New results towards the Duffin-Schaeffer conjecture, which is a fundamental unsolved problem in metric number theory, have been established recently assuming extra divergence. Given a non-negative function $\psi: \mathbb{N}\to\mathbb{R}$ we denote by $W(\psi)$ the set of all $x\in\mathbb{R}$ such that $|nx-a| 0$ such that $\sum_{n=2}^\infty\psi(n)\varphi(n)/(n(\log n)^\varepsilon)=\infty$. This result seems to be the best one can expect from the method used. Assuming the extra divergence $\sum_{n=2}^\infty\psi(n)/(\log n)^\varepsilon=\infty$ we prove that $W(\psi)$ is of full measure. This could also be deduced from the result in [1], but we believe that our proof is of independent interest, since its method is totally different from the one in [1]. As a further application of our method, we prove that a variant of Khintchine's theorem is true without monotonicity, subject to an additional condition on the set of divisors of the support of $\psi$.

中文翻译：

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度量丢番图近似中 Khintchine 定理的变体

Duffin-Schaeffer 猜想的新结果是度量数论中一个基本未解决的问题，最近已经在假设额外发散的情况下建立。给定一个非负函数 $\psi：\mathbb{N}\to\mathbb{R}$ 我们用 $W(\psi)$ 表示所有 $x\in\mathbb{R}$ 的集合，使得 $ |nx-a| 0$ 使得 $\sum_{n=2}^\infty\psi(n)\varphi(n)/(n(\log n)^\varepsilon)=\infty$。这一结果似乎是所用方法所能预期的最好结果。假设额外的散度 $\sum_{n=2}^\infty\psi(n)/(\log n)^\varepsilon=\infty$ 我们证明 $W(\psi)$ 是完全度量的。这也可以从[1]中的结果推导出来，但我们认为我们的证明是独立的，因为它的方法与[1]中的完全不同。作为我们方法的进一步应用，我们证明了 Khintchine' 的一个变体