论文标题
$λ$ - 泊松通用序列的结构
A construction of a $λ$- Poisson generic sequence
论文作者
论文摘要
多年前，Zeev Rudnick将$λ$ -Poisson通用序列定义为有限字母中符号的无限序列，其中最初段中长词的出现数量遵循Poisson分布，带有参数$λ$。尽管几乎所有序列（对于统一度量）都是普通的，但尚未给出明确的实例。在本说明中，我们为任何字母和任何正面的$λ$构造了明显的$λ$ -Poisson通用序列，除了两个符号字母的情况下，要求$λ$小于或等于自然对数$ 2 $。由于$λ$ -Poisson的通用性意味着Borel正态性，因此构造的序列是Borel正常的。相同的构造提供了不是$λ$ -Poisson Generic的Borel普通序列的明确实例。
Years ago Zeev Rudnick defined the $λ$-Poisson generic sequences as the infinite sequences of symbols in a finite alphabet where the number of occurrences of long words in the initial segments follow the Poisson distribution with parameter $λ$. Although almost all sequences, with respect to the uniform measure, are Poisson generic, no explicit instance has yet been given. In this note we give a construction of an explicit $λ$-Poisson generic sequence over any alphabet and any positive $λ$, except for the case of the two-symbol alphabet, in which it is required that $λ$ be less than or equal to the natural logarithm of $2$. Since $λ$-Poisson genericity implies Borel normality, the constructed sequences are Borel normal. The same construction provides explicit instances of Borel normal sequences that are not $λ$-Poisson generic.
