  Metric Diophantine approximation in its classical form is the study of how
well almost all real numbers can be approximated by rationals. There is a long
history of results which give partial answers to this problem, but there are
still questions which remain unknown. The Duffin-Schaeffer Conjecture is an
attempt to answer all of these questions in full, and it has withstood more
than fifty years of mathematical investigation. In this paper we establish a
strong connection between the Duffin-Schaeffer Conjecture and its p-adic
analogue. Our main theorems are transfer principles which allow us to go back
and forth between these two problems. We prove that if the variance method from
probability theory can be used to solve the p-adic Duffin-Schaeffer Conjecture
for even one prime p, then almost the entire classical Duffin-Schaeffer
Conjecture would follow. Conversely if the variance method can be used to prove
the classical conjecture then the p-adic conjecture is true for all primes.
Furthermore we are able to unconditionally and completely establish the higher
dimensional analogue of this conjecture in which we allow simultaneous
approximation in any finite number and combination of real and p-adic fields,
as long as the total number of fields involved is greater than one. Finally by
using a mass transference principle for Hausdorff measures we are able to
extend all of our results to their corresponding analogues with Haar measures
replaced by the Hausdorff measures associated with arbitrary dimension
functions.
