Asymptotic Density of Apollonian-Type Packings

TL;DR

This paper explores the deep connection between the generation of arithmetic groups by elementary matrices and the filling properties of circle and sphere packings, providing new insights, proofs, and conjectures.

Contribution

It establishes a novel link between group generation and geometric packing problems, offering new theoretical results and perspectives.

Findings

Arithmetic groups like SL(2,O) are generated by elementary matrices under certain conditions.

Circle and sphere packings can fill space in ways related to group properties.

New proofs and conjectures about the relationship between algebraic and geometric structures.

Abstract

We consider two seemingly unconnected problems: first, under what circumstances are arithmetic groups like SL(2,O) generated by elementary matrices; second, when do certain classes of circle/sphere packings fill up space? We show that these are in fact deeply related, leading to some new results, new proofs of old results, and interesting conjectures.