The fundamental principle underlying the geometry of numbers is that the number of lattice points in a round, compact subset $R \subset \mathbb{R}^n$ is roughly given by the volume of $R$. A very simple example of this principle is that the number of integers $n$ with $|n| < X$ is $2X$, up to an error of $O(1)$.
Question: Let $S$ be a finite set of places of $\mathbb{Q}$ including the archimedean place. Is there an analogue of the above principle for $S$-integer lattice points?
What I know: For simplicity, let $S = \{\infty, 2\}$. Then it's easy to see that the ring of $S$-integers of $\mathbb{Q}$ is $\mathbb{Z}[1/2]$, and that $\mathbb{Z}[1/2]$ is a lattice in the product $\mathbb{R} \times \mathbb{Q}_2$, where we regard $\mathbb{Z}[1/2]$ as a subset of the product via the obvious inclusions $\mathbb{Z}[1/2] \subset \mathbb{R}$ and $\mathbb{Z}[1/2] \subset \mathbb{Q}_2$. So just like $\mathbb{Z}$ is a lattice in $\mathbb{R}$, we have that $\mathbb{Z}[1/2]$ a lattice in $\mathbb{R} \times \mathbb{Q}_2$.
Why I ask: I'm wondering if such a geometry-of-numbers for $S$-integers could be used to give a direct proof that the class number of the ring of $S$-integers of a number field is finite. This result is usually proven sort of indirectly by observing that the $S$-integer class group is a quotient of the usual class group, and by applying geometry-of-numbers to prove the latter.