  New results towards the Duffin-Schaeffer conjecture, which is a fundamental
unsolved problem in metric number theory, have been established recently
assuming extra divergence. Given a non-negative function $\psi:
\mathbb{N}\to\mathbb{R}$ we denote by $W(\psi)$ the set of all $x\in\mathbb{R}$
such that $|nx-a|<\psi(n)$ for infinitely many $a,n$. Analogously, denote
$W'(\psi)$ if we additionally require $a,n$ to be coprime. Aistleitner et al.
[1] proved that $W'(\psi)$ is of full Lebesgue measure if there exist an
$\varepsilon>0$ such that $\sum_{n=2}^\infty\psi(n)\varphi(n)/(n(\log
n)^\varepsilon)=\infty$. This result seems to be the best one can expect from
the method used. Assuming the extra divergence $\sum_{n=2}^\infty\psi(n)/(\log
n)^\varepsilon=\infty$ we prove that $W(\psi)$ is of full measure. This could
also be deduced from the result in [1], but we believe that our proof is of
independent interest, since its method is totally different from the one in
[1]. As a further application of our method, we prove that a variant of
Khintchine's theorem is true without monotonicity, subject to an additional
condition on the set of divisors of the support of $\psi$.
