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

$$d_3(\infty) = \big( 2 \sqrt{3} \Lambda\big(\tfrac{\pi}{3}\big) \! \big)^{-1} \approx 0.853276,$$

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.