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| Format: | Recurso digital |
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Zenodo
2025
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| Accès en ligne: | https://doi.org/10.5281/zenodo.17749352 |
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| _version_ | 1866901816400150528 |
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| author | HI-AI |
| author_facet | HI-AI |
| contents | <p>We present a complete derivation establishing that the 4 lattice configuration achieves the<br>maximal density for a packing of congruent spheres in four-dimensional Euclidean space, this density be-<br>ing 2/16. The demonstration utilizes the linear programming bound methodology introduced by Cohn<br>and Elkies. We detail the necessary mathematical apparatus concerning lattice theory in Euclidean spaces,<br>Fourier analysis including the Poisson summation formula, and the requisite interpolation theorems derived<br>from the theory of quasi-modular forms pertinent to the specific dimension = 4 and the 4 lattice struc-<br>ture. We construct the specific auxiliary function required for the optimization of the bound, utilizing the<br>interpolation theorem specialized for the 4 lattice symmetries. We verify the necessary conditions on the<br>signs of the function and its Fourier transform, employing the variation diminishing property associated<br>with expansions in the basis of Laguerre polynomials.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17749352 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Derivation of the Maximal Sphere Packing Density in Four Dimensions via Modular Interpolation HI-AI <p>We present a complete derivation establishing that the 4 lattice configuration achieves the<br>maximal density for a packing of congruent spheres in four-dimensional Euclidean space, this density be-<br>ing 2/16. The demonstration utilizes the linear programming bound methodology introduced by Cohn<br>and Elkies. We detail the necessary mathematical apparatus concerning lattice theory in Euclidean spaces,<br>Fourier analysis including the Poisson summation formula, and the requisite interpolation theorems derived<br>from the theory of quasi-modular forms pertinent to the specific dimension = 4 and the 4 lattice struc-<br>ture. We construct the specific auxiliary function required for the optimization of the bound, utilizing the<br>interpolation theorem specialized for the 4 lattice symmetries. We verify the necessary conditions on the<br>signs of the function and its Fourier transform, employing the variation diminishing property associated<br>with expansions in the basis of Laguerre polynomials.</p> |
| title | A Derivation of the Maximal Sphere Packing Density in Four Dimensions via Modular Interpolation |
| url | https://doi.org/10.5281/zenodo.17749352 |