A thorough analysis of the discrepancy of shifted Hammersley and van der Corput point sets
If \(\omega=\{{\mathbf x}_n=(x_n,y_n)\), \(0\leq n\leq 2^m-1\}\) is a Hammersley net (\(x_n=n/2^m\), \(y_n=\sum_{i=0}^{m-1} a_i/2^{i+1}\), \(n=\sum_{i=0}^{m-1}a_i2^i\)), then a \(\sigma\)-shifted Hammersley net with \(\sigma=(\sigma_1,\dots,\sigma_m)^{m-1}\in{\mathbb Z}_2^m\) is the net \(\omega_\sigma=\{{\mathbf x}_n=(x_n,y_n)\), \(0\leq n\leq 2^m-1\}\) with \(y_n=\sum_{i=1}^m (a_i\oplus \sigma_{i+1})/2^i\), where \(\oplus\) is addition modulo 2. The shifted Hammersley nets are introduced because the digital shifts can improve the distribution properties of the classical Hammersley nets. The authors focus their investigation on the leading term in the estimations of the star discrepancy \(D^*\). For instance, while the leading term of the upper estimate of \(2^mD^*_{2^m}(\omega)\) is \(m/3\), there is a shifted Hammersley net \(\omega_\sigma\) with \(2^mD^*_{2^m}(\omega_\sigma)\leq m/6+O(1)\). A detailed investigation of the star discrepancy of shifted Hammersley nets contained in the paper shows that for the star discrepancy of any \(\sigma\)-shifted Hammersley net \(\omega_\sigma\) we have \(2^mD^*_{2^m}(\omega)\geq m/6-4/9\) and that the value of \(2^mD^*_{2^m}(\omega)\) is influenced by the number of ones in \(\sigma\) and by the number of changes from ones to zeros in \(\sigma\). The estimates of the star discrepancy of shifted Hammerley nets depend on certain sums of distances to the nearest integer, which are also studied in the paper. Finally, some interesting results concerning the star discrepancy of shifted van der Corput sequences are proved in the last section of the paper.
scientific article; zbMATH DE number 5177471
scientific article