Densities of integer sets represented by quadratic forms

Abstract

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 δ(f)∈Q∩(0,1) such that Df(X)∼δ(f)X and call δ(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.