Sharp general and metric bounds for the star discrepancy of perturbed Halton--Kronecker sequences

TL;DR

This paper derives sharp bounds for the star discrepancy of hybrid two-dimensional sequences combining Kronecker and perturbed Halton sequences, with implications for metric number theory and lacunary trigonometric products.

Contribution

It introduces new discrepancy bounds for hybrid sequences with digital perturbations and explores their metric properties and connections to lacunary trigonometric products.

Findings

Sharp discrepancy estimates for sequences with bounded continued fraction parameters

Metric bounds for lacunary trigonometric products related to the sequences

Enhanced understanding of discrepancy behavior in hybrid digital sequences

Abstract

We consider the star discrepancy of two-dimensional sequences made up as a hybrid between a Kronecker sequence and a perturbed Halton sequence in base 2, where the perturbation is achieved by a digital-sequence construction in the sense of Niederreiter whose generating matrix contains a periodic perturbing sequence of a given period length. Under the assumption that the Kronecker sequence involves a parameter with bounded continued fraction coefficients sharp discrepancy estimates are obtained. Furthermore, we study the problem from a metric point of view as well. Finally, we also present sharp general and tight metric bounds for certain lacunary trigonometric products which appear to be strongly related to these problems.