$L_p$-discrepancy of the symmetrized van der Corput sequence

TL;DR

This paper demonstrates that symmetrizing the van der Corput sequence achieves optimal $L_p$-discrepancy rates for all $p$ in (1,∞), extending known results from the $L_2$ case to all such $p$.

Contribution

It generalizes the optimal $L_2$-discrepancy result for the symmetrized van der Corput sequence to all $p$ in (1,∞) using Haar coefficients and Littlewood-Paley inequality.

Findings

Achieves optimal $L_p$-discrepancy rates for all $p eq 1, \, \infty$

Extends known $L_2$ results to a broader range of $p$

Uses Haar coefficient estimates and Littlewood-Paley inequality in proof

Abstract

It is well known that the $L_p$-discrepancy for $p \in [1,\infty]$ of the van der Corput sequence is of exact order of magnitude $O((\log N)/N)$. This however is for $p \in (1,\infty)$ not best possible with respect to the lower bounds according to Roth and Proinov. For the case $p=2$ it is well known that the symmetrization trick due to Davenport leads to the optimal $L_2$-discrepancy rate $O(\sqrt{\log N}/N)$ for the symmetrized van der Corput sequence. In this note we show that this result holds for all $p \in (1,\infty)$. The proof is based on an estimate of the Haar coefficients of the corresponding local discrepancy and on the use of the Littlewood-Paley inequality.