Comptes Rendus
Article de recherche - Analyse harmonique
Bounds on hyperbolic sphere packings: on a conjecture by Cohn and Zhao
[Limites des empilements sphériques hyperboliques : à propos d’une conjecture de Cohn et Zhao]
Maximilian Wackenhuth1
1KIT, Germany
Comptes Rendus. Mathématique, Volume 364 (2026), pp. 237-242

We prove sphere packing density bounds in hyperbolic space (and more generally irreducible symmetric spaces of noncompact type), which were conjectured by Cohn and Zhao and generalize Euclidean bounds by Cohn and Elkies. We work within the Bowen–Radin framework of packing density and replace the use of the Poisson summation formula in the proof of the Euclidean bound by Cohn and Elkies with an analogous formula arising from methods used in the theory of mathematical quasicrystals.

Nous démontrons les limites de densité d’empilement sphérique dans l’espace hyperbolique (et plus généralement dans les espaces symétriques irréductibles de type non compact), qui ont été conjecturées par Cohn et Zhao et généralisent les limites euclidiennes de Cohn et Elkies. Nous travaillons dans le cadre de Bowen–Radin sur la densité d’empilement et remplaçons l’utilisation de la formule de sommation de Poisson dans la preuve de la limite euclidienne de Cohn et Elkies par une formule analogue issue des méthodes utilisées dans la théorie des quasi-cristaux mathématiques.

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DOI : 10.5802/crmath.821
Classification : 52C17
Keywords: Sphere packings, harmonic analysis, spherical transform, quasicrystals, point processes
Mots-clés : Empilements sphériques, analyse harmonique, transformation sphérique, quasi-cristaux, processus ponctuels

Maximilian Wackenhuth  1

1 KIT, Germany
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Bounds on hyperbolic sphere packings: on a conjecture by {Cohn} and {Zhao}},
     journal = {Comptes Rendus. Math\'ematique},
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Maximilian Wackenhuth. Bounds on hyperbolic sphere packings: on a conjecture by Cohn and Zhao. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 237-242. doi: 10.5802/crmath.821

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