  A paradigm for a global algebraic number theory of the reals is formulated
with the purpose of providing a unified setting for algebraic and
transcendental number theory. This is achieved through the study of subgroups
of nonstandard models of Dedekind domains called diophantine approximation
groups. The arithmetic of diophantine approximation groups is defined in a way
which extends the ideal-theoretic arithmetic of algebraic number theory, using
the structure of an approximate ideal: a bi-filtration by subgroups along which
partial products may be performed.
