Digital nets in dimension two with the optimal order of $L_p$ discrepancy

TL;DR

This paper demonstrates that certain two-dimensional digital nets inherently achieve the optimal order of $L_p$ discrepancy without the need for symmetrization, extending previous results and confirming their optimal uniformity properties.

Contribution

It identifies specific digital nets in two dimensions that attain the optimal $L_p$ discrepancy order for all $p \, \in [1, \infty)$ without symmetrization, advancing discrepancy theory.

Findings

Certain digital nets achieve optimal $L_p$ discrepancy without symmetrization.

The results extend the class of digital nets known to have optimal discrepancy.

Connections are made to Diophantine properties influencing discrepancy bounds.

Abstract

We study the $L_p$ discrepancy of two-dimensional digital nets for finite $p$. In the year 2001 Larcher and Pillichshammer identified a class of digital nets for which the symmetrized version in the sense of Davenport has $L_2$ discrepancy of the order $\sqrt{\log N}/N$, which is best possible due to the celebrated result of Roth. However, it remained open whether this discrepancy bound also holds for the original digital nets without any modification. In the present paper we identify nets from the above mentioned class for which the symmetrization is not necessary in order to achieve the optimal order of $L_p$ discrepancy for all $p \in [1,\infty)$. Our findings are in the spirit of a paper by Bilyk from 2013, who considered the $L_2$ discrepancy of lattices consisting of the elements $(k/N,\{k \alpha\})$ for $k=0,1,\ldots,N-1$, and who gave Diophantine properties of $\alpha$ which guarantee the optimal order of $L_2$ discrepancy.