On the inverse of the discrepancy for infinite dimensional infinite sequences

In 2001 Heinrich, Novak, Wasilkowski and Woźniakowski proved the upper bound N∗(s,ε)≤cabssε−2 for the inverse of the star discrepancy N∗(s,ε)N∗(s,ε). This is equivalent to the fact that for any N≥1N≥1 and s≥1s≥1 there exists a set of NN points in the ss-dimensional unit cube whose star-discrepancy is bounded by cabss/N. Dick showed that there exists a double infinite matrix (xn,i)n≥1,i≥1(xn,i)n≥1,i≥1 of elements of [0,1][0,1] such that for any NN and ss the star discrepancy of the ss-dimensional NN-element sequence ((xn,i)1≤i≤s)1≤n≤N((xn,i)1≤i≤s)1≤n≤N is bounded by cabsslogNN. In the present paper we show that this upper bound can be reduced to cabss/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 NN and ss.