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K. Roth (1979)
On irregularities of distribution IV
J. Halton, S. Zaremba (1969)
The extreme and L2 discrepancies of some plane setsMonatshefte für Mathematik, 73
10F40: NUMBER THEORY; Diophantine ap-Imperial College, proximation, Distribution modulo one
K. Roth (1976)
On irregularities of distribution, IICommunications on Pure and Applied Mathematics, 29
K. Roth (1954)
On irregularities of distributionMathematika, 1
I. Vilenkin (1967)
Plane nets of integrationUssr Computational Mathematics and Mathematical Physics, 7
W. W. L. CHEN §1. Introduction. Let U = [0, 1) and U = (0,1]. Suppose we have a o l k +1 k +1 distribution & of N points in U , where, for k ^ l,U is the unit cube 0 0 consisting of the points y = (yi,-.,y ) with 0 ^ y, < 1 (i = 1,..., fc + 1). For k+1 x = (xj,..., x ) in U\ , let B{x) denote the box consisting of all y such that k+1 0 < yt < x (i = 1,..., k+1), and let Z\0>; B(x)] denote the number of points of & which lie in B(\). Write The irregularity of the distribution 3P can be measured in a number of ways by the behaviour of the function D[& ; B(x)]. One may consider the iT-norm \\D(n\w = ( ... \D[<?;B(x)Tdx...dx 1 k+1 u, y. Roth [3] obtained a lower bound for the L -norm, and Schmidt [7] established the following generalization, Roth's result being the special case W = 2. THEOREM 1. (Schmidt [7]). For every W> 1, there exists a positive number c {k, W), depending only on k and W, such that \\D(P)\\
Mathematika – Wiley
Published: Dec 1, 1980
Keywords: ; ;
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