Discrepancy estimates for sequences: new results and open problems

TL;DR

This paper reviews recent advances in discrepancy estimates for various sequences in the unit cube, including hybrid sequences combining different types, and discusses open problems in this active research area.

Contribution

It provides an overview of recent results on discrepancy estimates for sequences and introduces open problems, especially for hybrid sequences combining different sequence types.

Findings

Discrepancy estimates vary for different sequence classes.

Hybrid sequences' discrepancy analysis is a current research focus.

Open problems highlight gaps in understanding discrepancy behavior.

Abstract

In this paper we give an overview of recent results on (upper and lower) discrepancy estimates for (concrete) sequences in the unit-cube. In particular we also give an overview of discrepancy estimates for certain classes of hybrid sequences. Here by a hybrid sequence we understand an $(s+t)$-dimensional sequence which is a combination of an $s$-dimensional sequence of a certain type (e.g. Kronecker-, Niederreiter-, Halton-,... type) and a $t$-dimensional sequence of another type. The analysis of the discrepancy of hybrid sequences (and of their components) is a rather current and vivid branch of research. We give a collection of some challenging open problems on this topic.