The Local-Global Principle for Integral Soddy Sphere Packings

TL;DR

This paper proves that in integral Soddy sphere packings, all sufficiently large numbers that are locally representable are actually represented by the packing, establishing a local-global principle.

Contribution

It establishes a local-global principle for integral Soddy sphere packings, showing that large admissible numbers are globally represented.

Findings

Every sufficiently large admissible number is represented.

The result connects local conditions to global representation in sphere packings.

Provides a new understanding of number representation in geometric packings.

Abstract

Fix an integral Soddy sphere packing P. Let K be the set of all curvatures in P. A number n is called represented if n is in K, that is, if there is a sphere in P with curvature equal to n. A number n is called admissible if it is everywhere locally represented, meaning that n is in K(mod q) for all q. It is shown that every sufficiently large admissible number is represented.