The Duffin-Schaeffer Conjecture with extra divergence

Abstract

Given a nonnegative function $\psi :\mathbb{N}\to \mathbb{R}$, let $W(\psi )$ denote the set of real numbers $x$ such that $|nx-a|<\psi (n)$ for infinitely many reduced rationals $a/n(n>0)$. A consequence of our main result is that $W(\psi )$ is of full Lebesgue measure if there exists an $\epsilon >0$ such that $${\sum }_{n\in \mathbb{N}}(\frac{\psi (n)}{n}{)}^{1+\epsilon }\phi (n)=\infty .$$ The Duffin-Schaeffer Conjecture is the corresponding statement with $\epsilon =0$ and represents a fundamental unsolved problem in metric number theory. Another consequence is that $W(\psi )$ is of full Hausdorff dimension if the above sum with $\epsilon =0$ diverges; i.e. the dimension analogue of the Duffin-Schaeffer Conjecture is true.

13 pages -- a stronger theorem than in the original version is proved and connections to the work of Harman are made. Also the proof of the main theorem is split into two natural steps -- hopefully making it easier to see the overall strategy

References in corpus (1)

• Ubiquity and a general logarithm law for geodesics