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

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 $\Lambda$ 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.

21 pages, 3 figures, 8 tables. arXiv admin note: substantial text overlap with arXiv:1907.00595, arXiv:1809.05411, arXiv:1401.6084

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