Abstract
We quantify the density of rational points in the unit sphere S (n) , proving analogues of the classical theorems on the embedding of a"e (n) into a"e (n) . Specifically, we prove a Dirichlet theorem stating that every point alpha a 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.