The Champernowne constant is not Poissonian

Abstract

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 $s\ge 0$. In this note we study the pair correlation statistics for the sequence of shifts of $\alpha$, $x_n=\{2^n\alpha \},n=1,2,3,\dots$, where we choose $\alpha$ as the Champernowne constant in base $2$. Throughout this article $\{\cdot \}$ 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 $\alpha$ (in the sense of Lebesgue), but we will show that it does not have this property for all normal numbers $\alpha$, as it fails to be Poissonian for the Champernowne constant.

11 pages, several corrections and changes