Stochastically evolving ellipsoids with symmetries

Abstract

We prove that there is a universal constant $c>0$ such that, along an infinite sequence of dimensions $N$, there are lattice sphere packings in ${\mathbb{R}}^N$ of density at least $c{N}^2\log \log N{2}^{-N}$, improving the previous best bound due to Klartag by a $\log \log N$ factor. The proof follows Klartag's stochastic ellipsoid evolution process, subject to the cyclotomic symmetries introduced by Venkatesh.

Statement of AI use included. The first author is using a pseudonym. Chat log is in the ancillary files as a pdf