Extreme value laws for intrinsic Diophantine approximation on spheres

We study intrinsic Diophantine approximation on the sphere $S^d$, measured by the quality of the best rational approximation available at a given height. We show that, for $d\ge 3$ and along sufficiently sparse sequences of heights, these quantities have a Weibull limit distribution and the associated counting functions are asymptotically Poissonian.