The Duffin-Schaeffer conjecture with a moving target

We prove the inhomogeneous generalization of the Duffin-Schaeffer conjecture in dimension m ≥ 3. That is, given y∈ Rᵐ and ψ:N≥ 0 such that ∑ (φ(q)ψ(q)/q)ᵐ = ∞, we show that for almost every x ᵐ there are infinitely many rational vectors a/q such that qx - a - y<ψ(q) and such that each component of a is coprime to q. This is an inhomogeneous extension of a homogeneous conjecture of Sprind{z}uk which was itself proved in 1990 by Pollington and Vaughan. In fact, our main result generalizes Pollington-Vaughan not only to the inhomogeneous case, but also to the setting of moving targets, where the inhomogeneous parameter y is free to vary with q. In contrast, we show by an explicit construction that the (1-dimensional) inhomogeneous Duffin-Schaeffer conjecture fails to hold with a moving target, implying that any successful attack on the one-dimensional problem must use the fact that the inhomogeneous parameter is constant. We also introduce new questions regarding moving targets.