Sphere Packings in Higher Dimension (after Boaz Klartag)

Abstract

Let ${\delta }_n^L$ be the maximal density of a lattice sphere packing in the $n$-dimensional Euclidean space. We explain how Boaz Klartag proved the inequality ${\delta }_n^L\ge c{n}^2{2}^{-n}$ where $c>0$ is a universal constant. In higher dimension, even for non-lattice sphere packings, this new lower bound is a substantial improvement. Klartag's proof uses the probabilistic method in two different ways. The first, very standard, relies on the statistical properties of a uniformly chosen random lattice. The second, completely new, studies the stochastic evolution of an ellipsoid constrained to contain non nonzero lattice points in the interior.

This text is an English translation of the notes prepared (in French) for the Bourbaki seminar given by the author in June 2026