The weighted star discrepancy of Korobov's $p$-sets

TL;DR

This paper investigates the weighted star discrepancy of Korobov's $p$-sets, providing bounds that demonstrate their effectiveness in high-dimensional settings and conditions for polynomial tractability.

Contribution

It offers new bounds on the weighted star discrepancy of $p$-sets and establishes conditions for polynomial tractability in high dimensions.

Findings

Bounds on weighted star discrepancy for any weights

Conditions for dimension-independent discrepancy bounds

Weak conditions suffice for polynomial tractability

Abstract

We analyze the weighted star discrepancy of so-called $p$-sets which go back to definitions due to Korobov in the 1950s and Hua and Wang in the 1970s. Since then, these sets have largely been ignored since a number of other constructions have been discovered which achieve a better convergence rate. However, it has recently been discovered that the $p$-sets perform well in terms of the dependence on the dimension. We prove bounds on the weighted star discrepancy of the $p$-sets which hold for any choice of weights. For product weights we give conditions under which the discrepancy bounds are independent of the dimension $s$. This implies strong polynomial tractability for the weighted star discrepancy. We also show that a very weak condition on the product weights suffices to achieve polynomial tractability.