2
a/numbertheory posted by smh 3 months ago

Abstract: Let $$\psi:\mathbb{N} \to \mathbb{R}_{\ge0}$$ be an arbitrary function from the positive integers to the non-negative reals. Consider the set $$\mathcal{A}$$ of real numbers $$\alpha$$ for which there are infinitely many reduced fractions $$a/q$$ such that $$|\alpha - a/q|\le\psi(q)/q.$$ If $$\sum_{q=1}^\infty \psi(q)\phi(q)/q=\infty$$, we show that $$\mathcal{A}$$ has full Lebesgue measure. This answers a question of Duffin and Schaeffer. As a corollary, we also establish a conjecture due to Catlin regarding non-reduced solutions to the inequality $$|\alpha - a/q| \le \psi(q)/q$$, giving a refinement of Khinchin's Theorem.