Building on Viazovska's recent solution of the sphere packing problem in
eight dimensions, we prove that the Leech lattice is the densest packing of
congruent spheres in twenty-four dimensions and that it is the unique optimal
periodic packing. In particular, we find an optimal auxiliary function for the
linear programming bounds, which is an analogue of Viazovska's function for the
eight-dimensional case.Comment: 17 page
In this paper we prove the conjecture of Korevaar and Meyers: for each N ≥ c d t d there exists a spherical t-design in the sphere S d consisting of N points, where c d is a constant depending only on d.
We use weakly holomorphic modular forms for the Hecke theta group to construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set {0, ± √
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 f at the radii √ 2n for integers n ≥ 1 in R 8 and n ≥ 2 in 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.
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