Uniform distribution via lattices: from point sets to sequences

In this work we construct many sequences $S=S_{b,d}^{\mathrm{\square }}$, or $S=S_{b,d}^{⊞}$ 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 $\mathcal{D}$ of the multiset of projected lattice sets $P(b^{-m}{\mathbb{Z}}^d$), to bounds of $\mathcal{D}(Z_N)$, 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 $0<p\le \mathrm{\infty }$. In $d=2$ an application of our construction yields many sequences on the two-sphere, such that the initial segments $Z_N$ have low $L^{\mathrm{\infty }}$–discrepancy.