The Duffin-Schaeffer Conjecture for multiplicative Diophantine approximation

Given a monotonically decreasing $\psi :\mathbb{N}\to [0,\mathrm{\infty })$, 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 $\mid \alpha -\frac{p}{q}\mid \le \frac{\psi (q)}{q}$. 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 $k\ge 1$, any function $\psi :\mathbb{N}\to [0,1\mathrm{/}2]$ (not necessarily monotonic) and almost every $\alpha \in {\mathbb{R}}^k$, there exist infinitely many $q$ such that ${\prod }_{i=1}^k\mid {\alpha }_i-\frac{p_i}{q}\mid \le \frac{\psi (q)}{q^k},p_1,\dots ,p_k$ all coprime to $q$, if and only if $$\sum _{q\in \mathbb{N}}\psi (q){\left(\frac{\phi (q)}{q}\right)}^k\log {\left(\frac{q}{\phi (q)\psi (q)}\right)}^{k-1}=\mathrm{\infty }\mathrm{.}$$ This settles a conjecture of Beresnevich, Haynes, and Velani.