Discrepancy of generalized Hammersley type point sets in Besov spaces of dominating mixed smoothness

TL;DR

This paper demonstrates that a broad class of generalized Hammersley point sets asymptotically attain the optimal discrepancy bounds in Besov spaces of dominating mixed smoothness, extending known results in this area.

Contribution

It introduces a generalization of Hammersley point sets and proves their asymptotic optimality in Besov space discrepancy norms using a $b$-adic Haar system.

Findings

Generalized Hammersley point sets achieve the lower bounds in Besov space norms.

The proof employs a $b$-adic Haar system for analysis.

Results are a step towards understanding discrepancy in higher dimensions.

Abstract

The symmetrized Hammersley point set is known to achieve the best possible rate for the $L_2$-norm of the discrepancy function. Also lower bounds for the norm in Besov spaces of dominating mixed smoothness are known. In this paper a large class of point sets which are generalizations of the Hammersley type point sets are proved to asymptotically achieve the known lower bound of the Besov norm. The proof uses a $b$-adic generalization of the Haar system. This result can be regarded as a preparation for the proof in arbitrary dimension.