The Duffin-Schaeffer conjecture with a moving target

Abstract

We prove the inhomogeneous generalization of the Duffin-Schaeffer conjecture in dimension $m \geq 3$. That is, given $\mathbf{y}\in \mathbb{R}^m$ and $\psi:\mathbb{N}\to\mathbb{R}_{\geq 0}$ such that $\sum (\varphi(q)\psi(q)/q)^m = \infty$, we show that for almost every $\mathbf{x} \in\mathbb{R}^m$ there are infinitely many rational vectors $\mathbf{a}/q$ such that $\vert q\mathbf{x} - \mathbf{a} - \mathbf{y}\vert<\psi(q)$ and such that each component of $\mathbf{a}$ is coprime to $q$. This is an inhomogeneous extension of a homogeneous conjecture of Sprind\v{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 $\mathbf{y}$ 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.