L_2 discrepancy of linearly digit scrambled Zaremba point sets
Résumé
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. For the construction of such point sets one needs sequences of permutations of the form πl(k) = αk + l (mod b) for k, l ∈ {0, . . . , b − 1}. As a corollary we obtain a condition on these sequences which yields the best possible order of L_2 discrepancy of generalized Zaremba point sets in the sense of Roth’s lower bound, with very small leading constants.