[Paper Review] How to construct generalized van der Corput sequences

Ingrid Carbone|arXiv (Cornell University)|Apr 18, 2013

Mathematical Approximation and Integration8 references3 citations

TL;DR

This paper presents a simplified algorithm to construct generalized van der Corput sequences, known as LS-sequences, by representing natural numbers in base $L+S$ and inverting their digits. The method generalizes the classical van der Corput construction and provides a geometric explanation for the digit inversion process, proving that the resulting sequence matches the LS-radical inverse function, with the classical van der Corput sequence recovered when $L=b$ and $S=0$. The key contribution is a unified, computationally efficient construction with a clear geometric interpretation of the inversion operation.

ABSTRACT

The LS-sequences of points recently introduced by the author are a generalization of van der Corput sequences. They were constructed by reordering the points of the corresponding LS-sequences of partitions. Here we present another algorithm which coincides with the classical one for van der Corput sequences and is simpler to compute than the original construction. This algorithm is based on the representation of natural numbers in base $L+S$ and gives the van der Corput sequence in base $b$ if $L=b$ and S=0. In this construction, as well as in the van der Corput one, it is essential the inversion of digits of the representation in base $L+S$: in this paper we also give a nice geometrical explanation of this "magical" operation.

Motivation & Objective

• To develop a simpler and more efficient algorithm for constructing LS-sequences of points, which generalize van der Corput sequences.
• To unify the construction of generalized van der Corput sequences under a single framework based on base-$L+S$ representation.
• To provide a geometric explanation for the digit inversion process that underlies the construction of these sequences.
• To prove that the LS-sequence of points coincides with the LS-radical inverse function, ensuring uniform distribution and low discrepancy.
• To extend the classical van der Corput construction to a broader class of sequences while preserving desirable discrepancy properties.

Proposed method

• The algorithm represents each natural number $n$ in base $L+S$, yielding a digit sequence $a_k(n)$.
• Digits are inverted in position and mapped via a transformation $\tilde{a}_k(n)$, which depends on whether $a_k(n) < L$ or $a_k(n) \geq L$.
• The LS-radical inverse function $\phi_{L,S}(n)$ is defined as $\sum_{k=0}^{M} \tilde{a}_k(n) \gamma^{k+1}$, where $\gamma$ is a parameter related to the sequence.
• The sequence $\{\phi_{L,S}(n)\}$ for $n \in \mathbb{N}_{L,S}$ generates the LS-sequence of points.
• The construction ensures that the sequence is uniformly distributed and inherits low discrepancy from the underlying sequence of partitions.
• Geometric intuition is provided by interpreting the inversion process as a transformation on intervals, linking digit reversal to interval subdivision.

Experimental results

Research questions

• RQ1 How can generalized van der Corput sequences be constructed more efficiently than the original reordering method?
• RQ2 What is the geometric meaning behind the digit inversion step in the construction of van der Corput-type sequences?
• RQ3 Can the classical van der Corput sequence be recovered as a special case of a broader class of sequences defined via base-$L+S$ representation?
• RQ4 What conditions ensure that the resulting sequence of points is uniformly distributed and has low discrepancy?
• RQ5 How does the LS-radical inverse function relate to the structure of the underlying sequence of partitions?

Key findings

• The LS-sequence of points is exactly equal to the sequence $\{\phi_{L,S}(n)\}$ defined via the LS-radical inverse function on $\mathbb{N}_{L,S}$.
• When $L = b$ and $S = 0$, the construction reduces to the classical van der Corput sequence in base $b$, confirming consistency with known results.
• The digit inversion process is geometrically interpreted as a transformation that maps intervals in a way that preserves uniform distribution.
• The algorithm is computationally simpler than the original reordering method, as it avoids explicit sorting of partition points.
• The $1,1$-sequence (Kakutani-Fibonacci sequence) is shown to coincide with $\phi_{1,1}(n)$, with explicit computation of the first 12 points demonstrating the method's validity.
• The construction ensures that the resulting sequence of points is uniformly distributed and inherits low discrepancy from the underlying sequence of partitions.