  In this work we construct many sequences $S=S^\Box_{b,d}$, or
$S=S^\boxplus_{b,d}$ 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 $f$-sublinearity
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 functions $\mathscr{D}$ of the
multiset of projected lattice sets $P(b^{-m}\mathbb{Z}^d$), to bounds of
$\mathscr{D}(Z_N)$, 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 $d$, for
any map $P$ defined on a hypercube, and any discrepancy function as introduced
in this work for which bounds on $P(b^{-m}\mathbb{Z}^d+v$) 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
$0<p\leq \infty$. In $d=2$ an application of our construction yields many
sequences on the two-sphere, such that the initial segments $Z_N$ have low
$L^\infty$--discrepancy.
