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| < ψ(n) for infinitely many reduced rationals a/n (n > 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.