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

For all $k\ge 2$, 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 $\psi :\mathbb{N}\to [0,1\mathrm{/}2]$ such that ${\sum }_{q\in \mathbb{N}}(\psi (q)\phi (q)\mathrm{/}q{)}^k$ diverges, $Q\ge 1$ 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 $q\le Q$, $\gcd (a_i,q)=1$ for each $i\in \{1,\dots ,k\}$, satisfying $\mathrm{\parallel }q\alpha -a{\mathrm{\parallel }}_{\mathrm{\infty }}<\psi (q)$. Defining ${\mathrm{\Psi }}_k(Q)={\sum }_{q\le Q}(2\psi (q)\phi (q)\mathrm{/}q{)}^k$, we show that for all $\epsilon >0$ and almost all $\alpha$ one has $S_k(\alpha ,Q)={\mathrm{\Psi }}_k(Q)+O_{\epsilon ,k}(\mathrm{\Psi }(Q{)}^{1\mathrm{/}2+\epsilon })$.