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

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.