>>805 関連

https://annals.math.princeton.edu/articles/17703
Annals of Mathematics
Universal optimality of the E8 and Leech lattices and interpolation formulas
From to appear in forthcoming issues by Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna Viazovska

Abstract
We prove that the E8 root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions 8 and 24, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently shown optimality as sphere packings, and broadly generalizes it to allow for long-rang interactions.

The proof uses sharp linear programming bounds for energy. To construct the optimal auxiliary functions used to attain these bounds, we prove a new interpolation theorem, which is of independent interest. It reconstructs a radial Schwartz function f from the values and radial derivatives of f and its Fourier transform f^ at the radii 2n--√ for integers n?1 in R8 and n?2 in R24. To prove this theorem, we construct an interpolation basis using integral transforms of quasimodular forms, generalizing Viazovska’s work on sphere packing and placing it in the context of a more conceptual theory.