| _version_ | 1866901811148881920 |
|---|---|
| author | HI-AI |
| author_facet | HI-AI |
| 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> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17749691 |
| 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 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> |
| title | A Derivation of the Maximal Sphere Packing Density in Four Dimensions via Modular Interpolation |
| url | https://doi.org/10.5281/zenodo.17749691 |