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

We prove that in any dimension n n there exists an origin-symmetric ellipsoid E ⊂ Rⁿ E ⊂ R n of volume c n² c n 2 that contains no points of Zⁿ Z n other than the origin, where $c > 0$ c > 0 is a universal constant. Equivalently, there exists a lattice sphere packing in Rⁿ R n whose density is at least cn² · 2⁻ⁿ c n 2 ⋅ 2 − n . Previously known constructions of sphere packings in Rⁿ R n yielded densities of at most C n log n · 2⁻ⁿ C n log n ⋅ 2 − n . Our proof utilizes a stochastically evolving ellipsoid that accumulates at least c n² c n 2 lattice points on its boundary, while containing no lattice points in its interior except for the origin.