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Mathematics > Number Theory

Title: The Duffin-Schaeffer conjecture with extra divergence

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.
Comments: 8 pages
Subjects: Number Theory (math.NT)
Cite as: arXiv:1803.05703 [math.NT]
  (or arXiv:1803.05703v1 [math.NT] for this version)

Submission history

From: Christoph Aistleitner [view email]
[v1] Thu, 15 Mar 2018 12:07:40 GMT (7kb)