Linear programming bounds in homogeneous spaces, I: Optimal packing density

TL;DR

This work extends the classical Euclidean linear programming bounds for sphere packing to a broad, uniform framework of commutative spaces given by Gelfand pairs $(G,K)$. By combining harmonic analysis (spherical transforms, Bochner–Schwartz theory) with stochastic geometry (Bowen–Radin densities for random invariant packings), the authors derive general density bounds for packings in non-Euclidean geometries, including hyperbolic space and Heisenberg groups. A key achievement is a general bound of the form $\triangle(r,X) \le m_X(B(x_0,r))\frac{f(e)}{\widehat{f}(\mathbf{1})}$ for witness functions in $\mathcal{W}(r,G/K)$, which recovers the Euclidean LP bound and proves a hyperbolic-space conjecture of Cohn and Zhao without relying on periodic approximations. The paper develops a unified framework that covers Euclidean, hyperbolic, Riemannian symmetric, and nilpotent settings, providing explicit formulations in several examples and establishing a robust probabilistic–analytic bridge between random and deterministic sphere packings with broad implications for noncompact and compact symmetric spaces. Overall, the results significantly broaden the applicability of LP-type bounds in discrete geometry and connect them to deep structures in harmonic analysis and ergodic theory.

Abstract

In this article we obtain linear programming bounds for the maximal sphere packing density of commutative spaces. A special case of our results solves a conjecture by Cohn and Zhao on linear programming bounds for sphere packings in hyperbolic space.

Theorems & Definitions (91)

• Theorem A
• Remark 1.1
• Example 1.2
• Definition 1.3
• Theorem B
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Godement, Godement
• Definition 2.4
• Lemma 2.5