The sphere packing problem in dimension $24$

Pages 1017-1033 from Volume 185 (2017), Issue 3 by Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna Viazovska

Abstract

Building on Viazovska’s recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska’s function for the eight-dimensional case.

Computer code for verifying the calculations in this paper is is available at the following location:
https://doi.org/10.4007/annals.2017.185.3.8.code

Keywords
, , ,
Mathematical Subject Classification

Primary 2000: ,

DOI
https://doi.org/10.4007/annals.2017.185.3.8
MR
3664817
zbMATH
06731864
Milestones
Received: 23 May 2016
Accepted: 19 December 2016
Published online: 12 April 2017

Authors

Henry Cohn

Microsoft Research New England, Cambridge, MA

Abhinav Kumar

Stony Brook University, Stony Brook, NY

Stephen D. Miller

Rutgers University, Piscataway, NJ

Danylo Radchenko

Max Planck Institute for Mathematics, Bonn, Germany

Current address:

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy Maryna Viazovska

Berlin Mathematical School and Humboldt University of Berlin, Berlin, Germany

Current address:

École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland