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

2025/04/07 by Boaz Klartag, Bo’az Klartag, Klartag, Boaz · 1 voice · 4 citations
Mathematics · #Mathematical Approximation and Integration #Point processes and geometric inequalities #Stochastic processes and statistical mechanics #math.MG #math.NT #math.PR

paper · pdf · doi:10.48550/arxiv.2504.05042

openalex publication_date 2025/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that in any dimension n there exists an origin-symmetric ellipsoid E ⊂ ℝⁿ of volume c n² that contains no points of ℤⁿ other than the origin, where c > 0 is a universal constant. Equivalently, there exists a lattice sphere packing in ℝⁿ whose density is at least cn² ⋅ 2^-n. Previously known constructions of sphere packings in ℝⁿ yielded densities of at most C n log n ⋅ 2^-n. Our proof utilizes a stochastically evolving ellipsoid that accumulates at least c n² lattice points on its boundary, while containing no lattice points in its interior except for the origin.

Cited by

• A phase transition for the hard sphere model on the hyperbolic plane
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Discussions

• New record for density of sphere packings in ℝ^n. Pretty big breakthrough! arxiv.org/abs/2504.05042 [bsky, 32 points, 0 comments]

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