Explicit constructions of point sets and sequences with low discrepancy

TL;DR

This paper surveys recent explicit constructions of finite point sets and infinite sequences with optimal low discrepancy, addressing longstanding open problems and extending results to various discrepancy measures.

Contribution

It reviews explicit constructions of low discrepancy point sets and sequences that achieve optimal order, including recent solutions for infinite sequences.

Findings

Explicit constructions match Roth's lower bounds in order of magnitude.

Constructions extend to $ ext{L}_q$ discrepancy for $q eq 2$.

Addresses longstanding open problem in discrepancy theory.

Abstract

In this article we survey recent results on the explicit construction of finite point sets and infinite sequences with optimal order of $\mathcal{L}_q$ discrepancy. In 1954 Roth proved a lower bound for the $\mathcal{L}_2$ discrepancy of finite point sets in the unit cube of arbitrary dimension. Later various authors extended Roth's result to lower bounds also for the $\mathcal{L}_q$ discrepancy and for infinite sequences. While it was known already from the early 1980s on that Roth's lower bound is best possible in the order of magnitude, it was a longstanding open question to find explicit constructions of point sets and sequences with optimal order of $\mathcal{L}_2$ discrepancy. This problem was solved by Chen and Skriganov in 2002 for finite point sets and recently by the authors of this article for infinite sequences. These constructions can also be extended to give optimal order of the $\mathcal{L}_q$ discrepancy of finite point sets for $q \in (1,\infty)$. The main aim of this article is to give an overview of these constructions and related results.