Optimal order of $L_p$-discrepancy of digit shifted Hammersley point sets in dimension 2

Abstract

It is well known that the two-dimensional Hammersley point set consisting of $N=2^n$ elements (also known as Roth net) does not have optimal order of $L_p$-discrepancy for $p\in (1,\infty )$ in the sense of the lower bounds according to Roth (for $p\in [2,\infty )$) and Schmidt (for $p\in (1,2)$). On the other hand, it is also known that slight modifications of the Hammersley point set can lead to the optimal order $\sqrt{\log N}/N$ of $L_2$-discrepancy, where $N$ is the number of points. Among these are for example digit shifts or the symmetrization. In this paper we show that these modified Hammersley point sets also achieve optimal order of $L_p$-discrepancy for all $p\in (1,\infty )$.

References in corpus (1)

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

Cited by in corpus (2)

• From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules
• Digital nets in dimension two with the optimal order of $L_p$ discrepancy