论文标题
典型数字集的Borel复杂性
Borel complexity of the set of typical numbers
论文作者
论文摘要
在本说明中，我们研究所谓的典型数字和底座中正常数的典型数字和数字之间的相互关系。我们采用了Nakai和Shiokawa的结果，展示了属于一组但不属于另一组的数字的例子，反之亦然。此外，我们证明了Borel层次结构中的一组典型数字为$π_3^0 $，即，它可以使用Ki和Linton的结果来表达为许多$f_σ\ text {-set。} $的结合$Δ_4^0 $类。
In the present note we study the interrelations between the sets of so-called typical numbers and numbers that are normal in base two. Employing results by Nakai and Shiokawa, we exhibit examples of numbers that belong to one set but do not belong to the other and vice versa. Moreover, we demonstrate the set of typical numbers is $Π_3^0$ in the Borel hierarchy, i.e., it can be expressed as the union of countably many $F_σ\text{-sets.}$ Using the result by Ki and Linton that asserts the same for normal numbers, we examine the Borel complexity of the set of typical numbers that are not normal, proving that it belongs to the $Δ_4^0$ class.
