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

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

Document type :

Pré-publication ou Document de travail

Title :

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

Author(s) :

Böröczky, Károly J. [Auteur]
Alfréd Rényi Institute of Mathematics
Radchenko, Danylo [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Ramos, João P. G. [Auteur]
Department of Mathematics [ETH Zurich] [D-MATH]

Publication date :

2023-03-14

English keyword(s) :

Sphere Packings
Lattice Packings
Stability
Metric Geometry

HAL domain(s) :

Mathématiques [math]
Mathématiques [math]/Géométrie métrique [math.MG]

English abstract : [en]

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 ...
Show more >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 $\Lambda_{24}.$ 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 $\Lambda_{24}.$Show less >

Language :

Anglais

Comment :

24 pages

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