  The Duffin-Schaeffer conjecture is a central open problem in metric number
theory. Let $\psi~\mathbb{N} \mapsto \mathbb{R}$ be a non-negative function,
and set $\mathcal{E}_n :=\bigcup \left( \frac{a -
\psi(n)}{n},\frac{a+\psi(n)}{n} \right)$, where the union is taken over all $a
\in \{1, \dots, n\}$ which are co-prime to $n$. Then the conjecture asserts
that almost all $x \in [0,1]$ are contained in infinitely many sets
$\mathcal{E}_n$, provided that the series of the measures of $\mathcal{E}_n$ is
divergent. At the core of the conjecture is the problem of controlling the
measure of the pairwise overlaps $\mathcal{E}_m \cap \mathcal{E}_n$, in
dependence on $m, n, \psi(m)$ and $\psi(n)$. In the present paper we prove
upper bounds for the measures of these overlaps, which show that globally the
degree of dependence in the set system $(\mathcal{E}_n)_{n \geq 1}$ is
significantly smaller than supposed. As applications, we obtain significantly
improved "extra divergence" and "slow divergence" variants of the
Duffin-Schaeffer conjecture.
