First Moment of Distances Between Centres of Ford Spheres

TL;DR

This paper extends the theory of Ford spheres by deriving an asymptotic estimate for the average distance between centers of consecutive Ford spheres with bounded radius, advancing the understanding of their geometric distribution.

Contribution

It establishes the first moment estimate for distances between centers of consecutive Ford spheres, connecting the geometric properties of Ford spheres with number theoretic estimates.

Findings

Derived an asymptotic estimate for the sum over distances between consecutive Ford spheres.

Connected geometric properties of Ford spheres with number theoretic functions.

Extended the theory of Ford spheres in line with Ford circles.

Abstract

This paper aims to develop the theory of Ford spheres in line with the current theory for Ford circles laid out in a recent paper by S. Chaubey, A. Malik and A. Zaharescu. As a first step towards this goal, we establish an asymptotic estimate for the first moment $\mathcal{M}_{1,I_2} (S) = \sum\limits_{\substack{\frac{r}{s},\frac{r'}{s'}\in \mathcal{G}_S \\ consec}} \frac{1}{2\lvert s \rvert ^2} + \frac{1}{2\lvert s' \rvert ^2}$ where the sum is taken over pairs of fractions associated with `consecutive' Ford spheres of radius less than or equal to $\frac{1}{2S^2}$.