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

  • Károly J. Böröczky ORCID logo EMAIL logo , Danylo Radchenko und João P. G. Ramos
Veröffentlicht/Copyright: 30. Januar 2024

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 ε close to satisfying the optimal density, then it is, in a suitable sense, 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 in [M. S. Viazovska, The sphere packing problem in dimension 8, Ann. of Math. (2) 185 2017, 3, 991–1015] in dimension 8 and in [H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. Viazovska, The sphere packing problem in dimension 24, Ann. of Math. (2) 185 2017, 3, 1017–1033] in dimension 24, together with results of independent interest on the abstract stability of the lattices E 8 and Λ 24 .

Funding statement: J. P. G. Ramos acknowledges support by the ERC grant RSPDE 721675, and K. J. Böröczky ackowledges the hospitality of ETH Zurich where part of the research was done and the support by NKFIH grant 132002.

Acknowledgements

All authors would like to express their deepest gratitude towards the anonymous referee for indicating how to prove the current of version of Theorems 1.3 and 1.4 through adapting the techniques from [6].

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Received: 2023-02-27
Revised: 2023-07-14
Published Online: 2024-01-30
Published in Print: 2024-03-01

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