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Irregularities of point distribution relative to half-planes I

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

Published online by Cambridge University Press:  26 February 2010

J. Beck  and
W. W. L. Chen
J. Beck
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A.
W. W. L. Chen
Affiliation:
School of MPCE, Macquarie University, Sydney, NSW 2109, Australia.
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Extract

Suppose that is a distribution of N points in U0, the closed disc of unit area and centred at the origin 0. For every measurable set B in ℝ2, let Z[; B] denote the number of ponts of in B, and write

where µ denotes the usual measure in ℝ2

MSC classification

Secondary: 11K38: Irregularities of distribution, discrepancy
Type
Research Article
Information
Mathematika , Volume 40 , Issue 1 , June 1993 , pp. 102 - 126
Copyright
Copyright © University College London 1993

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References

1.Alexander, R.. Geometric methods in the study of irregularities of distribution. Combinatorica, 10 (1990), 115–136.CrossRefGoogle Scholar
2.Beck, J.. On a problem of K. F. Roth concerning irregularities of point distribution. Invent. Math., 74 (1983), 477–487.CrossRefGoogle Scholar
3.Beck, J. and Chen, W. W. L.. Irregularities of Distribution (Cambridge Tracts in Mathematics 89, Cambridge University Press, Cambridge, 1987).CrossRefGoogle Scholar
4.Hardy, G. H.. On the expression of a number as the sum of two squares. Quart. J. Math., 46 (1915), 263–283.Google Scholar
5.Landau, E.. Vorlesungen über Zahlentheorie, vol. 2 (Chelsea Publishing Company, New York, 1969).Google Scholar
6.Roth, K. F.. On irregularities of distribution III. Ada Arith., 35 (1979), 373–384.CrossRefGoogle Scholar
7.Schmidt, W. M.. Lectures on Irregularities of Distribution (Tata Institute of Fundamental Research, Bombay, 1977).Google Scholar