A cyclotomic gluing construction for lattice sphere packings

Abstract

For a positive integer $d$, let ${\Delta }_d^L$ denote the supremum of the densities of lattice sphere packings in ${\mathbb{R}}^d$. We prove that, for every sufficiently large $d$, ${\Delta }_d^L\gg d^2\log \log d{2}^{-d}$. Abuya-Gargava-Zhao obtained the same bound along a sparse sequence of dimensions by combining Venkatesh's cyclotomic symmetries with Klartag's stochastic ellipsoid method. We introduce a multi-cyclotomic gluing construction that allows this argument to work in every sufficiently large dimension.

arXiv comment39 pages, 3 figures