Optimal Horoball Packing Densities for Koszul-type tilings in Hyperbolic $3$-space
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866917233491443712 |
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| author | Kozma, Robert T. Szirmai, Jenő |
| author_facet | Kozma, Robert T. Szirmai, Jenő |
| contents | We determine the optimal horoball packing densities for Koszul-type Coxeter simplex tilings in hyperbolic $3$-space. Using a parametrization of horoballs by the Busemann function and the symmetry of the tilings, we obtain families of packings that attain the universal simplicial density upper bound \[ d_3(\infty) \;=\; \left( 2 \sqrt{3}\,Λ\!\left(\tfracπ{3}\right) \right)^{-1}
\;\approx\; 0.853276, \] where $Λ$ denotes the Lobachevsky function. These results show that extremal packing densities in $\mathbb{H}^3$ are realized by multiple explicit Coxeter tilings and are closely tied to special values of $L$-functions and hyperbolic manifold volumes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_03945 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Optimal Horoball Packing Densities for Koszul-type tilings in Hyperbolic $3$-space Kozma, Robert T. Szirmai, Jenő Metric Geometry 52C17 52C22 52B15 We determine the optimal horoball packing densities for Koszul-type Coxeter simplex tilings in hyperbolic $3$-space. Using a parametrization of horoballs by the Busemann function and the symmetry of the tilings, we obtain families of packings that attain the universal simplicial density upper bound \[ d_3(\infty) \;=\; \left( 2 \sqrt{3}\,Λ\!\left(\tfracπ{3}\right) \right)^{-1} \;\approx\; 0.853276, \] where $Λ$ denotes the Lobachevsky function. These results show that extremal packing densities in $\mathbb{H}^3$ are realized by multiple explicit Coxeter tilings and are closely tied to special values of $L$-functions and hyperbolic manifold volumes. |
| title | Optimal Horoball Packing Densities for Koszul-type tilings in Hyperbolic $3$-space |
| topic | Metric Geometry 52C17 52C22 52B15 |
| url | https://arxiv.org/abs/2205.03945 |