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

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 {\sim\varepsilon} 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}} .