Computability questions in the sphere packing problem

TL;DR

This paper investigates the computability of optimal sphere packings in various dimensions, demonstrating that these sets are oracle-computable when provided with an oracle ordering spherical codes by Kolmogorov complexity.

Contribution

It introduces the concept of oracle-computability for sets of dimensions with optimal sphere packings and connects it to Kolmogorov complexity of spherical codes.

Findings

Sets of dimensions with optimal packings are oracle-computable.

Computability depends on an oracle ordering spherical codes by Kolmogorov complexity.

The approach links sphere packing problems to algorithmic information theory.

Abstract

We consider the sets of dimensions for which there is an optimal sphere packing with special regularity properties (respectively, a lattice, or a periodic set with a given bound on the number of translations, or an arbitrary periodic set). We show that all these sets are oracle-computable, given an oracle that orders an associated set of spherical codes by increasing Kolmogorov complexity.