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

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