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 $p \in (1,\infty)$.