  The Duffin-Schaeffer conjecture is a fundamental unsolved problem in metric
number theory. It asserts that for every non-negative function
$\psi:~\mathbb{N} \rightarrow \mathbb{R}$ for almost all reals $x$ there are
infinitely many coprime solutions $(a,n)$ to the inequality $|nx - a| <
\psi(n)$, provided that the series $\sum_{n=1}^\infty \psi(n) \varphi(n) /n$ is
divergent. In the present paper we prove that the conjecture is true under the
"extra divergence" assumption that divergence of the series still holds when
$\psi(n)$ is replaced by $\psi(n) / (\log n)^\varepsilon$ for some $\varepsilon
> 0$. This improves a result of Beresnevich, Harman, Haynes and Velani, and
solves a problem posed by Haynes, Pollington and Velani.
