On the uniform distribution modulo 1 of multidimensional LS-sequences

TL;DR

This paper investigates the distribution properties of multidimensional LS-sequences, revealing that under certain conditions, their coordinatewise combinations do not produce dense or uniformly distributed points in the unit square.

Contribution

It proves that combining low-discrepancy LS-sequences coordinatewise does not always yield a dense or uniformly distributed sequence in two dimensions.

Findings

Certain parameter choices prevent the combined sequence from being dense in [0,1]^2.

Not all low-discrepancy LS-sequences can be extended to multidimensional uniform distributions.

The paper establishes specific number-theoretic conditions affecting distribution properties.

Abstract

Ingrid Carbone introduced the notion of so-called LS-sequences of points, which are obtained by a generalization of Kakutani's interval splitting procedure. Under an appropriate choice of the parameters $L$ and $S$, such sequences have low discrepancy, which means that they are natural candidates for Quasi-Monte Carlo integration. It is tempting to assume that LS-sequences can be combined coordinatewise to obtain a multidimensional low-discrepancy sequence. However, in the present paper we prove that this is not always the case: if the parameters $L_1,S_1$ and $L_2,S_2$ of two one-dimensional low-discrepancy LS-sequences satisfy certain number-theoretic conditions, then their two-dimensional combination is not even dense in $[0,1]^2$.