Universal optimality of the $E_8$ and Leech lattices and interpolation formulas Article Swipe
Henry Cohn
,
Abhinav Kumar
,
Stephen D. Miller
,
Danylo Radchenko
,
Maryna Viazovska
·
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.4007/annals.2022.196.3.3
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.4007/annals.2022.196.3.3
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 longrange 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 fb at the radii root 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.
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- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.4007/annals.2022.196.3.3
- OA Status
- green
- Cited By
- 72
- References
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- OpenAlex ID
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https://openalex.org/W2913291429Canonical identifier for this work in OpenAlex
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https://doi.org/10.4007/annals.2022.196.3.3Digital Object Identifier
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Universal optimality of the $E_8$ and Leech lattices and interpolation formulasWork title
- Type
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articleOpenAlex work type
- Language
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enPrimary language
- Publication year
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2022Year of publication
- Publication date
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2022-10-31Full publication date if available
- Authors
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Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna ViazovskaList of authors in order
- Landing page
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https://doi.org/10.4007/annals.2022.196.3.3Publisher landing page
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YesWhether a free full text is available
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greenOpen access status per OpenAlex
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https://infoscience.epfl.ch/record/298718Direct OA link when available
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Mathematics, Euclidean geometry, Lattice (music), Combinatorics, Interpolation (computer graphics), Euclidean space, Discrete mathematics, Geometry, Physics, Computer science, Animation, Acoustics, Computer graphics (images)Top concepts (fields/topics) attached by OpenAlex
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72Total citation count in OpenAlex
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2025: 9, 2024: 10, 2023: 16, 2022: 9, 2021: 10Per-year citation counts (last 5 years)
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10Other works algorithmically related by OpenAlex
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| publication_date | 2022-10-31 |
| publication_year | 2022 |
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