缺项系统诱导的双无限矩阵的星离散界
Star Discrepancy Bounds of Double Infinite Matrices induced by Lacunary Systems
摘要：
在2001年Heinrich，Novak，Wasilkowski和Woonniakowski证明了星差的逆满足$n(d,\varepsilon)\leq c_{\abs}d \varepsilon^{-2}$，证明了在$[0,1)^d$中存在一组点，其星差以$c_{\abs}\sqrt{d/N}$为界。Aistleitner推广了这一结果，他证明了存在一个双无限随机矩阵，其元素在$[0,1)$中，部分是Halton序列元素的坐标，部分是独立的均匀分布随机变量，使得任何$N\times d$ -本文考虑一个类似的双无限矩阵，其中元素不是独立随机变量，而是随机变量，从一个特定的多元缺项序列，并证明了高概率的每个投影定义了一组点，它有一些常数相同的上界上的明星差异，但只需要一个显着较低的数字模拟。
英文摘要：
In 2001 Heinrich, Novak, Wasilkowski and Woźniakowski proved that the inverse of the star discrepancy satisfies $n(d,\varepsilon)\leq c_{\abs}d \varepsilon^{-2}$ by showing that there exists a set of points in $[0,1)^d$ whose star-discrepancy is bounded by $c_{\abs}\sqrt{d/N}$. This result was generalized by Aistleitner who showed that there exists a double infinite random matrix with elements in $[0,1)$ which partly are coordinates of elements of a Halton sequence and partly independent uniformly distributed random variables such that any $N\times d$-dimensional projection defines a set $\{x_1,\ldots,x_N\}\subset [0,1)^d$ with \begin{equation*} D^*_N(x_1,\ldots,x_N)\leq c_{\abs}\sqrt{d/N}. \end{equation*} In this paper we consider a similar double infinite matrix where the elements instead of independent random variables are taken from a certain multivariate lacunary sequence and prove that with high probability each projection defines a set of points which has up to some constant the same upper bound on its star-discrepancy but only needs a significantly lower number of digits to simulate.
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