Almost-Sharp Quantitative Duffin-Schaeffer without GCD Graphs

TL;DR

This paper presents a new proof of the almost-sharp quantitative bound for the Duffin-Schaeffer conjecture that entirely avoids the use of GCD graphs, building on recent advances and simplifying previous methods.

Contribution

It extends existing arguments to incorporate new elements from recent proofs, providing a GCD graph-free proof of the bound.

Findings

New GCD graph-free proof of the Duffin-Schaeffer bound

Unified approach combining recent and previous techniques

Simplification of the proof structure

Abstract

In recent work, Koukoulopoulos, Maynard and Yang proved an almost sharp quantitative bound for the Duffin-Schaeffer conjecture, using the Koukoulopoulos-Maynard technique of GCD graphs. This coincided with a simplification of the previous best known argument by Hauke, Vazquez and Walker, which avoided the use of the GCD graph machinery. In the present paper, we extend this argument to the new elements of the proof of Koukoulopoulos-Maynard-Yang. Combined with the work of Hauke-Vazquez-Walker, this provides a new proof of the almost sharp bound for the Duffin-Schaeffer conjecture, which avoids the use of GCD graphs entirely.