November 2022 Universal optimality of the $E_8$ and Leech lattices and interpolation formulas
Henry Cohn, Abhinav Kumar, Stephen Miller, Danylo Radchenko, Maryna Viazovska
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Ann. of Math. (2) 196(3): 983-1082 (November 2022). DOI: 10.4007/annals.2022.196.3.3


We prove that the $E_8$ 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 $\hat{f}$ at the radii $\sqrt{2n}$ for integers $n\ge 1$ in $\mathbb{R}^8$ and $n\ge 2$ 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.


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Henry Cohn. Abhinav Kumar. Stephen Miller. Danylo Radchenko. Maryna Viazovska. "Universal optimality of the $E_8$ and Leech lattices and interpolation formulas." Ann. of Math. (2) 196 (3) 983 - 1082, November 2022.


Published: November 2022
First available in Project Euclid: 30 October 2022

Digital Object Identifier: 10.4007/annals.2022.196.3.3

Primary: 52C17
Secondary: 31C20 , 82B05

Keywords: energy minimization , Fourier interpolation , modular forms , universal optimality

Rights: Copyright © 2022 Department of Mathematics, Princeton University


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Vol.196 • No. 3 • November 2022
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