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.

💡 Research Summary

The paper establishes new upper bounds for sphere‑packing densities in hyperbolic space and, more generally, in any irreducible symmetric space of non‑compact type. The authors work within the Bowen–Radin framework of “nice” packings, which defines density via ergodic invariant random packings (r‑IRPs). In Euclidean space, Cohn and Elkies obtained linear‑programming bounds by applying the Poisson summation formula to periodic packings and then extending the result to all packings. In hyperbolic space, however, the lack of a suitable Poisson formula and the unresolved question of whether optimal densities can be approximated by periodic packings prevent a direct transfer of that method.

To overcome these obstacles, the authors replace the Poisson summation argument with tools from the theory of mathematical quasicrystals. They introduce the autocorrelation measure (\eta_\mu^+) and the reduced autocorrelation (\eta_\mu) associated with an r‑IRP (\mu). Both measures are shown to be positive‑definite, which implies that their spherical transforms are positive Borel measures on the spherical spectrum of the underlying Lie group ((G,K)). By the Plancherel theorem, for any rapidly decreasing radial function (f) on the symmetric space (more precisely a Harish‑Chandra (L^1)‑Schwartz function) with non‑negative spherical transform (\widehat f) and (\widehat f(1)>0), one obtains the inequality \