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| Format: | Recurso digital |
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Zenodo
2025
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| Online Access: | https://doi.org/10.5281/zenodo.17749352 |
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Table of 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>