  Roughly speaking, the problem of geography asks for the existence of
varieties of general type after we fix some invariants. In dimension $1$, where
we fix the genus, the geography question is trivial, but already in dimension
$2$, it becomes a hard problem in general. In higher dimensions, this problem
is essentially wide open. In this paper, we focus on geography in dimension
$3$. We generalize the techniques which compare the geography of surfaces with
the geography of arrangements of curves via asymptotic constructions. In
dimension $2$ this involves resolutions of cyclic quotient singularities and a
certain asymptotic behavior of the associated Dedekind sums and continued
fractions. We discuss the general situation with emphasis on dimension $3$,
analyzing the singularities and various resolutions that show up, and proving
results about the asymptotic behavior of the invariants we fix.
