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 $d.$ 



covering by spherical caps
There exist coverings such that each point is covered at most $400 d \log d$ times, and you can improve this bound a little if you look at the covering density, i.e., the average number of times each point is covered. See the "Covering the sphere by equal spherical balls" by Boroczky and Wintsche (available at <http://www.renyi.hu/~carlos/spherecover.ps>) and Chapter 6 of Boroczky's book "Finite packing and covering". In the other direction, it is widely believed that the covering density grows at least linearly in $d$, but I don't think this has been proved. It's listed as an open problem on page 199 of Boroczky's book, which was presumably up to date when it was published in 2004.


