On the $L_p$ discrepancy of two-dimensional folded Hammersley point sets

TL;DR

This paper constructs two-dimensional folded Hammersley point sets with optimal $L_p$ discrepancy for all $1 \,\le\, p \le \infty$, using the $b$-adic baker's transformation to achieve best possible uniformity.

Contribution

It provides an explicit construction of point sets with optimal discrepancy properties for all $L_p$ norms, extending previous work to a broader class of transformations.

Findings

Folded Hammersley point sets achieve optimal $L_p$ discrepancy for all $p$.

The $b$-adic baker's transformation effectively enhances uniformity.

The minimum Niederreiter-Rosenbloom-Tsfasman and Dick weights are sufficiently large.

Abstract

We give an explicit construction of two-dimensional point sets whose $L_p$ discrepancy is of best possible order for all $1\le p\le \infty$. It is provided by folding Hammersley point sets in base $b$ by means of the $b$-adic baker's transformation which has been introduced by Hickernell (2002) for $b=2$ and Goda, Suzuki and Yoshiki (2013) for arbitrary $b\in \mathbb{N}$, $b\ge 2$. We prove that both the minimum Niederreiter-Rosenbloom-Tsfasman weight and the minimum Dick weight of folded Hammersley point sets are large enough to achieve the best possible order of $L_p$ discrepancy for all $1\le p\le \infty$.