Let the surface $S_n$ of the unit ball in $\mathbb{R}^n$ centered at the origin $O$ be defined as the set of points $P(x_1,x_2,…,x_n )$ such that $x_1^2+x_2^2+⋯+x_n^2=1$. Let the spherical cap $C(α)$ of angular radius $\alpha≤π$ centered at $T$ on $S_n$ be defined as the set of all points $Q$ in $S$ such that $∠QOT≤α$. Let $N(α)$ be the maximum number of non-overlapping spherical caps $C(α)$ that can be placed on $S_n$ for $n≥2$.
Is there some function $\Upsilon (n)$ such that $N(α)\sim\Upsilon(n)\alpha^{1-n}$ as $\alpha→0$?
EDIT: An estimate for the first few values of $\Upsilon$ is $\Upsilon(n)\approx1.83,1.46,1.3$ when $n=3,4,5$. The only exact value so far is $\Upsilon(2)=\pi$.