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 $c n^2$ lattice points on its boundary, while containing no lattice points in its interior except for the origin.
Short Bio:
Boaz Klartag is a professor in the Department of Mathematics at the Weizmann Institute of Science. He obtained his PhD in 2004 from Tel Aviv University under the supervision of Prof. Vitali Milman. His research interests lie in convex geometry, analysis, and high-dimensional phenomena. Klartag's contributions include his work on the central limit theorem for convex sets, Bourgain's slicing problem, needle decompositions in Riemannian geometry, and the approximation of data by C^m-smooth functions. He has received numerous accolades, including the European Mathematical Society (EMS) Prize, the Salem Prize, and the Erdős Prize. Before joining Weizmann, he held faculty positions at Princeton University and Tel Aviv University and was a Clay Research Fellow as well as a member of the Institute for Advanced Study. He has also held visiting positions at the Fondation Sciences Mathématiques de Paris (FSMP), the Mathematical Sciences Research Institute (MSRI), the Fields Institute at the University of Toronto, Princeton University, and ETH Zurich.
