## Functiones et Approximatio Commentarii Mathematici

- Funct. Approx. Comment. Math.
- Volume 60, Number 2 (2019), 253-262.

### The Champernowne constant is not Poissonian

Ísabel Pirsic and Wolfgang Stockinger

#### Abstract

We say that a sequence $(x_n)_{n \in \mathbb{N}}$ in $[0,1)$ has Poissonian pair correlations if $$ \lim_{N \to \infty} \frac{1}{N} \# \left\lbrace 1 \leq l \neq m \leq N: \| x_l - x_m \| \leq \frac{s}{N} \right\rbrace = 2s $$ 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.

#### Article information

**Source**

Funct. Approx. Comment. Math., Volume 60, Number 2 (2019), 253-262.

**Dates**

First available in Project Euclid: 26 June 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.facm/1529978436

**Digital Object Identifier**

doi:10.7169/facm/1749

**Mathematical Reviews number (MathSciNet)**

MR3964264

**Zentralblatt MATH identifier**

07068535

**Subjects**

Primary: 11K06: General theory of distribution modulo 1 [See also 11J71]

Secondary: 11K31: Special sequences

**Keywords**

Poissonian pair correlation normal numbers

#### Citation

Pirsic, Ísabel; Stockinger, Wolfgang. The Champernowne constant is not Poissonian. Funct. Approx. Comment. Math. 60 (2019), no. 2, 253--262. doi:10.7169/facm/1749. https://projecteuclid.org/euclid.facm/1529978436