Abstract We prove that in any dimension n n there exists an origin-symmetric ellipsoid E R^n E ⊂ R n of volume c n^2 c n 2 that contains no points of 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 R^n R n whose density is at least cn^2 2^-n c n 2 ⋅ 2 − n. Previously known constructions of sphere packings in R^n R n yielded densities of at most C n n 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.
Boaz Klartag (Wed,) studied this question.