The Champernowne constant is not Poissonian

TL;DR

This paper investigates the pair correlation properties of sequences derived from the Champernowne constant, demonstrating that unlike almost all normal numbers, this specific constant does not exhibit Poissonian pair correlations.

Contribution

It proves that the sequence of shifts of the Champernowne constant in base 2 does not have Poissonian pair correlations, contrasting with the behavior for almost all normal numbers.

Findings

Champernowne constant sequence is not Poissonian

Almost all normal numbers have Poissonian pair correlations

Specific counterexample to typical behavior

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 $s \geq 0$. In this note we study the pair correlation statistics for the sequence of shifts of $\alpha$, $x_n = \lbrace 2^n \alpha \rbrace, \ n=1, 2, 3, \ldots$, where we choose $\alpha$ 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 $\alpha$ (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.