Optimal Order of L -Discrepancy of Digit P Shifted Hammersley Point Sets in Dimension 2

It is well known that the two-dimensional Hammersley point set consisting of N = 2n elements (also known as Roth net) does not have optimal order of Lp-discrepancy for p [1, ) in the sense of the lower bounds according ∈ ∞ to Roth (for p [2, )), Schmidt (for p (1, 2)) and Hala´sz (for p = 1). On the ∈ ∞ ∈ other hand, it is also known that slight modifications of the Hammersley point set can lead to the optimal order √log N/N of L2-discrepancy, where N is the number of points. Among these are for example digit shifts or the symmetrization. In this paper we show that these modified Hammersley point sets also achieve optimal order of Lp-discrepancy for all p (1, ). ∈ ∞