The asymptotic distribution function of the 4-dimensional shifted van der corput sequence

Vladimír Baláž; Jana Fialová; Markus Hofer; Maria R. Iacò; Oto Strauch

doi:10.1515/tmmp-2015-0043

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The asymptotic distribution function of the 4-dimensional shifted van der corput sequence

Tatra Mountains Mathematical Publications

Volume 64 (2015): Issue 1 (September 2015)

By: Vladimír Baláž, Jana Fialová, Markus Hofer, Maria R. Iacò and Oto Strauch

Open Access

|Feb 2016

Abstract

Let γ_q(n) be the van der Corput sequence in the base q and g(x, y, z, u) be an asymptotic distribution function of the 4-dimensional sequence

In this paper we find an explicit formula for g(x, x, x, x) and then as an example we find the limit

for the base q = 4, 5, 6, . . . Also we find an explicit form of sth iteration T^(s)(x) of the von Neumann-Kakutani transformation defined by T(γ_q(n)) = γ_q(n + 1).

References

[1] AISLEITNER, CH.-HOFER, M.: On the limit distribution of consecutive elements of the van der Corput sequence, Unif. Distrib. Theory 8 (2013), 89-96.
Search in Google Scholar
[2] FIALOVÁ, J.-MIŠÍK, L.-STRAUCH, O.: An asymptotic distribution functions of the three-dimensional shifted van der Corput sequence, Appl. Math. 5 (2014), 2334-2359, http://dx.doi.org/10.4236/am.2014.51522710.4236/am.2014.515227
Search in Google Scholar
[3] FIALOVÁ, J.-STRAUCH, O.: On two-dimensional sequences composed of one-dimensional uniformly distributed sequences, Unif. Distrib. Theory 6 (2011), 101-125.
Search in Google Scholar
[4] STRAUCH, O.: Unsolved Problems, Tatra Mt. Math. Publ. 56 (2013), 109-229.
Search in Google Scholar
[5] KUIPERS, L.-NIEDERREITER, H.: Uniform Distribution of Sequences, in: Pure Appl. Math. (N.Y.), John Wiley, New York, 1974 (reprint ed. published by Dover Publ., Inc. Mineola, New York, 2006).
Search in Google Scholar
[6] DRMOTA, M.-TICHY, R. F.: Sequences, Discrepancies and Applications. Springer Verlag, Berlin, 1997.10.1007/BFb0093404
Search in Google Scholar
[7] STRAUCH, O.-PORUBSKÝ, Š.: Distribution of Sequences: A Sampler. Peter Lang, Frankfurt am Main, 2005, https://math.boku.ac.at/udt
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.1515/tmmp-2015-0043 | Journal eISSN: 1338-9750 | Journal ISSN: 1210-3195

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Language: English

Page range: 75 - 92

Submitted on: Oct 27, 2015

Published on: Feb 19, 2016

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

uniform distribution,

distribution function

Related subjects:

Mathematics,

General mathematics

© 2016 Vladimír Baláž, Jana Fialová, Markus Hofer, Maria R. Iacò, Oto Strauch, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 64 (2015): Issue 1 (September 2015)