[Paper Review] A van der Corput-type algorithm for LS-sequences of points

Ingrid Carbone|arXiv (Cornell University)|Sep 17, 2012

Mathematical Approximation and Integration4 references6 citations

TL;DR

This paper introduces a van der Corput-type algorithm to generate LS-sequences of points by reordering points in LS-sequences of partitions using a custom LS-radical inverse function. The method leverages base-$L+S$ representations of natural numbers and affine geometric mappings, extending classical van der Corput sequences to a broader class of low-discrepancy sequences with explicit construction and uniform distribution properties.

ABSTRACT

In this paper we associate to any $LS$-sequence of partitions ${ρ_{L,S}^n}$ the corresponding $LS$-sequence of points ${ξ_{L,S}^n}$ obtained reordering the points of each partition with an explicit algorithm. The procedure begins with the representation in base $L+S$ of natural numbers, $[n]_{L+S}$, and ends with the $LS$-radical inverse function $ϕ_{L,S}$, introduced ad hoc, evaluated at an appropriate subsequence of natural numbers depending on $L$ and $S$. This construction is deeply related to the geometric representation of the points of ${ξ_{L,S}^n}$ by suitable affine functions and reminds the van der Corput sequences in base $b$. Keywords: Uniform distribution, sequences of partitions, van der Corput sequences, discrepancy.

Motivation & Objective

• To develop a systematic algorithm for generating LS-sequences of points from LS-sequences of partitions.
• To extend the classical van der Corput construction to a broader class of sequences parameterized by $L$ and $S$.
• To establish a connection between geometric affine mappings and the distribution of LS-sequences.
• To define and utilize a novel $LS$-radical inverse function for point generation.
• To ensure uniform distribution and low discrepancy in the resulting sequences.

Proposed method

• The construction begins with the base-$L+S$ representation of natural numbers, $[n]_{L+S}$.
• An appropriate subsequence of natural numbers is selected based on parameters $L$ and $S$.
• The $LS$-radical inverse function $ϕ_{L,S}$ is applied to this subsequence to generate point coordinates.
• Points from each partition $ρ_{L,S}^n$ are reordered using this function to form the sequence $ξ_{L,S}^n$.
• Geometric representation via affine functions is used to analyze and construct the sequence.
• The method generalizes the classical van der Corput sequence construction to $LS$-sequences.

Experimental results

Research questions

• RQ1 How can a van der Corput-type algorithm be adapted to generate LS-sequences of points?
• RQ2 What is the role of the $LS$-radical inverse function in ensuring uniform distribution?
• RQ3 How do affine geometric mappings relate to the structure of $LS$-sequences?
• RQ4 What properties of base-$L+S$ representations enable low-discrepancy point sequences?
• RQ5 Can the classical van der Corput construction be generalized to $LS$-sequences with parameters $L$ and $S$?

Key findings

• The proposed algorithm successfully generates $LS$-sequences of points by reordering partitions using the $LS$-radical inverse function.
• The construction ensures uniform distribution of points through explicit geometric and number-theoretic mappings.
• The method generalizes the classical van der Corput sequence to a broader class of sequences via parameterization by $L$ and $S$.
• The $LS$-radical inverse function is defined specifically for this construction and depends on a subsequence of natural numbers determined by $L$ and $S$.
• The geometric interpretation via affine functions provides insight into the distribution and structure of the resulting sequences.
• The sequence $ξ_{L,S}^n$ achieves low discrepancy through a systematic transformation of base-$L+S$ digits.