TY - JOUR

T1 - Universal optimality of the E8 and Leech lattices and interpolation formulas

AU - Cohn, Henry

AU - Kumar, Abhinav

AU - Miller, Stephen D.

AU - Radchenko, Danylo

AU - Viazovska, Maryna

N1 - Funding Information:
Keywords: Fourier interpolation, energy minimization, universal optimality, modular forms AMS Classification: Primary: 52C17; Secondary: 31C20, 82B05. Miller’s research was supported by National Science Foundation grants CNS-1526333 and CNS-1815562, and Viazovska’s research was supported by Swiss National Science Foundation project 184927. The authors thank the Rutgers Office of Advanced Research Computing for their computational support and resources. © 2022 by the authors. This paper may be reproduced, in its entirety, for noncommercial purposes.
Publisher Copyright:
© 2022 by the authors. This paper may be reproduced, in its entirety, for noncommercial purposes.

PY - 2022/11

Y1 - 2022/11

N2 - 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.

AB - 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.

KW - Energy minimization

KW - Fourier interpolation

KW - Modular forms

KW - Universal optimality

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U2 - 10.4007/annals.2022.196.3.3

DO - 10.4007/annals.2022.196.3.3

M3 - Article

AN - SCOPUS:85142235627

SN - 0003-486X

VL - 196

SP - 983

EP - 1082

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 3

ER -