One of the main goals of classical metric Diophantine approximation is to quantify the denseness of the rational numbers in the real numbers, or more generally, of Q^d in R^d. An equally natural problem is to quantify the denseness of the rational points on the sphere, and more generally, rational points in other compact and non-compact algebraic sub-varieties in R^d. We will describe a solution to this problem for a large class of homogeneous varieties.

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