Improvements on the star discrepancy of \((t,s)\)-sequences
For an infinite sequence \(X\), \(D^*(N,X)\) denotes the star discrepancy. A low-discrepancy sequence \(X\) satisfies an inequality of the form \(D^*(N, X)\leq c_s(\log N)^s+O((\log N)^{s-1})\). The constant \(c_s\) is the main object of this paper. \textit{H. Niederreiter} [Monatsh. Math. 104, 273--337 (1987; Zbl 0626.10045)] established the discrepancy of \((t,s)\)-sequences in the narrow sense with some constant \(c_s^{\text{Ni}}\). For \((t,s)\)-sequence in narrow sense and in dimension \(s\geq 2\), \textit{P. Kritzer} [J. Complexity 22, No. 3, 336--347 (2006; Zbl 1155.11336)] obtained the improved constants \(c_s^{\text{Kr}}\). In the same paper, when the base \(b\) is even, the smallest value of the constant \(c_s\) for \(s\geq 2\), \(c_s^{\text{conj}}\) is conjectured.NEWLINENEWLINEThe authors provide a new result for the discrepancy bound for \((t,1)\)-sequences. For example they show that the conjecture of Kritzer is true for \(s=1\). For \(s\geq 2\) the authors obtain the discrepancy bound for \((t,s)\)-sequences in the broad sense by using the method of \textit{E. I. Atanassov} [Math. Balk., New Ser. 18, No. 1--2, 15--32 (2004; Zbl 1088.11058)] for Halton sequences. Moreover, for even bases an improvement of the constant \(c_s^{\text{Kr}}\) is given, along with some numerical comparison.
scientific article; zbMATH DE number 6043108