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

Böröczky, Károly (Ifj.) and Radchenko, Danylo and Ramos, Joao P. G. (2024) A quantitative stability result for the sphere packing problem in dimensions 8 and 24. JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK. ISSN 0075-4102

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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 E8 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 E8 or Λ24. Our methods make explicit use of the magic functions constructed in [31] in dimension 8 and in [7] in dimension 24, together with results of independent interest on the abstract stability of the lattices E8 and Λ24.

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