An almost sharp quantitative version of the Duffin-Schaeffer conjecture

Abstract

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

41 pages. Final version. To appear in Duke Math. J. arXiv admin note: text overlap with arXiv:1907.04593