Levels of distribution for sieve problems in prehomogeneous vector spaces

In a companion paper, we developed an efficient algebraic method for computing the Fourier transforms of certain functions defined on prehomogeneous vector spaces over finite fields, and we carried out these computations in a variety of cases. Here we develop a method, based on Fourier analysis and algebraic geometry, which exploits these Fourier transform formulas to yield level of distribution results, in the sense of analytic number theory. Such results are of the shape typically required for a variety of sieve methods. As an example of such an application we prove that there are $\gg$ X/log(X) quartic fields whose discriminant is squarefree, bounded above by X, and has at most eight prime factors.