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arXiv · 2505.10929

Minimal dispersion on the sphere

Alexander E. Litvak·Mathias Sonnleitner·Tomasz Szczepanski
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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.

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Alexander E. Litvak, Mathias Sonnleitner, Tomasz Szczepanski. 2025-05-16. Minimal dispersion on the sphere. https://arxiv.org/abs/2505.10929

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