We prove versions of Khintchine's Theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity assumption can be removed. Roughly speaking, we show that if the number of available fractions for each denominator grows too fast, then the monotonicity assumption cannot be removed. There are questions in this random setting which may be seen as cognates of the Duffin-Schaeffer Conjecture (1941), and are likely to be more accessible. We point out that the direct random analogue of the Duffin-Schaeffer Conjecture, like the Duffin-Schaeffer Conjecture itself, implies Catlin's Conjecture (1976). It is not obvious whether the Duffin-Schaeffer Conjecture and its random version imply one another, and it is not known whether Catlin's Conjecture implies either of them. The question of whether Catlin implies Duffin-Schaeffer has been unsettled for decades.

中文翻译：

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带有随机分数的欣钦定理

我们证明了 Khintchine 定理 (1924) 的版本，用于通过其分子位于随机选择的整数集的有理数逼近，并探索单调性假设可以消除的程度。粗略地说，我们表明，如果每个分母的可用分数数量增长得太快，则无法消除单调性假设。这种随机设置中的一些问题可能被视为 Duffin-Schaeffer 猜想 (1941) 的同源词，并且可能更容易理解。我们指出，Duffin-Schaeffer 猜想的直接随机类似物，就像 Duffin-Schaeffer 猜想本身一样，暗示了 Catlin 猜想（1976）。Duffin-Schaeffer 猜想及其随机版本是否相互暗示尚不明显，也不知道 Catlin' s 猜想暗示了其中任何一个。Catlin 是否暗示 Duffin-Schaeffer 的问题几十年来一直悬而未决。