L_p- and S_{p,q}^rB-discrepancy of the symmetrized van der Corput sequence and modified Hammersley point sets in arbitrary bases

TL;DR

This paper analyzes the discrepancy of symmetrized van der Corput sequences and modified Hammersley point sets in arbitrary bases, providing optimal bounds in Besov spaces and generalizing previous results.

Contribution

It introduces new bounds on the local discrepancy in Besov spaces for these sequences and point sets, extending prior work and achieving optimal order results.

Findings

Achieves optimal order bounds for $L_p$-discrepancy.

Provides sharp upper bounds for $S_{pq}^rB$-discrepancy in one dimension.

Generalizes previous discrepancy estimates to arbitrary bases.

Abstract

We study the local discrepancy of a symmetrized version of the well-known van der Corput sequence and of modified two-dimensional Hammersley point sets in arbitrary base $b$. We give upper bounds on the norm of the local discrepancy in Besov spaces of dominating mixed smoothness $S_{pq}^rB([0,1)^s)$, which will also give us bounds on the $L_p$-discrepancy. Our sequence and point sets will achieve the known optimal order for the $L_p$- and $S_{pq}^rB$-discrepancy. The results in this paper generalize several previous results on $L_p$- and $S_{pq}^rB$-discrepancy estimates and provide a sharp upper bound on the $S_{pq}^rB$-discrepancy of one-dimensional sequences for $r>0$. We will use the $b$-adic Haar function system in the proofs.