Peter Kritzer: Two-dimensional point sets with low $L_{p}$ discrepency

This is a joint work with Friedrich Pillichshammer.

We study the $L_{p}$-discrepancy of the digitally shifted Hammersley point set in base 2. It has been shown by Pillichshammer that for any natural number $p$ the $L_{p}$-discrepancy of the unshifted Hammersley point set $H$ with $N=2^{m}$ points satisfies

$$\big(NL_{p,N}(H)\big)^{p}=\frac{m^{p}}{2^{3p}}+O\big((\log N)^{p-1}\big),$$

i. e., the $L_{p}$-discrepancy of the unshifted Hammersley point set is of order $\log N/N$. Here we show that, for even integers $p$, there always exists a digital shift of the Hammersley point set such that its $L_{p}$-discrepancy is of order $\sqrt{\log N}/N$, which is best possible by a result of Schmidt. For the special case $p=2$, we give very tight lower and upper bounds on the $L_{2}$-discrepancy of digitally shifted Hammersley point sets. These bounds show that the value of the $L_{2}$-discrepancy mostly depends on the number of zeros in the shift vector and not so much on the position of these. We compare our results to an existing result by Halton and Zaremba and draw some interesting consequences.

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