Extreme and periodic $L_2$ discrepancy of plane point sets

Abstract: In this paper we study the extreme and the periodic $L_2$ discrepancy of plane point sets. The extreme discrepancy is based on arbitrary rectangles as test sets whereas the periodic discrepancy uses "periodic intervals", which can be seen as intervals on the torus. The periodic $L_2$ discrepancy is, up to a multiplicative factor, also known as diaphony. The main results are exact formulas for these kinds of discrepancies for the Hammersley point set and for rational lattices. In order to value the obtained results we also prove a general lower bound on the extreme $L_2$ discrepancy for arbitrary point sets in dimension $d$, which is of order of magnitude $(\log N)^{(d-1)/2}$, like the standard and periodic $L_2$ discrepancies, respectively. Our results confirm that the extreme and periodic $L_2$ discrepancies of the Hammersley point set are of best possible asymptotic order of magnitude. This is in contrast to the standard $L_2$ discrepancy of the Hammersley point set. Furthermore our exact formulas show that also the $L_2$ discrepancies of the Fibonacci lattice are of the optimal order. We also prove that the extreme $L_2$ discrepancy is always dominated by the standard $L_2$ discrepancy, a result that was already conjectured by Morokoff and Caflisch when they introduced the notion of extreme $L_2$ discrepancy in the year 1994.