A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures
A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures
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DOI：
10.4007/annals.2006.164.971
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发表时间：
2004-12
影响因子：
4.9
通讯作者：
V. Beresnevich;S. Velani
中科院分区：
文献类型：
--
作者：
V. Beresnevich;S. Velani
介绍并讨论了度量数论中 Duffin-Schaeffer 猜想的 Hausdorff 测度版本。一般猜想是对原始猜想求模而成立的。关键结果是质量转移原理，它允许我们将 R k 的 lim sep 子集的勒贝格测度理论陈述转移到豪斯多夫测度理论陈述。鉴于此，勒贝格的 lim sup 集理论被证明是一般豪斯多夫理论的基础。这是相当令人惊讶的，因为后一种理论被认为是前一种理论的微妙改进。
A Hausdorff measure version of the Duffin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for lim sup subsets of R k to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of lim sup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.