  We establish effective counting and equidistribution results for lattice
points in families of domains in hyperbolic spaces, of any dimension and over
any field. The domains we focus on are defined as product sets with respect to
the Iwasawa decomposition. Several classical Diophantine problems can be
reduced to counting lattice points in such domains, including distribution of
shortest solution to the gcd equation, and angular distribution of primitive
vectors in the plane. We give an explicit and effective solution to these
problems, and extend them to imaginary quadratic number fields. Further
applications include counting lifts of closed horospheres to hyperbolic
manifolds and establishing an equidistribution property of integral solutions
to the Diophantine equation defined by a Lorentz form.
