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arXiv · 1803.05703

The Duffin-Schaeffer conjecture with extra divergence

Christoph Aistleitner·Thomas Lachmann·Marc Munsch·Niclas Technau·Agamemnon Zafeiropoulos
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Abstract

The Duffin-Schaeffer conjecture is a fundamental unsolved problem in metric number theory. It asserts that for every non-negative function $\psi:~\mathbb{N} \rightarrow \mathbb{R}$ for almost all reals $x$ there are infinitely many coprime solutions $(a,n)$ to the inequality $|nx - a| < \psi(n)$, provided that the series $\sum_{n=1}^\infty \psi(n) \varphi(n) /n$ is divergent. In the present paper we prove that the conjecture is true under the "extra divergence" assumption that divergence of the series still holds when $\psi(n)$ is replaced by $\psi(n) / (\log n)^\varepsilon$ for some $\varepsilon > 0$. This improves a result of Beresnevich, Harman, Haynes and Velani, and solves a problem posed by Haynes, Pollington and Velani.

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Christoph Aistleitner, Thomas Lachmann, Marc Munsch, Niclas Technau, Agamemnon Zafeiropoulos. 2018-03-15. The Duffin-Schaeffer conjecture with extra divergence. https://arxiv.org/abs/1803.05703

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