Irregularities of Point Distribution Relative to Convex Polygons III

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.