A note on the Duffin-Schaeffer conjecture with slow divergence

Abstract

For a non-negative function $\psi :\mathbb{N}\mapsto \mathbb{R}$, let $W(\psi )$ denote the set of real numbers $x$ for which the inequality $|nx-a|<\psi (n)$ has infinitely many coprime solutions $(a,n)$. The Duffin--Schaeffer conjecture, one of the most important unsolved problems in metric number theory, asserts that $W(\psi )$ has full measure provided {equation} \label{dsccond} \sum_{n=1}^\infty \frac{ψ(n) φ(n)}{n} = \infty. {equation} Recently Beresnevich, Harman, Haynes and Velani proved that $W(\psi )$ has full measure under the \emph{extra divergence} condition $$\sum _{n=1}^{\infty }\frac{\psi (n)\phi (n)}{n\exp (c(\log \log n)(\log \log \log n))}=\infty \text{for some c>0}.$$ In the present note we establish a \emph{slow divergence} counterpart of their result: $W(\psi )$ has full measure, provided\eqref{dsccond} holds and additionally there exists some $c>0$ such that $$\sum _{n=2^{2^h}+1}^{2^{2^{h+1}}}\frac{\psi (n)\phi (n)}{n}\le \frac{c}{h}\text{for all h≥1.}$$

4 pages; for version 2 some typos have been fixed and a corollary has been added; for version 3, some further minor changes have been made. The manuscript has been accepted for publication by Bull. London Math. Soc

References in corpus (1)

• Classical metric Diophantine approximation revisited

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