Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-02-11T12:22:58.367Z Has data issue: false hasContentIssue false
  • English
  • Français

Irregularities of distributions with respect to polytopes

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

Published online by Cambridge University Press:  26 February 2010

Michael Drmota
Michael Drmota
Affiliation:
Department of Discrete Mathematics, Technical University of Vienna, Wiedner Hauptstrasse 8-10/118, A-1040 Vienna, Austria.
Get access

Abstract

In the first part of the paper we show that the L2-discrepancy with respect to squares is of the same order of magnitude as the usual L2- discrepancy for point distributions in the K-dimensional torus. In the second part we adapt this method to obtain a generalization of Roth's [7] lower bound (log N)(k-1)/2 (for the usual discrepancy) to the discrepancy with respect to homothetic simple convex poly topes.

MSC classification

Secondary: 11K38: Irregularities of distribution, discrepancy
Type
Research Article
Information
Mathematika , Volume 43 , Issue 1 , June 1996 , pp. 108 - 119
Copyright
Copyright © University College London 1996

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.)

References

1.Beck, J. and Chen, W. W. L.. Irregularities of Distribution (Cambridge University Press, Cambridge, 1987).CrossRefGoogle Scholar
2.Drmota, M.. Irregularities of distribution and convex sets. Grazer Math. Berichte, 318 (1992), 9–16.Google Scholar
3.Hlawka, E.. The Theory of Uniform Distribution (Academic Publishers, Berkhamsted, 1984).Google Scholar
4.Károlyi, G.. Geometric discrepancy theorems in higher dimensions. Studia Scientiarum Mathematicorum Hungarica, 30 (1995), 59–94.Google Scholar
5.Kuipers, L. and Niederreiter, H.. Uniform Distribution of Sequences (John Wiley and Sons, London, 1974).Google Scholar
6.Niederreiter, H., Tichy, R. F. and Turnwald, G.. An unequality for differences of distribution functions. Arch. Math., 54 (1990), 166–172.CrossRefGoogle Scholar
7.Roth, K. F.. On irregularities of distribution. Mathematika, 1 (1954), 73–79.CrossRefGoogle Scholar
8.Ruzsa, I. Z.. The discrepancy of rectangles and squares, Grazer Math. Bericht, 318 (1992), 135–140.Google Scholar
9.Schmidt, W. M.. Irregularities of Distribution VII. Ada Arith., 21 (1972), 63–74.Google Scholar