Extreme and periodic L_2 discrepancy of plane point sets
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
... btained 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.