Uniformly distributed sequences generated by a greedy minimization of the $L_2$ discrepancy

Ralph Kritzinger

Published 2021-09-13Version 1

The $L_2$ discrepancy is a quantitative measure for the irregularity of distribution of point sets in $d$-dimensional $[0,1]^d$. We construct sequences in a greedy way such that the inclusion of a new element always minimizes the $L_2$ discrepancy. We will do so for the classical star $L_2$ discrepancy where the test sets are intervals anchored in the origin and the extreme and periodic $L_2$ discrepancy, where arbitrary unanchored subintervals of $[0,1]^d$ and periodic intervals modulo 1 are used as test sets, respectively. We will prove that the sequences we obtain by these greedy algorithms are uniformly distributed modulo 1. In dimension 1, we prove results on the structure of the resulting sequences, where we observe that a greedy minimization of the star $L_2$ discrepancy yields a novel sequence in discrepancy theory with interesting properties, while a greedy minimization of the extreme or periodic $L_2$ discrepancy yields the wellknown van der Corput sequence. The latter follows directly from a recent result by Pausinger.