The Duffin–Schaeffer Conjecture with extra divergence

Abstract

Given a nonnegative function $${\psi\mathbb{N} \to \mathbb{R}}$$ , let W(ψ) denote the set of real numbers x such that |nx − a| 0). A consequence of our main result is that W(ψ) is of full Lebesgue measure if there exists an $${\epsilon > 0}$$ such that $$ \textstyle \sum_{n\in\mathbb{N}}\left(\frac{\psi(n)}{n}\right)^{1+\epsilon}\varphi (n)=\infty. $$ 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(ψ) 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.