  Given a nonnegative function $\psi : \N \to \R $, let $W(\psi)$ denote the
set of real numbers $x$ such that $|nx -a| < \psi(n) $ for infinitely many
reduced rationals $a/n (n>0) $. A consequence of our main result is that
$W(\psi)$ is of full Lebesgue measure if there exists an $\epsilon > 0 $ such
that $$ \textstyle \sum_{n\in\N}(\frac{\psi(n)}{n})^{1+\epsilon}\varphi
(n)=\infty . $$ The Duffin-Schaeffer Conjecture is the corresponding statement
with $\epsilon = 0$ and represents a fundamental unsolved problem in metric
number theory. Another consequence is that $W(\psi)$ is of full Hausdorff
dimension if the above sum with $\epsilon = 0$ diverges; i.e. the dimension
analogue of the Duffin-Schaeffer Conjecture is true.
