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June 2019 The Champernowne constant is not Poissonian
Ísabel Pirsic, Wolfgang Stockinger
Funct. Approx. Comment. Math. 60(2): 253-262 (June 2019). DOI: 10.7169/facm/1749

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.

Citation

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Ísabel Pirsic. Wolfgang Stockinger. "The Champernowne constant is not Poissonian." Funct. Approx. Comment. Math. 60 (2) 253 - 262, June 2019. https://doi.org/10.7169/facm/1749

Information

Published: June 2019
First available in Project Euclid: 26 June 2018

zbMATH: 07068535
MathSciNet: MR3964264
Digital Object Identifier: 10.7169/facm/1749

Subjects:
Primary: 11K06
Secondary: 11K31

Keywords: Normal numbers , Poissonian pair correlation

Rights: Copyright © 2019 Adam Mickiewicz University

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Vol.60 • No. 2 • June 2019
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