Decoupling theorems for the Duffin-Schaeffer problem

Abstract

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.