We prove a Central Limit Theorem and an effective estimate for the counting function of Diophantine approximants on the sphere S$^n$ using homogeneous dynamics on the space of orthogonal lattices.
Diophantine approximation on subspaces of $\mathbb{R}^n$ and dynamics on homogeneous spaces
A Central Limit Theorem for Counting Functions Related to Symplectic Lattices and Bounded Sets
Exceptional sets in dynamical systems and Diophantine approximation
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