The inverse of the star discrepancy of a union of randomly shifted Korobov rank-1 lattice point sets depends polynomially on the dimension

TL;DR

This paper demonstrates that unions of Korobov rank-1 lattice point sets with random shifts achieve near-optimal star discrepancy bounds, with the inverse discrepancy depending quadratically on the dimension, using Fourier analysis in integer arithmetic.

Contribution

It extends previous probabilistic results to classical integer settings, providing explicit bounds and reducing the search space for optimal point sets.

Findings

Star discrepancy bounded by O(s log(N_tot) / sqrt(N_tot)) with high probability

Inverse star discrepancy depends quadratically on the dimension s

Results apply to various construction scenarios with random or fixed generators and shifts

Abstract

The inverse of the star discrepancy, $N(\epsilon, s)$, defined as the minimum number of points required to achieve a star discrepancy of at most $\epsilon$ in dimension $s$, is known to depend linearly on $s$. However, explicit constructions achieving this optimal linear dependence remain elusive. Recently, Dick and Pillichshammer (2025) made significant progress by showing that a multiset union of randomly digitally shifted Korobov polynomial lattice point sets almost achieve the optimal dimension dependence with high probability. In this paper, we investigate the analog of this result in the setting of classical integer arithmetic using Fourier analysis. We analyze point sets constructed as multiset unions of Korobov rank-1 lattice point sets modulo a prime $N$. We provide a comprehensive analysis covering four distinct construction scenarios, combining either random or fixed integer generators with either continuous torus shifts or discrete grid shifts. We prove that in all four cases, the star discrepancy is bounded by a term of order $O(s \log(N_{tot}) / \sqrt{N_{tot}})$ with high probability, where $N_{tot}$ is the total number of points. This implies that the inverse of the star discrepancy for these structured sets depends quadratically on the dimension $s$. While the proofs are probabilistic, our results significantly reduce the search space for optimal point sets from a continuum to a finite set of candidates parameterized by integer generators and random shifts.