Sphere Packings in Higher Dimension (after Boaz Klartag)

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.