a/numbertheory posted by smh 1 month 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. 

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