Discrepancy bounds for low-dimensional point sets

TL;DR

This paper discusses discrepancy bounds for low-dimensional $(t,m,s)$-nets and $(t,s)$-sequences, which are crucial in quasi-Monte Carlo methods for integration and approximation, highlighting their theoretical properties and structural insights.

Contribution

It provides new bounds and analysis for discrepancy in low-dimensional point sets and sequences, enhancing understanding of their structural properties.

Findings

Established improved discrepancy bounds for low-dimensional nets and sequences

Analyzed structural properties of Hammersley point sets and van der Corput sequences

Enhanced theoretical understanding of low-dimensional quasi-Monte Carlo point sets

Abstract

The class of $(t,m,s)$-nets and $(t,s)$-sequences, introduced in their most general form by Niederreiter, are important examples of point sets and sequences that are commonly used in quasi-Monte Carlo algorithms for integration and approximation. Low-dimensional versions of $(t,m,s)$-nets and $(t,s)$-sequences, such as Hammersley point sets and van der Corput sequences, form important sub-classes, as they are interesting mathematical objects from a theoretical point of view, and simultaneously serve as examples that make it easier to understand the structural properties of $(t,m,s)$-nets and $(t,s)$-sequences in arbitrary dimension. For these reasons, a considerable number of papers have been written on the properties of low-dimensional nets and sequences.