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arXiv · 2404.15123

Proving the Duffin-Schaeffer conjecture without GCD graphs

Manuel Hauke·Santiago Vazquez Saez·Aled Walker
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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 $(\log Ψ(N))^{-C}$ error-term of Aistleitner-Borda and the first named author to $\exp(-(\log Ψ(N))^{\frac{1}{2} - \varepsilon})$.

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Manuel Hauke, Santiago Vazquez Saez, Aled Walker. 2026-07-10. Proving the Duffin-Schaeffer conjecture without GCD graphs. https://arxiv.org/abs/2404.15123

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