The Duffin-Schaeffer Conjecture for multiplicative Diophantine approximation

Given a monotonically decreasing ψ: N → [0,∞), Khintchine's Theorem provides an efficient tool to decide whether, for almost every α ∈ R, there are infinitely many (p,q) ∈ Z² such that α - p/q ≤ ψ(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≥ 1, any function ψ: N → [0,1/2] (not necessarily monotonic) and almost every α ∈ Rᵏ, there exist infinitely many q such that ∏ᵢ₌₁ᵏ αᵢ - pᵢ/q ≤ ψ(q)/qᵏ, p₁,…,pₖ all coprime to q, if and only if \[∑q ∈ N ψ(q) (φ(q)/q )^klog (q/φ(q)ψ(q))ᵏ⁻¹ = ∞.\] This settles a conjecture of Beresnevich, Haynes, and Velani.