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

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

close to satisfying the optimal density, then it is, in a suitable sense, close to the

E 8 {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 {E_{8}}

or

Λ 24 {\Lambda_{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 {E_{8}}

and

Λ 24 {\Lambda_{24}}

.