Abstract: We prove that in any dimension n there exists an origin-symmetric ellipsoid of volume c n^2 that contains no points of Z^n other than the origin. Here c > 0 is a universal constant. Equivalently, there exists a lattice sphere packing in R^n whose density is at least c n^2 \/ 2^n. Previously known constructions of sphere packings in R^n had densities of the order of magnitude of n \/ 2^n, up to logarithmic factors. Our proof utilizes a stochastically evolving ellipsoid that accumulates at least\u00a0 c n^2 lattice points on its boundary, while containing no lattice points in its interior except for the origin.<\/p>\n","protected":false},"excerpt":{"rendered":"
Abstract: We prove that in any dimension n there exists an origin-symmetric ellipsoid of volume c n^2 that contains no points of Z^n other than the origin. Here c > 0 is a universal constant. Equivalently, there exists a lattice…
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