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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.17749691 |
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Table of Contents:
- <p>We present a derivation establishing that the 4 lattice configuration achieves the maximal<br>density for a packing of congruent spheres in four-dimensional Euclidean space, this density being 2/16.<br>The demonstration utilizes the linear programming bound technique. We detail the mathematical apparatus<br>concerning lattice theory, Fourier analysis on Euclidean spaces, and the theory of quasi-modular forms per-<br>tinent to this specific dimension and lattice structure. We employ an exact interpolation formula applicable<br>to radial functions in the Schwartz space on R4. This formula is derived from the properties of a specific<br>three-dimensional space of quasi-modular forms of weight 2 related to the congruence subgroup Γ0 (4) and<br>its associated symmetries. We construct the specific auxiliary function required for the optimization of the<br>bound, utilizing the basis derived from this interpolation theorem, and verify the necessary conditions on<br>the signs of the function and its Fourier transform, following the methodology established in the literature<br>concerning universal optimality.</p>