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On irregularities of distribution

Published online by Cambridge University Press:  26 February 2010

W. W. L. Chen
W. W. L. Chen
Affiliation:
Imperial collegeLondon, SW7.
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Extract

Let U0 = [(0, 1); and U1 = (0,1)]. Suppose we have a distribution of N points in , where, for k ≥ 1, is the unit cube consisting of the points y = (y1, … , yk+1) with 0 ≤ yi < 1 (i = 1 , … , k + 1). For X = (x1 ,…, xk + 1) in , let B(x) denote the box consisting of all y such that 0 ≤ yi < xi (i = 1 ,…, k + 1), and let denote the number of points of which lie in B(x).

Type
Research Article
Information
Mathematika , Volume 27 , Issue 2 , December 1980 , pp. 153 - 170
Copyright
Copyright © University College London 1980

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References

1.Davenport, H.. “Note on irregularities of distribution”, Mathematika, 3 (1956), 131–135.CrossRefGoogle Scholar
2.Halton, J. H. and Zaremba, S. K.. “The extreme and L2 discrepancies of some plane sets”, Monatsh.fur Math., 73 (1969), 316–328.CrossRefGoogle Scholar
3.Roth, K. F.. “On irregularities of distribution”, Mathematika, 1 (1954), 73–79.CrossRefGoogle Scholar
4.Roth, K. F.. “On irregularities of distribution. II”, Communications on Pure and Applied Math., 29 (1976), 749–754.CrossRefGoogle Scholar
5.Roth, K. F.. “On irregularities of distribution. III”, Ada Aritk, 35 (1979), 373–384.Google Scholar
6.Roth, K. F.. “On irregularities of distribution. IV”, to appear in Ada Arith.Google Scholar
7.Schmidt, W. M.. “Irregularities of distribution. X”, Number theory and algebra, pp. 311–329 (AcademicPress, New York, 1977).Google Scholar
8.Vilenkin, I. V.. “Plane nets of integration” (Russian), Ž. Vyčisl. Mat. i Mat. Fiz., 7 (1967), 189–196; English translation in U.S.S.R. Comp. Math, and Math. Phys., 7(1) (1967), 58–267.Google Scholar