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 $\sim \varepsilon$ close to satisfying the optimal density, then it is, in a suitable sense, $O(\varepsilon^{1/2})$ 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 $\Lambda_{24}.$ Our methods make explicit use of the magic functions constructed by M. Viazovska in dimension 8 and by H. Cohn, A. Kumar, S. Miller, the second author, and M. Viazovska in dimension 24, together with results of independent interest on the abstract stability of the lattices $E_8$ and $\Lambda_{24}.$