L_2 discrepancy of generalized Zaremba point sets

Abstract

We give an exact formula for the $L_2$ discrepancy of a class of generalized two-dimensional Hammersley point sets in base $b$, namely generalized Zaremba point sets. These point sets are digitally shifted Hammersley point sets with an arbitrary number of different digital shifts in base $b$. The Zaremba point set introduced by White in 1975 is the special case where the $b$ shifts are taken repeatedly in sequential order, hence needing at least $b^b$ points to obtain the optimal order of $L_2$ discrepancy. On the contrary, our study shows that only one non-zero shift is enough for the same purpose, whatever the base $b$ is.