Discrepancy Bounds for Mixed Sequences

A common measure for the uniformity of point distributions is the star discrepancy. Let λ s denote the s-dimensional Lebesgue measure. Then the star discrepancy of a mul-tiset P = {p 1 ,. .. , p N } ⊂ [0, 1] s is given by D * N (P) := sup α∈[0,1] s λ s ([0, α)) − 1 N N k=1 1 [0,α) (p k) ; here [0, α) denotes the s-dimensional axis-parallel box [0, α 1) × · · · × [0, α s) and 1 [0,α) its characteristic function. For an infinite sequence p in [0, 1] s we denote by D * N (p) the discrepancy of its first N points. The smallest star discrepancy of any N-point set is and the inverse of the star discrepancy is given by There are bounds known describing the behavior of the star discrepancy in the number of points N and in the dimension s