In this work we construct many sequences \(S=S^\Box _{b,d}\) and \(S=S^\boxplus _{b,d}\) in the d-dimensional unit hypercube, which for \(d=1\) are (generalized) van der Corput sequences or Niederreiter’s (0, 1)-sequences in base b, respectively. Further, we introduce the notion of f-subadditivity and use it to define discrepancy functions which subsume the notion of \(L^p\) -discrepancy, Wasserstein p-distance, and many more methods to compare empirical measures to an underlying base measure. We will relate bounds for a given discrepancy function \(\mathscr {D}\) of the multiset of projected lattice sets (treated as empirical measures), \(P(b^{-m}\mathbb {Z}^d\) ), to bounds of \(\mathscr {D}(E_{Z_N})\) , i.e. the initial segments of the sequence \(Z=P(S)\) for any \(N\in \mathbb {N}\) . We show that this relation holds in any dimension d and for any map P defined on a hypercube, for which bounds on \(\mathscr {D}(E_{P(b^{-m}\mathbb {Z}^d+v)})\) can be obtained. We apply this theorem in \(d=1\) to obtain bounds for the \(L^p\) -discrepancy of van der Corput and Niederreiter (0,1) sequences in terms of digit sums for all \(0<p\le \infty \) . In \(d=2\) an application of our construction yields many sequences on the two-sphere, such that the initial segments \(Z_N\) have small \(L^\infty \) -discrepancy.
