Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-23T15:05:04.966Z Has data issue: false hasContentIssue false
  • English
  • Français

Note on irregularities of distribution

Published online by Cambridge University Press:  26 February 2010

J. Beck  and
W. W. L. Chen
J. Beck
Affiliation:
Eötvös Loránd University, Budapest, Hungary.
W. W. L. Chen
Affiliation:
Imperial College, London, SW7 2BZ.
Get access

Extract

Let UK = [0, 1)K be the K-dimensional unit cube, where K ≥ 2. Suppose that we have a distribution ℘ of N points in UK. For × = ( x1, … , xK) ε UK, let A(x) denote the box

and write

Note that since N is the cardinality of ℘ and x1 … xK is the K-dimensional volume of A(x), the term Nx1 … xK represents the “expected number” of, points of ℘ in A(x).

Type
Research Article
Information
Mathematika , Volume 33 , Issue 1 , June 1986 , pp. 148 - 163
Copyright
Copyright © University College London 1986

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.Some upper bounds in the theory of irregularities of distribution. Ada Arith., 43 (1984), 115–130.CrossRefGoogle Scholar
2.Beck, J.. Irregularities of distribution I. To appear in Acta Math.Google Scholar
3.Chung, K. L.. A course in probability theory (Academic Press, New York, 1974).Google Scholar
4.Davenport, H.. Note on irregularities of distribution. Mathematika, 3 (1956), 131–135.CrossRefGoogle Scholar
5.Montgomery, H. L.. Irregularities of distribution by means of power sums. Manuscript submitted to the Proceedings of the Congress de Teoria de los Numeros, Bilbao, 1984.Google Scholar
6.Roth, K. F.. On irregularities of distribution. Mathematika, 1 (1954), 73–79.CrossRefGoogle Scholar
7.Roth, K. F.. On irregularities of distribution III. Ada Arith., 35 (1979), 373–384.CrossRefGoogle Scholar
8.Roth, K. F.. On irregularities of distribution IV. Ada Arith., 37 (1980), 67–75.CrossRefGoogle Scholar
9.Schmidt, W. M.. Irregularities of distribution IV. Invent. Math., 7 (1969), 55–82.CrossRefGoogle Scholar
10.Schmidt, W. M.. Lectures on irregularities of distribution (Tata Institute of Fundamental Research, Bombay, 1977).Google Scholar