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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关于达芬-谢弗猜想

令 $\psi:\mathbb{N}\to\mathbb{R}_{\ge0}$ 是从正整数到非负实数的任意函数。考虑实数 $\alpha$ 的集合 $\mathcal{A}$，其中存在无限多个约简分数 $a/q$，使得 $|\alpha-a/q|\le \psi(q)/q $. 如果 $\sum_{q=1}^\infty \psi(q)\phi(q)/q=\infty$，我们证明 $\mathcal{A}$ 具有完整的 Lebesgue 测度。这回答了达芬和谢弗的问题。作为推论，我们还建立了由于 Catlin 的关于不等式 $|\alpha - a/q|\le \psi(q)/q$ 的非约解解的猜想，从而对 Khinchin 定理进行了改进。