The Duffin-Schaeffer Conjecture for multiplicative Diophantine approximation

Abstract

Given a monotonically decreasing $\psi: \mathbb{N} \to [0,\infty)$, Khintchine's Theorem provides an efficient tool to decide whether, for almost every $\alpha \in \mathbb{R}$, there are infinitely many $(p,q) \in \mathbb{Z}^2$ such that $\left\lvert \alpha - \frac{p}{q}\right\rvert \leq \frac{\psi(q)}{q}$. The recent result of Koukoulopoulos and Maynard provides an elegant way of removing monotonicity when only counting reduced fractions. Gallagher showed a multiplicative higher-dimensional generalization to Khintchine's Theorem, again assuming monotonicity. In this article, we prove the following Duffin-Schaeffer-type result for multiplicative approximations: For any $k\geq 1$, any function $\psi: \mathbb{N} \to [0,1/2]$ (not necessarily monotonic) and almost every $\alpha \in \mathbb{R}^k$, there exist infinitely many $q$ such that $\prod\limits_{i=1}^k \left\lvert \alpha_i - \frac{p_i}{q}\right\rvert \leq \frac{\psi(q)}{q^k}, p_1,\ldots,p_k$ all coprime to $q$, if and only if \[\sum\limits_{q \in \mathbb{N}} \psi(q) \left(\frac{\varphi(q)}{q} \right)^k\log \left(\frac{q}{\varphi(q)\psi(q)}\right)^{k-1} = \infty.\] This settles a conjecture of Beresnevich, Haynes, and Velani.