Effective estimate and Central Limit Theorem for Diophantine approximation on spheres

We study the counting function of rational approximations with given bounds on the denominator and satisfying the critical Dirichlet exponent on the sphere $S^d$, $d\geq 3$. We give an effective estimate for this counting function, with an error term of square root order, analogous to the optimal estimate in the Euclidean setting. We also show that the counting function has vanishing third and higher correlations and derive a Central Limit Theorem describing its fluctuations. We prove these results using arguments from homogeneous dynamics on the space of orthogonal lattices, in particular effective multiple equidistribution of all orders, which we establish for spherical averages and which could be useful for other applications.