Geometry and Arithmetic of Crystallographic Sphere Packings

Alex Kontorovich; Kei Nakamura

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arXiv · 1712.00147

Geometry and Arithmetic of Crystallographic Sphere Packings

Alex Kontorovich·Kei Nakamura

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Abstract

We introduce the notion of a "crystallographic sphere packing," defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit for the first time an infinite family of conformally-inequivalent such with all radii being reciprocals of integers. We then prove a result in the opposite direction: the "superintegral" ones exist only in finitely many "commensurability classes," all in dimensions below 30.

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Alex Kontorovich, Kei Nakamura. 2017-12-01. Geometry and Arithmetic of Crystallographic Sphere Packings. https://arxiv.org/abs/1712.00147

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