Moments of distances between centres of Ford spheres

Given any positive integer k, we establish asymptotic formulas for the k-moments of the distances between the centres of ‘consecutive’ Ford spheres with radius less than 1/2S^2 for any positive integer S. This extends to higher dimensions the work on Ford circles by Chaubey, Malik and Zaharescu in their 2014 paper k-Moments of Distances Between Centres of Ford Circles. To achieve these estimates we bring the current theory of Ford spheres in line with the existing more developed theory for Ford circles and Farey fractions. In particular, we see (i) that a variant of the mediant operation can be used to generate Gaussian rationals analogously to the Stern-Brocot tree construction for Farey fractions and (ii) that two Ford spheres may be considered ‘consecutive’ for some order S if they are tangent and there is some Ford sphere with radius greater than 1/2S^2 that is tangent to both of them. We also establish an asymptotic estimate for a version of the Gauss Circle Problem in which we count Gaussian integers in a subregion of a circle in the complex plane that are coprime to a given Gaussian integer.