An almost sharp quantitative version of the Duffin-Schaeffer conjecture

2024/04/22 by Koukoulopoulos, Dimitris, Maynard, James, Yang, Daodao · 3 citations
#FOS: Mathematics #Number Theory (math.NT) #Primary: 11J83. Secondary: 05C40

paper · doi:10.48550/arxiv.2404.14628

Abstract

We prove a quantitative version of the Duffin-Schaeffer conjecture with an almost sharp error term. Precisely, let ψ:ℕ→[0,1/2] be a function such that the series ∑_q=1^∞ φ(q)ψ(q)/q diverges. In addition, given α∈ℝ and Q\geqslant1, let N(α;Q) be the number of coprime pairs (a,q)∈ℤ×ℕ with q\leqslant Q and |α-a/q|<ψ(q)/q. Lastly, let Ψ(Q)=∑_{q\leqslant Q}2φ(q)ψ(q)/q, which is the expected value of N(α;Q) when α is uniformly chosen from [0, 1]. We prove that N(α;Q)=Ψ(Q)+O_α,ε(Ψ(Q)^1/2+ε) for almost all α (in the Lebesgue sense) and for every fixed ε>0. This improves upon results of Koukoulopoulos-Maynard and of Aistleitner-Borda-Hauke.

Cited by

• Twisted Diophantine approximation for matrix transformations of tori
• Toward Khintchine's theorem with a moving target: extra divergence or finitely centered target
• A century of metric Diophantine approximation and half a decade since Koukoulopoulos-Maynard

Related

• The Duffin-Schaeffer conjecture with extra divergence
• On the metric theory of approximations by reduced fractions: a quantitative Koukoulopoulos-Maynard theorem
• A note on the Duffin-Schaeffer conjecture
• On the Duffin-Schaeffer conjecture
• The Duffin-Schaeffer Conjecture for multiplicative Diophantine approximation