ 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.