The Duffin-Schaeffer conjecture with extra divergence II
The Duffin-Schaeffer conjecture with extra divergence II
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具有额外散度的 Duffin-Schaeffer 猜想 II
DOI：
10.1007/s00209-012-1126-5
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发表时间：
2012
影响因子：
0.8
通讯作者：
Beresnevich V
中科院分区：
文献类型：
--
作者：
Beresnevich V
在2012年，作者提出了一个方案来证明Duffin-Schaeffer猜想的措施任意接近勒贝格措施。在本文中，我们在这个方向上迈出了新的一步。给定一个非负函数，记为一组真实的数，使得对于无穷多个约化有理数。我们的主要结果是：如果存在$$\开始{aligned} \sum _{n\ge 16} \，\frac{\varphi（n）\psi（n）}{n \exp（c（\log \log n）（\log \log \log n））} \，= \，\infty \，. \end{aligned}$$
In 2012 the authors set out a programme to prove the Duffin–Schaeffer conjecture for measures arbitrarily close to Lebesgue measure. In this paper we take a new step in this direction. Given a non-negative function, letdenote the set of real numberssuch thatfor infinitely many reduced rationals. Our main result is thatis of full Lebesgue measure if there exists asuch that $$\begin{aligned} \sum _{n\ge 16} \, \frac{\varphi (n) \psi (n)}{n \exp (c(\log \log n)(\log \log \log n))} \, = \, \infty \, . \end{aligned}$$