Irregularities of Point Distribution Relative to Convex Polygons

Abstract

Let U ² = [0, l]² denote the unit square in R ². Suppose that P is a distribution of N points in U ². For any Lebesgue measurable set A in U ², denote by Z[P; A] the number of points P in A. We are interested in the discrepancy function

$$<Emphasis Type="Italic">D[P;A]=Z[P;A]-N\mu (A)</Emphasis>$$

where μ denotes, as usual, the 2-dimensional Lebesgue measure.