Geometry and Arithmetic of Crystallographic Sphere Packings

TL;DR

This paper introduces crystallographic sphere packings linked to hyperbolic reflection groups, demonstrating infinite families with integer reciprocals and finitely many superintegral classes below dimension 30.

Contribution

It defines crystallographic sphere packings, constructs infinite conformally-inequivalent examples with integer reciprocals, and proves finiteness of superintegral classes in low dimensions.

Findings

Infinite family of conformally-inequivalent packings with reciprocal integer radii

Finiteness of superintegral packings in dimensions below 30

Connection between sphere packings and hyperbolic reflection groups

Abstract

We introduce the notion of a "crystallographic sphere packing," defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit for the first time an infinite family of conformally-inequivalent such with all radii being reciprocals of integers. We then prove a result in the opposite direction: the "superintegral" ones exist only in finitely many "commensurability classes," all in dimensions below 30.