Sphere Packings in Euclidean Space with Forbidden Distances

TL;DR

This paper investigates constrained sphere packings in high-dimensional Euclidean spaces, proving optimality of certain extremal lattices under specific distance restrictions and providing results up to 1200 dimensions.

Contribution

It establishes the optimality of extremal lattices for constrained sphere packings in 48 and higher dimensions, extending to dimensions up to 1200, and introduces an algorithm for one-dimensional cases.

Findings

Optimal density bounds for constrained packings in 48 dimensions.

Extremal lattices are uniquely optimal for certain constraints up to 1200 dimensions.

A new algorithm for one-dimensional packing configurations based on domino problems.

Abstract

We study the sphere packing problem in Euclidean space where we impose additional constraints on the separations of the center points. We prove that any sphere packing in dimension $48$, with spheres of radii $r$, such that no two centers $x_1$ and $x_2$ satisfy $\sqrt{\tfrac{4}{3}} < \frac{1}{2r}|x_1-x_2| <\sqrt{\tfrac{5}{3}}$, has center density less or equal than $(3/2)^{24}$. Equality occurs for periodic packings if and only if the packing is given by a $48$-dimensional even unimodular extremal lattice. This shows that any of the lattices $P_{48p},P_{48q},P_{48m}$ and $P_{48n}$ are optimal for this constrained packing problem, and gives evidence towards the conjecture that extremal lattices are optimal unconstrained sphere packings in $48$ dimensions. We also provide results for packings up to dimension $d\leq 1200$, where we impose constraints on the distance between centers and on the minimal norm of the spectrum, showing that even unimodular extremal lattices are again uniquely optimal. Moreover, in the one-dimensional case, where it is not at all clear that periodic packings are among those with largest density, we nevertheless give a condition on the set of constraints that allows this to happen, and we develop an algorithm to find these periodic configurations by relating the problem to a question about dominos.