关于Borel-Cantelli引理、Erdens-Rényi定理和Kochen-Stone定理
On the Borel-Cantelli Lemmas, the Erdős-Rényi Theorem, and the Kochen-Stone Theorem
摘要：
在本文中，我们提出了一个定量分析的第一和第二Borel-Cantelli引理和他们的两个推广：Erdens-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公式，给出了假设的定量公式与结论之间的明确关系。然而，对于Kochen-Stone定理，我们可以证明直接定量公式的数值界一般是不可计算的。尽管如此，我们得到了一个定量的公式的Kochen-Stone定理使用陶的亚稳性的概念。
英文摘要：
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.
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