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 $cn^2 \cdot 2^{-n}$. Previously known constructions of sphere packings in ${\mathbb{R}}^n$ yielded densities of at most $C n \log n \cdot 2^{-n}$. 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.