The Duffin-Schaeffer Conjecture for multiplicative Diophantine approximation

Given a monotonically decreasing: N 0, ), Khintchine's Theorem provides an efficient tool to decide whether, for almost every R, there are infinitely many (p, q) Z² such that - pq (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: N [0, 1/2 (not necessarily monotonic) and almost every Rᵏ, there exist infinitely many q such that ₈=₁ᵏ ᵢ - pᵢq (q) qᵏ, p₁, , pₖ all coprime to q, if and only if \ₐ ₍ (q) ( (q) q) ᵏ (q (q) (q) ) ^k-1 =. \ This settles a conjecture of Beresnevich, Haynes, and Velani.