$S_n$-extensions with prescribed norms

Given a number field $k$, a finitely generated subgroup $\mathcal{A}\subseteq k^\times$, and an integer $n\geq 3$, we study the distribution of $S_n$-extensions of $k$ such that the elements of $\mathcal{A}$ are norms. For $n\leq 5$, and conjecturally for $n \geq 6$, we show that the density of such extensions is the product of so-called ``local masses'' at the places of $k$. When $n$ is an odd prime, we give formulas for these local masses, allowing us to express the aforementioned density as an explicit Euler product. For $n=4$, we determine almost all of these masses exactly and give an efficient algorithm for computing the rest, again yielding an explicit Euler product.