On M. B. Levin’s Proofs for The Exact Lower Discrepancy Bounds of Special Sequences and Point Sets (A Survey)

Abstract

The goal of this overview article is to give a tangible presentation of the breakthrough works in discrepancy theory [3, 5] 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.

A lower bound on the star discrepancy of generalized Halton sequences in rational bases

“…In the starting paper [7] Levin ensured lim sup N →∞ N D * N (H b1,...,bs (n))/ log s N > 0 by using an elementary method of proof. (This proof can be found with more details in the survey article [6]). This method was generalized in [8] and applied to several modified Halton sequences that are using Cantor's expansions, Neumann-Kakutani's b-adic adding machines, and digital permutations.…”

Discrepancy of Digital Sequences: New Results on a Classical QMC Topic

“…. For a survey of Levin's result we also refer to [48]. It should also be mentioned that there is one single result by Faure [32] from the year 1995 who already gave a lower bound for a particular digital (0, 2)-sequence (in dimension 2) which is also of order (log N ) 2 /N .…”