Uniform distribution via lattices: from point sets to sequences

TL;DR

This paper develops a unified framework for constructing low-discrepancy sequences in multi-dimensional spaces using lattice-based methods, relating bounds of lattice point sets to sequence discrepancy, with applications to sequences on the sphere.

Contribution

Introduces the notion of $f$-sublinearity and relates lattice discrepancy bounds to sequence discrepancy in arbitrary dimensions.

Findings

Constructed new low-discrepancy sequences in hypercubes and on the sphere.

Established bounds for $L^p$-discrepancy of classical sequences using lattice methods.

Unified discrepancy analysis applicable to various discrepancy measures.

Abstract

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.