Functiones et Approximatio Commentarii Mathematici

The Champernowne constant is not Poissonian

Ísabel Pirsic and Wolfgang Stockinger

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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


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