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

We prove that in any dimension n there exists an origin-symmetric ellipsoid E⊂Rn of volume cn2 that contains no points of Zn other than the origin, where c>0 is a universal constant. Equivalently, there exists a lattice sphere packing in Rn whose density is at least cn2⋅2−n. Previously known constructions of sphere packings in Rn yielded densities of at most Cnlogn⋅2−n. Our proof utilizes a stochastically evolving ellipsoid that accumulates at least cn2 lattice points on its boundary, while containing no lattice points in its interior except for the origin.