L2discrepancy of generalized two-dimensional Hammersley point sets scrambled with arbitrary permutations

Abstract

The L2 discrepancy is a quantitative measure for the irregularity of distribution of a finite point set. In this paper we consider the L2 discrepancy of so-called generalized Hammersley point sets which can be obtained from the classical Hammersley point sets by introducing some permutations on the base b digits. While for the classical Hammersley point set it is not possible to achieve the optimal order of L2 discrepancy with respect to a general lower bound due to Roth this disadvantage can be overcome with the generalized version thereof. For special permutations we obtain an exact formula for the L2 discrepancy from which we obtain two-dimensional finite point sets with the lowest value of L2 discrepancy known so far. AMS subject classification: 11K06, 11K38.