On the inverse of the discrepancy for infinite dimensional infinite sequences

In 2001 Heinrich, Novak, Wasilkowski and Wozniakowski proved the upper bound N* (s, epsilon) <= c(abs)s epsilon(-2) for the inverse of the star discrepancy N* (s, epsilon). This is equivalent to the fact that for any N >= 1 and s >= 1 there exists a set of N points in the s-dimensional unit cube whose star-discrepancy is bounded by c(abs)root s/root N. Dick showed that there exists a double infinite matrix (x(n,i))(n >= 1,i >= 1) of elements of [0, 1] such that for any N and s the star discrepancy of the s-dimensional N-element sequence ((x(n,i))(1 <= i <= s))(1 <= n <= N) is bounded by c(abs)root s log N/root N. In the present paper we show that this upper bound can be reduced to c(abs)root s/root N, which is (up to the value of the constant) the same upper bound as the one obtained by Heinrich et al. in the case of fixed N and s. (c) 2012 Elsevier Inc. All rights reserved.