For all s ? 1 and N ? 1 there exist sequences (z1,..., zN) in [0, 1]s such that the star-discrepancy of these points can be bounded by D* N(z1,..., zN) ? c ?s/?N . In practice it is desirable to obtain low values of c. The best known value for the constant is c = 10 as has been calculated by Aistleitner. In this paper we improve the bound to c = 9.

We prove a generalized Faulhaber inequality to bound the sums of the j j -th powers of the first n n (possibly shifted) natural numbers. With the help of this inequality we are able to improve the known bounds for bracketing numbers of d d -dimensional axis-parallel boxes anchored in 0 0 (or, put differently, of lower left orthants intersected with the d d -dimensional unit cube [ 0 , 1 ] d [0,1]^d ). We use these improved bracketing numbers to establish new bounds for the star-discrepancy of negatively dependent random point sets and its expectation. We apply our findings also to the weighted star-discrepancy.

For all s ≥ 1 and N ≥ 1 there exist sequences (z 1 , . . . , z N ) in [0, 1] s such that the star-discrepancy of these points can be bounded byThe best known value for the constant is c = 10 as has been calculated by Aistleitner in [Ais11]. In this paper we improve the bound to c = 9.

No abstract