According to Aistleitner and Weimar, there exist two-dimensional (double) infinite matrices whose star-discrepancy $D_N^{*s}$ of the first $N$ rows and $s$ columns, interpreted as $N$ points in $[0,1]^s$, satisfies an inequality of the form $$D_N^{*s} \leq \sqrt{\alpha} \sqrt{A+B\frac{\ln(\log_2(N))}{s}}\sqrt{\frac{s}{N}}$$ with $\alpha = \zeta^{-1}(2) \approx 1.73, A=1165$ and $B=178$. These matrices are obtained by using i.i.d sequences, and the parameters $s$ and $N$ refer to the dimension and the sample size respectively. In this paper, we improve their result in two directions: First, we change the character of the equation so that the constant $A$ gets replaced by a value $A_s$ dependent on the dimension $s$ such that for $s>1$ we have $A_s<A$. Second, we generalize the result to the case of the (extreme) discrepancy. The paper is complemented by a section where we show numerical results for the dependence of the parameter $A_s$ on $s$.

翻译：根据Aistleitner与Weimar的研究，存在二维（双）无限矩阵，其前N行与前s列构成的星差异$D_N^{*s}$（视为$[0,1]^s$中的N个点）满足以下形式的不等式：$$D_N^{*s} \leq \sqrt{\alpha} \sqrt{A+B\frac{\ln(\log_2(N))}{s}}\sqrt{\frac{s}{N}}$$其中$\alpha = \zeta^{-1}(2) \approx 1.73$，$A=1165$，$B=178$。这些矩阵通过使用独立同分布序列获得，参数$s$和$N$分别表示维度与样本量。本文从两个方向改进了他们的结果：首先，我们改变了方程的性质，将常数$A$替换为依赖于维度$s$的值$A_s$，使得当$s>1$时有$A_s<A$。其次，我们将结果推广至（极值）差异情形。文末附有一节，展示了参数$A_s$随$s$变化的数值结果。