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A quantitative stability result for the sphere packing problem in dimensions 8 and 24

Károly J. Böröczky, Danylo Radchenko, João P. G. Ramos

TL;DR

The paper proves explicit quantitative stability for sphere packings in dimensions $n=8$ and $n=24$, showing that near-optimal lattice packings are $O(\varepsilon^{1/2})$-close (up to isometry) to the $E_8$ and Leech lattices, respectively, and that near-optimal periodic packings admit a large frame in which the local arrangement matches $E_8$ or $\Lambda_{24}$ up to a small Hausdorff error. The approach combines Viazovska's magic functions $g_8$ and $g_{24}$ with a detailed stability analysis of lattices and a probabilistic framing for periodic packings, including a robust basis-approximation toolkit based on near-integer Gram matrices and Cholesky factor stability. The authors extend the core results to bin packings and general packings, obtaining parallel stability statements and showing that almost-optimal configurations are rigidly modeled by the corresponding exceptional lattices at large scales. These results provide rigorous, computable rigidity statements for the densest packings in these dimensions and pave the way for precise structural characterizations of near-optimal packings in high dimensions.

Abstract

We prove explicit stability estimates for the sphere packing problem in dimensions 8 and 24, showing that, in the lattice case, if a lattice is $\sim \varepsilon$ close to satisfying the optimal density, then it is, in a suitable sense, $O(\varepsilon^{1/2})$ close to the $E_8$ and Leech lattices, respectively. In the periodic setting, we prove that, under the same assumptions, we may take a large 'frame' through which our packing locally looks like $E_8$ or $Λ_{24}.$ Our methods make explicit use of the magic functions constructed by M. Viazovska in dimension 8 and by H. Cohn, A. Kumar, S. Miller, the second author, and M. Viazovska in dimension 24, together with results of independent interest on the abstract stability of the lattices $E_8$ and $Λ_{24}.$

A quantitative stability result for the sphere packing problem in dimensions 8 and 24

TL;DR

The paper proves explicit quantitative stability for sphere packings in dimensions and , showing that near-optimal lattice packings are -close (up to isometry) to the and Leech lattices, respectively, and that near-optimal periodic packings admit a large frame in which the local arrangement matches or up to a small Hausdorff error. The approach combines Viazovska's magic functions and with a detailed stability analysis of lattices and a probabilistic framing for periodic packings, including a robust basis-approximation toolkit based on near-integer Gram matrices and Cholesky factor stability. The authors extend the core results to bin packings and general packings, obtaining parallel stability statements and showing that almost-optimal configurations are rigidly modeled by the corresponding exceptional lattices at large scales. These results provide rigorous, computable rigidity statements for the densest packings in these dimensions and pave the way for precise structural characterizations of near-optimal packings in high dimensions.

Abstract

We prove explicit stability estimates for the sphere packing problem in dimensions 8 and 24, showing that, in the lattice case, if a lattice is close to satisfying the optimal density, then it is, in a suitable sense, close to the and Leech lattices, respectively. In the periodic setting, we prove that, under the same assumptions, we may take a large 'frame' through which our packing locally looks like or Our methods make explicit use of the magic functions constructed by M. Viazovska in dimension 8 and by H. Cohn, A. Kumar, S. Miller, the second author, and M. Viazovska in dimension 24, together with results of independent interest on the abstract stability of the lattices and
Paper Structure (8 sections, 20 theorems, 148 equations)

This paper contains 8 sections, 20 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and properties of packings
  4. Properties of the functions $g_8$ and $g_{24}$
  5. Approximating a basis of $\mathbb{R}^n$
  6. Stability for lattice packings
  7. Stability for periodic packings
  8. Corollaries about bin packings and general packings

Key Result

Theorem 1.1

If $\Xi+\frac{\sqrt{2}}{2}\,B^8$ is a periodic packing in $\mathbb{R}^8$, then its center density is at most $1$, with equality if and only if $\Xi$ is congruent to $E_8$.

Theorems & Definitions (30)

  • Theorem 1.1: Viazovska
  • Theorem 1.2: Cohn, Kumar, Miller, Radchenko, Viazovska
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Example 1.8
  • Proposition 2.1
  • Lemma 2.2
  • ...and 20 more