An almost sharp quantitative version of the Duffin-Schaeffer conjecture

We prove a quantitative version of the Duffin-Schaeffer conjecture with an almost sharp error term. Precisely, let $\psi :\mathbb{N}\to [0,1\mathrm{/}2]$ be a function such that the series ${\sum }_{q=1}^{\mathrm{\infty }}\phi (q)\psi (q)\mathrm{/}q$ diverges. In addition, given $\alpha \in \mathbb{R}$ and $Q⩾1$, let $N(\alpha ;Q)$ be the number of coprime pairs $(a,q)\in \mathbb{Z}\times \mathbb{N}$ with $q⩽Q$ and $\mathrm{\mid }\alpha -a\mathrm{/}q\mathrm{\mid }<\psi (q)\mathrm{/}q$. Lastly, let $\mathrm{\Psi }(Q)={\sum }_{q⩽Q}2\phi (q)\psi (q)\mathrm{/}q$, which is the expected value of $N(\alpha ;Q)$ when $\alpha$ is uniformly chosen from $[0,1]$. We prove that $N(\alpha ;Q)=\mathrm{\Psi }(Q)+O_{\alpha ,\epsilon }(\mathrm{\Psi }(Q{)}^{1\mathrm{/}2+\epsilon })$ for almost all $\alpha$ (in the Lebesgue sense) and for every fixed $\epsilon >0$. This improves upon results of Koukoulopoulos-Maynard and of Aistleitner-Borda-Hauke.