Hostname: page-component-77f85d65b8-4lrz4 Total loading time: 0.001 Render date: 2026-03-26T12:11:33.918Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE N-POINT CORRELATION OF VAN DER CORPUT SEQUENCES

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

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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Drmota, M. and Tichy, R., Sequences, Discrepancies and Applications, Lecture Notes in Mathematics, 1651 (Springer, Berlin, 1997).CrossRefGoogle Scholar
El-Baz, D., Marklof, J. and Vinogradov, I., ‘The two-point correlation function of the fractional parts of $\sqrt{n}$ is Poisson’, Proc. Amer. Math. Soc. 143(7) (2015), 2815–2828.CrossRefGoogle Scholar
Kuipers, L. and Niederreiter, H., Uniform Distribution of Sequences (Wiley, New York, 1974).Google Scholar
Lutsko, C., ‘Directions in orbits of geometrically finite hyperbolic subgroups’, Math. Proc. Cambridge Philos. Soc. 171(2) (2020), 277–316.CrossRefGoogle Scholar
Lutsko, C., Sourmelidis, A. and Technau, N., ‘Pair correlation of the fractional parts of $\alpha {n}^{\theta }$ ’, Preprint, 2023, arXiv:2106.09800.CrossRefGoogle Scholar
Lutsko, C. and Technau, N., ‘Correlations of the fractional parts of $\alpha {n}^{\theta }$ ’, Preprint, 2021, arXiv:2112.11524.Google Scholar
Lutsko, C. and Technau, N., ‘Full Poissonian local statistics of slowly growing sequences’, Preprint, 2022, arXiv:2206.07809.Google Scholar
Marklof, J., ‘Distribution modulo one and Ratner’s theorem’, in: Equidistribution in Number Theory, An Introduction, NATO Science Series, 237 (eds. Granville, A. and Rudnick, Z.) (Springer, Dordrecht, 2007).Google Scholar
Marklof, J. and Strömbergsson, A., ‘Gaps between logs’, Bull. Lond. Math. Soc. 45(6) (2013), 1267–1280.CrossRefGoogle Scholar
Niederreiter, H., Random Number Generation and Quasi-Monte Carlo Methods, CBMS-NSF Series in Applied Mathematics, 63 (SIAM, Philadelphia, PA, 1992).CrossRefGoogle Scholar
Sayous, R., ‘Effective pair correlations of fractional powers of integers’, Preprint, 2023, arXiv:2306.00793.Google Scholar
Weiß, C., ‘Some connections between discrepancy, finite gap properties and pair correlations’, Monatsh. Math. 199 (2022), 909–927.CrossRefGoogle Scholar
Weiß, C., ‘An explicit non-Poissonian pair correlation function’, Preprint, 2023, arXiv:2304.14202.Google Scholar
Weiß, C. and Skill, T., ‘Sequences with almost Poissonian pair correlations’, J. Number Theory 236 (2022), 116–127.CrossRefGoogle Scholar