论文标题
在Borel-Cantelli引理，Erdős-rényi定理和Kochen-Stone定理上
On the Borel-Cantelli Lemmas, the Erdős-Rényi Theorem, and the Kochen-Stone Theorem
论文作者
论文摘要
在本文中，我们介绍了对第一和第二Borel-cantelli引理的定量分析以及它们的两个概括：Erdős-rényi定理和Kochen-Stone定理。我们将看到，前三个结果具有直接的定量表述，从而在假设的定量表述与结论之间具有明确的关系。但是，对于Kochen-Stone定理，我们可以证明直接定量公式的数值界限一般不可计算。尽管如此，我们使用陶的亚稳定性概念获得了高兴定理的定量公式。
In this paper we present a quantitative analysis of the first and second Borel-Cantelli Lemmas and of two of their generalisations: the Erdős-Rényi Theorem, and the Kochen-Stone Theorem. We will see that the first three results have direct quantitative formulations, giving an explicit relationship between quantitative formulations of the assumptions and the conclusion. For the Kochen-Stone theorem, however, we can show that the numerical bounds of a direct quantitative formulation are not computable in general. Nonetheless, we obtain a quantitative formulation of the Kochen-Stone Theorem using Tao's notion of metastability.
