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ON THE N-POINT CORRELATION OF VAN DER CORPUT SEQUENCES

Part of: Multivariate analysis Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  15 September 2023

CHRISTIAN WEIß Open the ORCID record for CHRISTIAN WEIß [Opens in a new window]
CHRISTIAN WEIß*
Affiliation:
Ruhr West University of Applied Sciences, Duisburger Str. 100, D-45479 Mülheim an der Ruhr, Germany
*
e-mail: christian.weiss@hs-ruhrwest.de
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Abstract

We derive an explicit formula for the N-point correlation $F_N(s)$ of the van der Corput sequence in base $2$ for all $N \in \mathbb {N}$ and $s \geq 0$. The formula can be evaluated without explicit knowledge about the elements of the van der Corput sequence. This constitutes the first example of an exact closed-form expression of $F_N(s)$ for all $N \in \mathbb {N}$ and all $s \geq 0$ which does not require explicit knowledge about the involved sequence. Moreover, it can be immediately read off that $\lim _{N \to \infty } F_N(s)$ exists only for $0 \leq s \leq 1/2$.

Keywords

pair correlation statisticvan der Corput sequenceequidistribution

MSC classification

Primary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Secondary: 11K38: Irregularities of distribution, discrepancy 62H20: Measures of association (correlation, canonical correlation, etc.)

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 3 , June 2024 , pp. 471 - 475
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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