Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid

Abstract

We prove that in any dimension $n$ there exists an origin-symmetric ellipsoid $\mathcal{E}\subset {\mathbb{R}}^n$ of volume $c{n}^2$ that contains no points of ${\mathbb{Z}}^n$ other than the origin, where $c>0$ is a universal constant. Equivalently, there exists a lattice sphere packing in ${\mathbb{R}}^n$ whose density is at least $c{n}^2\cdot 2^{-n}$. Previously known constructions of sphere packings in ${\mathbb{R}}^n$ yielded densities of at most $Cn\log n\cdot 2^{-n}$. Our proof utilizes a stochastically evolving ellipsoid that accumulates at least $c{n}^2$ lattice points on its boundary, while containing no lattice points in its interior except for the origin.

29 pages