I have been puzzling for a while on a problem that I managed to reduce to the following question. Suppose that we have a real random $n$-vector $X$ that is uniformly distributed on the unit sphere in dimension $n$. Let $a$ be a real number. What is, for each $a$, the probability distribution of the number of elements of $X$ that exceed $a$? That is, what is the distribution of $\#\{i : X_i \geq a\}$.
I am especially interested in small $n$, so large-$n$ asymptotics do not seem to be a way out.
My attempts so far:
My initial strategy was to try use the fact that if the $n$-vector $Z$ is i.i.d. normally distributed then $Z / \|Z\|_2$ is uniformly distributed on the unit sphere, so that $X \overset{d}{=} Z / \|Z\|_2$. The idea was then somehow try to use the 'i.i.d.-ness' of $Z$ to relate this to a binomial distribution with $n$ draws and a probability depending on $a$. Unfortunately, this seems like a dead end due to the dependence induced by division through $\|Z\|_2$ if $a \neq 0$.
The special case of $a = 0$
For $a = 0$, this approach does work, as $P(Z_i / \|Z\|_2 \geq 0) = P(Z_i \geq 0) = .5$ (so the $\|Z\|_2$ term and therefore the dependence drops out). As all the $Z_i$'s are independent, this reduces the distribution of interest to a binomial distribution with draws-parameter $n$ and probability-parameter $.5$.
Geometric approach
I also attempted to approach it geometrically, from which it becomes clear that the distribution changes as a crosses certain boundaries depending on $n$. For example, if $a \geq 1$ the distribution is degenerate at 0. Furthermore, if $1 / \sqrt{n} \leq a \leq 1$, then the distribution is supported on (a subset of) $\{0, 1, \dots, n-1\}$, as not all elements can exceed $1 / \sqrt{n}$ since X is a unit vector. I managed to solve the case for $n = 2$ entirely geometrically, but generalising to $n = 3$ is already quite tricky, though a lot of symmetries exist.
Discussion
Overall, the problem is somewhat elegant in that it only uses an $(n-1)$-sphere and $(n-1)$ dimensional hyperplanes positioned by the parameter $a$. So it feels like this may relate to some well-known distribution, and it is very straightforward to obtain draws from the distribution by transforming a normally distributed random vector. However I have so far been unable to classify it.