# Optimal discrepancy rate of point sets in Besov spaces with negative smoothness

## TL;DR

This paper investigates the discrepancy of symmetrized Hammersley point sets in the unit square within Besov spaces of negative smoothness, demonstrating improved rates and implications for numerical integration.

## Contribution

It shows that symmetrized Hammersley point sets achieve optimal discrepancy rates in Besov spaces with negative smoothness, overcoming previous limitations.

## Key findings

- Symmetrized Hammersley points attain optimal discrepancy rates in negative smoothness Besov spaces.
- Results impact the understanding of irregularity measures for point distributions.
- Findings have consequences for quasi-Monte Carlo numerical integration.

## Abstract

We consider the local discrepancy of a symmetrized version of Hammersley type point sets in the unit square. As a measure for the irregularity of distribution we study the norm of the local discrepancy in Besov spaces with dominating mixed smoothness. It is known that for Hammersley type points this norm has the best possible rate provided that the smoothness parameter of the Besov space is nonnegative. While these point sets fail to achieve the same for negative smoothness, we will prove in this note that the symmetrized versions overcome this defect. We conclude with some consequences on discrepancy in further function spaces with dominating mixed smoothness and on numerical integration based on quasi-Monte Carlo rules.