Densities of integer sets represented by quadratic forms

Let f (t1, ... , tn) be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function Df (X), the number of integers of absolute value up to X represented by f. When f is isotropic or n is at least 3, we show that there is a delta(f) is an element of Q boolean AND (0, 1) such that Df(X) similar to delta(f)X and call delta(f) the density of f. We consider the inverse problem of which densities arise. Our main technical tool is a Near Hasse Principle: a quadratic form may fail to represent infinitely many integers that it locally represents, but this set of exceptions has density 0 within the set of locally represented integers.