The minimal spherical dispersion

TL;DR

This paper establishes improved bounds on the minimal spherical dispersion, revealing that its inverse scales linearly with dimension for fixed dispersion, and provides optimal bounds for expected dispersion of random points on the sphere.

Contribution

It offers new upper and lower bounds on spherical dispersion, refining previous estimates and analyzing the behavior of the inverse and expected dispersion in high dimensions.

Findings

Inverse spherical dispersion is linear in dimension for fixed epsilon.

Bounds on expected dispersion are optimal with respect to epsilon.

Provides tighter estimates compared to previous work by Rote and Tichy.

Abstract

We prove upper and lower bounds on the minimal spherical dispersion, improving upon previous estimates obtained by Rote and Tichy [Spherical dispersion with an application to polygonal approximation of curves, Anz. \"Osterreich. Akad. Wiss. Math.-Natur. Kl. 132 (1995), 3--10]. In particular, we see that the inverse $N(\varepsilon,d)$ of the minimal spherical dispersion is, for fixed $\varepsilon>0$, linear in the dimension $d$ of the ambient space. We also derive upper and lower bounds on the expected dispersion for points chosen independently and uniformly at random from the Euclidean unit sphere. In terms of the corresponding inverse $\widetilde{N}(\varepsilon,d)$, our bounds are optimal with respect to the dependence on $\varepsilon$.