Universal optimality of the E_8 and Leech lattices and interpolation formulas

Wednesday, 13 February, 2019

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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-range 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 \widehat{f} at the radii \sqrt{2n} for integers n\ge1 in \mathbb{R}^8 and n\ge2 in \mathbb{R}^{24}. 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.


Henry Cohn
Abhinav Kumar
Stephen D. Miller
Danylo Radchenko
Maryna Viazovska