A variant of Atanassov’s method for (t,s)(t,s)-sequences and (t,e,s)-sequences

The term low-discrepancy sequences is widely used to refer to ss-dimensional sequences XX for which the bound D∗(N,X)≤cs(logN)s+O((logN)s−1)D∗(N,X)≤cs(logN)s+O((logN)s−1) is satisfied, where D∗D∗ denotes the usual star discrepancy. In this paper, we study such bounds for (t,s)(t,s)-sequences and a newer class of low-discrepancy sequences called (t,e,s)-sequences, introduced recently by Tezuka (2013). In the first case, by using a combinatorial argument coupled with a careful worst-case analysis, we are able to improve the discrepancy bounds from Faure and Lemieux (2012) for (t,s)(t,s)-sequences. In the second case, an adaptation of the same pair of arguments allows us to improve the asymptotic behavior of the bounds from Tezuka (2013) in the case of even bases.