Minimal dispersion on the sphere

TL;DR

This paper analyzes minimal spherical cap dispersion on the unit sphere S^d, relating it to sphere coverings and convex-body approximation. It establishes precise links between dispersion, covering radius, and covering density, obtaining asymptotics lim_{n→∞} n·disp_C(n,d)=θ_d and providing both upper and lower bounds, including a sublogarithmic ln ln improvement for moderate n. The authors develop two complementary frameworks to bound dispersion: (i) geodesic-ε-net methods for constructive upper bounds, and (ii) VC-dimension and ε-traversal techniques for probabilistic bounds, yielding bounds for caps and lens intersections and extending to k-fold intersections. Collectively, the work clarifies the dispersion-covering-density landscape on spheres, connects it to inscribed-polytope approximation, and delivers actionable bounds in high dimensions with implications for spherical coverings and discrepancy theory.

Abstract

The minimal spherical cap dispersion ${\rm disp}_{\mathcal{C}}(n,d)$ is the largest number $\varepsilon\in (0,1]$ such that, for every $n$ points on the $d$-dimensional Euclidean unit sphere $\mathbb{S}^d$, there exists a spherical cap with normalized area $\varepsilon$ not containing any of these points. We study the behavior of ${\rm disp}_{\mathcal{C}}(n,d)$ as $n$ and $d$ grow to infinity. We develop connections to the problems of sphere covering and approximation of the Euclidean unit ball by inscribed polytopes. Existing and new results are presented in a unified way. Upper bounds on ${\rm disp}_{\mathcal{C}}(n,d)$ result from choosing the points independently and uniformly at random and possibly adding some well-separated points to close large gaps. Moreover, we study dispersion with respect to intersections of caps.

Theorems & Definitions (58)

• Lemma 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Proposition 1.5
• Proposition 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9
• Remark 1.10