Proving the Duffin-Schaeffer conjecture without GCD graphs

TL;DR

This paper provides a simplified and strengthened proof of the Duffin-Schaeffer conjecture in metric Diophantine approximation, avoiding GCD graphs and improving error-term bounds.

Contribution

It introduces a new proof that simplifies previous methods and enhances error-term estimates in the metric theory of Diophantine approximation.

Findings

Proves the Duffin-Schaeffer conjecture.

Improves error bounds from polynomial to exponential decay.

Avoids the use of GCD graphs in the proof.

Abstract

We present a novel proof of the Duffin-Schaeffer conjecture in metric Diophantine approximation. Our proof is heavily motivated by the ideas of Koukoulopoulos-Maynard's breakthrough first argument, but simplifies and strengthens several technical aspects. In particular, we avoid any direct handling of GCD graphs and their `quality'. We also consider the metric quantitative theory of Diophantine approximations, improving the $(\log \Psi(N))^{-C}$ error-term of Aistleitner-Borda and the first named author to $\exp(-(\log \Psi(N))^{\frac{1}{2} - \varepsilon})$.