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

TL;DR

The paper addresses the $k$-dimensional Duffin–Schaeffer problem for $k\ge 2$ by establishing a quantitative law of large numbers for the number of good approximations $S_k(\alpha,Q)$ for almost all $\alpha$ under the natural divergence condition $\sum_q(\psi(q)\varphi(q)/q)^k=\infty$. Building on the 1D advances, the authors develop a higher-dimensional variance framework that leverages overlap control and bilinear bounds within a GCD-graph machinery to obtain a near-optimal error term. The main result shows $S_k(\alpha,Q)=\Psi_k(Q)+O_{\varepsilon,k}(\Psi_k(Q)^{1/2+\varepsilon})$ for almost every $\alpha$ as $Q\to\infty$, where $\Psi_k(Q)=\sum_{q\le Q}(2\psi(q)\varphi(q)/q)^k$, matching the best-known 1D exponent in higher dimensions. This work thus extends sharp quantitative Duffin–Schaeffer outcomes from 1D to $k\ge 2$, enabling quasi-independence analyses and reinforcing the broader understanding of metric Diophantine approximation in multiple dimensions.

Abstract

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 $ψ:\mathbb{N}\to[0,1/2]$ such that $\sum_{q\in \mathbb{N}}(ψ(q)\varphi(q)/q)^k$ diverges, $Q\geq 1$ and $α\in\mathbb{R}$, we denote by $S_k(α, Q)$ 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 $\|qα-a\|_{\infty}<ψ(q)$. Defining $Ψ_k(Q)=\sum_{q\leq Q}(2ψ(q)\varphi(q)/q)^k$, we show that for all $\varepsilon>0$ and almost all $α$ one has $S_k(α,Q)=Ψ_k(Q)+O_{\varepsilon,k}(Ψ(Q)^{1/2+\varepsilon})$.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 3.1
• Lemma 3.2
• Proposition 3.3
• proof