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arXiv · 2205.03945

Optimal Horoball Packing Densities for Koszul-type tilings in Hyperbolic $3$-space

Robert T. Kozma·Jenő Szirmai
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Abstract

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.

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Robert T. Kozma, Jenő Szirmai. 2025-10-30. Optimal Horoball Packing Densities for Koszul-type tilings in Hyperbolic $3$-space. https://doi.org/10.1007/s10474-026-01581-3

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