The sphere packing problem in dimension \(24\)
type: Item
instance of: scholarly article
title: The sphere packing problem in dimension \(24\)
zbMATH Open document ID: 1370.52037
author: Henry Cohn
author: Abhinav Kumar
author: Stephen D. Miller
author: Danylo V. Radchenko
author: Maryna Viazovska
published in: Annals of Mathematics. Second Series
publication date: 2017-06-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1603.06518
review text: The sphere packing problem is probably the most intriguing problem of discrete geometry. This problem is closely related to geometry, number theory and information theory (eg. optimal codes). Despite the numerous partial results in this field, up to 2016 the final solution was obtained only in dimensions 2 and 3. In [Ann. Math. (2) 185, No. 3, 991--1015 (2017; Zbl 1373.52025)], \textit{M. S. Viazovska} proposed a new breakthrough approach to sphere packing problem and proved that no packing of unit balls in Euclidean space \(\mathbb{R}^8\) has density greater than that of the \(E_8\)-lattice packing. Viazovska's idea was to use modular and quasimodular forms to construct radial eigenfunctions of Fourier transform in \(\mathbb{R}^8\) with eigenvalues \(\pm 1\) and to use some of their linear combination as auxiliary function in the Cohn-Elkies version of the linear programming bound. In this paper, the analogue of Viazovska's approach in the dimension 24 is presented. The main result states that the Leech lattice achieves the optimal sphere packing density in \(\mathbb{R}^{24}\), and it is the only periodic packing in \(\mathbb{R}^{24}\) with that density, up to scaling and isometries.
Mathematics Subject Classification ID: 52C17
zbMATH DE Number: 6731864
zbMATH Keywords: Fourier analysis
zbMATH Keywords: Leech lattice
zbMATH Keywords: sphere packing
zbMATH Keywords: modular forms
reviewed by: Anton Shutov
MaRDI profile type: Publication
OpenAlex ID: W3100495456
Wikidata QID: Q54152779
arXiv ID: 1603.06518
DOI: 10.4007/ANNALS.2017.185.3.8
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description: scientific article