Quantitative results on the $k$-dimensional Duffin-Schaeffer conjecture

For all $k\geq 2$, we provide almost-sharp quantitative results for the $k$-dimensional Duffin-Schaeffer conjecture, analogous to recent developments in the 1-D case of Koukoulopoulos-Maynard-Yang. In particular, for $\psi:\mathbb{N}\to[0,1/2]$ such that $\sum_{q\in \mathbb{N}}(\psi(q)\varphi(q)/q)^k$ diverges, $Q\geq 1$ and $\alpha\in\mathbb{R}$, we denote by $S_k(\alpha, Q)$ the number of pairs $(a,q)\in\mathbb{Z}^k\times \mathbb{N}$ with $q\leq Q$, $\gcd(a_i,q)=1$ for each $i\in\{1,\dots,k\}$, satisfying $\|q\alpha-a\|_{\infty}<\psi(q)$. Defining $\Psi_k(Q)=\sum_{q\leq Q}(2\psi(q)\varphi(q)/q)^k$, we show that for all $\varepsilon>0$ and almost all $\alpha$ one has $S_k(\alpha,Q)=\Psi_k(Q)+O_{\varepsilon,k}(\Psi(Q)^{1/2+\varepsilon})$.