TITLE: covering by spherical caps
QUESTION [16 upvotes]: Consider the unit sphere $\mathbb{S}^d.$ Pick now some $\alpha$ (I am thinking of $\alpha \ll 1,$ but I don't know how germane this is). The question is: how many spherical caps of angular radius $\alpha$ are needed to cover $\mathbb{S}^d$ completely? There is an obvious bound coming from dividing the volume of the sphere by the volume of the cap, but I am assuming this is far from sharp. I assume that the coding theorists among us have considered this sort of question at great length... One can consider either fixed $d$ or asymptotic results for large $d.$

REPLY [2 votes]: I think that when $d$ is fixed, the first order term when $\alpha\to0$ is given by the case of a suitably renormalized cube of the same dimension $d$. This is what happens when you compute the minimal $L^p$ average distance from a uniform point on a manifold to a set of $N$ points, see Approximation by finitely supported measures.
One important lesson I learned while writing this article is that for this kind of questions, it is worth asking statisticians (see e.g. S. Graf & H. Luschgy Foundations of quantization for probability distributions, Lecture Notes in Mathematics, vol. 1730). Often they work on domains of $\mathbb{R}^d$, but then a cutting method as in the above paper can extend the result to manifolds.
You should get an estimate on the number $N$ of $\alpha$-caps needed to cover $S^d$ of the form $N=\theta(d)\alpha^{-d} + o(\alpha^{-d})$ where $\theta(d)$ is explicit in terms of the volume of $S^d$ and the covering constant of $\mathbb{R}^d$, which is unknown when $d>3$.