Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid

TL;DR

This work addresses high-dimensional sphere packing by proving the existence of an origin-symmetric ellipsoid with volume $c n^2$ that is free of nonzero lattice points, implying a lattice packing density of at least $c n^2 2^{-n}$. The authors introduce a stochastically evolving ellipsoid driven by Dyson Brownian motion on symmetric matrices, constrained to preserve lattice boundary contact points; they exploit Itô calculus to relate volume growth to the number of boundary contacts and show the ellipsoid can reach a large volume while remaining lattice-free. By averaging over random lattices via the Siegel summation formula, they show the existence of favorable lattices and bound the determinant of the evolving matrix to achieve the target volume, thereby improving the known asymptotic lower bound from $\Theta(n)$ to $\Theta(n^2)$. The approach yields a randomized, practical path to constructing dense high-dimensional lattice packings and deepens understanding of how random lattices interact with evolving convex bodies in high dimensions.

Abstract

We prove that in any dimension $n$ there exists an origin-symmetric ellipsoid ${\mathcal{E}} \subset {\mathbb{R}}^n$ of volume $ c n^2 $ that contains no points of ${\mathbb{Z}}^n$ other than the origin, where $c > 0$ is a universal constant. Equivalently, there exists a lattice sphere packing in ${\mathbb{R}}^n$ whose density is at least $cn^2 \cdot 2^{-n}$. Previously known constructions of sphere packings in ${\mathbb{R}}^n$ yielded densities of at most $C n \log n \cdot 2^{-n}$. 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.

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Corollary 2.4
• proof
• Lemma 3.1