# On irregularities of distribution

On irregularities of distribution 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)\\ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematika Wiley

# On irregularities of distribution

, Volume 27 (2) – Dec 1, 1980
18 pages

/lp/wiley/on-irregularities-of-distribution-hUfFP0vD0l

# References (6)

Publisher
Wiley
ISSN
0025-5793
eISSN
2041-7942
DOI
10.1112/S0025579300010044
Publisher site
See Article on Publisher Site

### Abstract

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)\\

### Journal

MathematikaWiley

Published: Dec 1, 1980

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