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

23 pages, 3 figures, comments are welcome!