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Rational Approximation on Spheres

We quantify the density of rational points in the unit sphere $S^n$, proving analogues of the classical theorems on the embedding of $\q^n$ into $\r^n$. Specifically, we prove a Dirichlet theorem stating that every point $α\in S^n$ is sufficiently approximable, the optimality of this approximation via the existence of badly approximable points, and a Khintchine theorem showing that the Lebesgue measure of approximable points is either zero or full depending on the convergence or divergence of a certain sum. These results complement and improve on previous results, particularly recent theorems of Ghosh, Gorodnik and Nevo.