Covering numbers, dyadic chaining and discrepancy

In 2001 Heinrich, Novak, Wasilkowski and Wozniakowski proved that for every s >= 1 and N >= 1 there exists a sequence (z(1) . . . . . . z(N)) of elements of the s-dimensional unit cube such that the star-discrepancy D-N* of this sequence satisfies D-N*(z(1) . . . . . z(N)) <= c root s/root N for some constant c independent of s and N. Their proof uses deep results from probability theory and combinatorics, and does not provide a concrete value for the constant c. In this paper we give a new simple proof of this result, and show that we can choose c = 10. Our proof combines Gnewuch's upper bound for covering numbers, Bernstein's inequality and a dyadic partitioning technique. (C) 2011 Elsevier Inc. All rights reserved.