Abstract
Suppose that [Pscr ] is a distribution of N points in the
unit square
U=[0, 1]2. For
every x=(x1, x2)∈U,
let
B(x)=[0, x1]×[0,
x2] denote the aligned rectangle containing all
points
y=(y1, y2)∈U
satisfying 0[les ]y1[les ]x1 and
0[les ]y2[les ]x2. Denote by
Z[[Pscr ]; B(x)]
the number of points of [Pscr ] that lie in B(x),
and consider the discrepancy function D[[Pscr ]; B(x)]=Z[[Pscr ];
B(x)]−Nμ(B(x)), where μ denotes the usual area measure.