A survey on M. B. Levin's proofs for the exact lower discrepancy bounds of special sequences and point sets

TL;DR

This survey reviews recent breakthroughs by M. B. Levin on exact lower discrepancy bounds for specific sequences, highlighting proof techniques, implications, and extensions in discrepancy theory.

Contribution

It provides a comprehensive overview of Levin's proofs for lower discrepancy bounds and discusses their significance and extensions in the field.

Findings

Levin proved exact lower discrepancy bounds for Halton's sequence.

Levin established bounds for a class of $(t, s)$-sequences.

The survey discusses proof ideas and potential extensions.

Abstract

The goal of this overview article is to give a tangible presentation of recent breakthrough works in discrepancy theory by M. B. Levin. These works provide proofs for the exact lower discrepancy bounds of Halton's sequence and a certain class of $(t, s)$-sequences. Our survey aims at highlighting the major ideas of the proofs and we discuss further implications of the employed methods. Moreover, we derive extensions of Levin's results.