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 \[ d_3(\infty) \;=\; \left( 2 \sqrt{3}\,\Lambda\!\left(\tfrac{\pi}{3}\right) \right)^{-1} \;\approx\; 0.853276, \] 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.