Uniform distribution via lattices: from point sets to sequences

TL;DR

This work develops a lattice-based framework to generate low-discrepancy sequences in $[0,1)^d$ by constructing the $S^{\Box}_{b,d}$ and $S^{\boxplus}_{b,d}$ sequences and proving a general transfer theorem: discrepancy bounds for empirical measures of perturbed lattice images $P(T_v(\omega))$ bound the discrepancy of initial segments $Z_N$ of the mapped sequence. It unifies and extends classical one-dimensional results (van der Corput and Niederreiter) to higher dimensions and to sphere domains using projections, yielding explicit bounds in terms of digit sums and $L^p$ norms. The paper also provides refined quantitative bounds on the size of index sets with favorable discrepancy and extends the construction to self-similar sets via Hutchinson's framework. These results offer a versatile, computationally efficient approach to produce highly uniform point sequences on $[0,1)^d$ and $\mathbb{S}^2$ with practical implications for numerical integration and quasi-Monte Carlo methods.

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.

Figures (7)

• Figure 1: The first 11 elements of a possible choice for $S^{\Box}_{2,2}$ in a). In b) we see the underlying construction via shifting $S_{2^2}$ clearly, and its order is seen in a).
• Figure 2: Here we see how Lemma \ref{['lem_latticeTracing']} works for $N=39=3^3+3^2+3$ and $S=S^\Box_{3,1}$ with $\pi_1=\pi_2=\pi_4$ the identity permutation of elements $(1,2)$, and $\pi_3\neq \pi_1$.
• Figure 3: Let $S_N$ be like in Figure \ref{['fig:Lem8Visualization']} and $\mathscr{D}$ be 1-subadditive. Here we see how Lemma \ref{['lem_latticeTracing']} and 1-subadditivity work hand in hand to bound $\mathscr{D}(E_{S_N})$.
• Figure 4: We see the first 11 elements of a possible choice for $S^{\boxplus}_{2,2}$, and how lattice filling and the $S^\Box$-guidance conditions are satisfied, where $S^\Box_{2,2}$ is as in Figure \ref{['fig:Sbox']}, which is also the base for the color scheme. The point $q_{11}$ could be placed in one of the boxes to the right of $q_3$ or $q_7$; the point $q_{12}$ must be placed in the box between $q_9$ and $q_8$; $q_{13}$ must be placed between $q_5$ and $q_4$, etc.
• Figure 5: This visual aid intends to help the reader verify that $M^\mathds{1}(b^m)\leq 1$ in the proof of Theorem B. The curve represents $L^{-1}(\partial C)$ of a spherical cap $C$. The dots represent $\omega_{b^m}$ and $T_v(\omega_{b^m})$ respectively.

Theorems & Definitions (40)

• Definition 1
• Definition 2
• Definition 3
• Remark
• Theorem 1: Main result
• Remark
• Definition 4
• Remark
• Corollary 2
• proof