Point sets with low L p-discrepancy

In this paper we study the L p-discrepancy of digitally shifted Hammersley point sets. While it is known that the (unshifted) Hammersley point set (which is also known as Roth net) with N points has L p-discrepancy (p an integer) of order (log N)/N, we show that there always exists a shift such that the digitally shifted Hammersley point set has L p-discrepancy (p an even integer) of order $$\sqrt {\log N} /N$$ which is best possible by a result of W. Schmidt. Further we concentrate on the case p = 2. We give very tight lower and upper bounds for the L 2-discrepancy of digitally shifted Hammersley point sets which show that the value of the L 2-discrepancy of such a point set mostly depends on the number of zero coordinates of the shift and not so much on the position of these.