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Irregularities of Point Distribution Relative to Convex Polygons III
Author(s) -
Beck J.,
Chen W. W. L.
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610797005267
Subject(s) - combinatorics , rectangle , mathematics , distribution (mathematics) , regular polygon , unit (ring theory) , measure (data warehouse) , geometry , mathematical analysis , mathematics education , database , computer science
Suppose that P is a distribution of N points in the unit square U =[0, 1] 2 . For every x =( x 1 , x 2 )∈ U , let B ( x )=[0, x 1 ]×[0, x 2 ] denote the aligned rectangle containing all points y =( y 1 , y 2 )∈ U satisfying 0⩽ y 1 ⩽ x 1 and 0⩽ y 2 ⩽ x 2 . Denote by Z [P; B ( x )] the number of points of P that lie in B ( x ), and consider the discrepancy function D [P; B ( x )]= Z [P; B ( x )]− N μ( B ( x )), where μ denotes the usual area measure.