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

Abstract

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 $\ddot{I}ˆ:\mathbb{N}\to [0,1/2]$ such that ${\sum }_{q\in \mathbb{N}}(\ddot{I}ˆ(q)\ddot{I}†(q)/q{)}^k$ diverges, $Q\ge 1$ and $\hat{I}\pm \in \mathbb{R}$, we denote by $S_k(\hat{I}\pm ,Q)$ 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 $\Vert q\hat{I}\pm -a{\Vert }_{\infty }<\ddot{I}ˆ(q)$. Defining $\hat{I}{¨}_k(Q)={\sum }_{q\le Q}(2\ddot{I}ˆ(q)\ddot{I}†(q)/q{)}^k$, we show that for all $\epsilon >0$ and almost all $\hat{I}\pm$ one has $S_k(\hat{I}\pm ,Q)=\hat{I}{¨}_k(Q)+O_{\epsilon ,k}(\hat{I}¨(Q{)}^{1/2+\epsilon })$.

10 pages