Affiliation:
1. Hungarian Academy of Sciences , [ 119496]Alfred Renyi Institute of Mathematics, Budapest , Hungary
2. Laboratoire Paul Painlevé , [ 129787]Université de Lille, Villeneuve-d’Ascq , France
3. [ 27219]ETH Zürich, Zürich , Switzerland
Abstract
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
{\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}}
.