Uniform Distribution via Lattices: From Point Sets to Sequences

Abstract

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