Bounds on hyperbolic sphere packings: On a conjecture by Cohn and Zhao

TL;DR

This paper establishes new bounds on sphere packings in hyperbolic and symmetric spaces, confirming a conjecture by Cohn and Zhao and extending Euclidean packing bounds using novel mathematical techniques.

Contribution

It proves hyperbolic sphere packing bounds conjectured by Cohn and Zhao, generalizing Euclidean bounds with a new approach based on quasicrystal methods.

Findings

Proved hyperbolic sphere packing density bounds

Generalized Euclidean packing bounds to noncompact symmetric spaces

Introduced a novel mathematical framework replacing Poisson summation

Abstract

We prove sphere packing density bounds in hyperbolic space (and more generally irreducible symmetric spaces of noncompact type), which were conjectured by Cohn and Zhao and generalize Euclidean bounds by Cohn and Elkies. We work within the Bowen-Radin framework of packing density and replace the use of the Poisson summation formula in the proof of the Euclidean bound by Cohn and Elkies with an analogous formula arising from methods used in the theory of mathematical quasicrystals.