  We study two polynomial counting questions in arithmetic statistics via a
combination of Fourier analytic and arithmetic methods. First, we obtain new
quantitative forms of Hilbert's Irreducibility Theorem for degree $n$
polynomials $f$ with $\mathrm{Gal}(f) \subseteq A_n$. We study this both for
monic polynomials and non-monic polynomials. Second, we study lower bounds on
the number of degree $n$ monic polynomials with almost prime discriminants, as
well as the closely related problem of lower bounds on the number of degree $n$
number fields with almost prime discriminants.
