Champernowne常数不是泊松常数
The Champernowne constant is not Poissonian
摘要：
如果对每个$s \geq 0$都有\begin{equation*} \lim_{N \to \infty} \frac{1}{N} \# \lbrace 1 \leq l \neq m \leq N: \| x_l - x_m \| \leq \frac{s}{N} \rbrace = 2s \end{equation*}，则我们称$[0,1)$中的序列$(x_n)_{n \in \mathbb{N}}$具有泊松对相关.本文研究了$α$，$x_n = \lbrace 2^n α\rbrace, \ n=1, 2, 3, \ldots$的位移序列的对相关统计，其中我们选择$α$作为基$2$中的Champernowne常数.在本文中，$\lbrace \cdot \rbrace$表示实数的小数部分。众所周知，$(x_n)_{n \in \mathbb{N}}$对几乎所有的正规数$α$（在勒贝格意义下）都具有泊松对相关性，但我们将证明它对所有的正规数$α$都不具有这种性质，因为它对Champernowne常数不是泊松的。
英文摘要：
We say that a sequence $(x_n)_{n \in \mathbb{N}}$ in $[0,1)$ has Poissonian pair correlations if \begin{equation*} \lim_{N \to \infty} \frac{1}{N} \# \lbrace 1 \leq l \neq m \leq N: \| x_l - x_m \| \leq \frac{s}{N} \rbrace = 2s \end{equation*} for every $s \geq 0$. In this note we study the pair correlation statistics for the sequence of shifts of $α$, $x_n = \lbrace 2^n α\rbrace, \ n=1, 2, 3, \ldots$, where we choose $α$ as the Champernowne constant in base $2$. Throughout this article $\lbrace \cdot \rbrace$ denotes the fractional part of a real number. It is well known that $(x_n)_{n \in \mathbb{N}}$ has Poissonian pair correlations for almost all normal numbers $α$ (in the sense of Lebesgue), but we will show that it does not have this property for all normal numbers $α$, as it fails to be Poissonian for the Champernowne constant.
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