Universal optimality of the E8 and Leech lattices and interpolation formulas

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

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


We prove that the E8 root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions eight and twenty-four, 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 b f at the radii p 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.

Original languageEnglish (US)
Pages (from-to)983-1082
Number of pages100
JournalAnnals of Mathematics
Issue number3
StatePublished - Nov 2022
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)


  • Energy minimization
  • Fourier interpolation
  • Modular forms
  • Universal optimality


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