L2 discrepancy of generalized Zaremba point sets

We give an exact formula for the L2 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 points to obtain the optimal order of L2 discrepancy. On the contrary, our study shows that only one non-zero shift is enough for the same purpose, whatever the base b is.