论文标题
具有小随机误差项的多维低距离序列序列的配对相关函数
The Pair Correlation Function of Multi-Dimensional Low-Discrepancy Sequences with Small Stochastic Error Terms
论文作者
论文摘要
在任何尺寸$ d \ geq 2 $中，都没有一个具有托儿所对相关性的低静止序列的示例。从某种意义上说，这是令人惊讶的，因为低拨序序列总是具有$β$ -Poissonian对的相关性，用于所有$ 0 <β<\ tfrac {1} {d} $，因此任意接近Poissonian Pairessonian对相关性（与情况相对应（$β= \ tfrac $β= \ tfrac {dffrac）。在本文中，我们进一步阐述了这两个概念的亲密关系。我们表明，$ d $二维的kronecker序列，用于近似近似矢量的$ \vecα$，具有任意的小均匀分布的随机误差项，通常具有$β= \ tfrac {1} {d} {d} $ - poissonian对相关性。
In any dimension $d \geq 2$, there is no known example of a low-discrepancy sequence which possess Poisssonian pair correlations. This is in some sense rather surprising, because low-discrepancy sequences always have $β$-Poissonian pair correlations for all $0 < β< \tfrac{1}{d}$ and are therefore arbitrarily close to having Poissonian pair correlations (which corresponds to the case $β= \tfrac{1}{d}$). In this paper, we further elaborate on the closeness of the two notions. We show that $d$-dimensional Kronecker sequences for badly approximable vectors $\vecα$ with an arbitrary small uniformly distributed stochastic error term generically have $β= \tfrac{1}{d}$-Poissonian pair correlations.
