论文标题
恒星差异的下限的基本证明
An elementary proof of a lower bound for the inverse of the star discrepancy
论文作者
论文摘要
差异理论中的一个中心问题是均匀分布点$ \ left \ {x_1，\ dots，x_n \ right \} $中的挑战。假设一组是如此规律，以至于对于某些$ \ varepsilon> 0 $，以及所有$ y \ in [0,1]^d $ sub-grigion $ [0，y] = [0，y_1] \ times \ times \ dots \ times \ times \ times [0，y_d] $包含与其体积和$ $ \ forall〜y y \ y y y \ y y \ y y \ y y \ y y \ y y \ y y \ y \ y \ y的， \ frac {1} {n} \＃\ left \ {1 \ leq i \ leq n：x_i \ in [0，y] \ right \} - \ mbox {vol}（vol}（[0，y]）（[0，y]）\ right | \ leq \ varepsilon，$ $ n $必须取决于$ d $和$ \ varepsilon $？由于Hinrichs，我们给出了当前最著名的结果的基本证明，这表明$ n \ gtrsim d \ cdot \ varepsilon^{ - 1} $。
A central problem in discrepancy theory is the challenge of evenly distributing points $\left\{x_1, \dots, x_n \right\}$ in $[0,1]^d$. Suppose a set is so regular that for some $\varepsilon> 0$ and all $y \in [0,1]^d$ the sub-region $[0,y] = [0,y_1] \times \dots \times [0,y_d]$ contains a number of points nearly proportional to its volume and $$\forall~y \in [0,1]^d \qquad \left| \frac{1}{n} \# \left\{1 \leq i \leq n: x_i \in [0,y] \right\} - \mbox{vol}([0,y]) \right| \leq \varepsilon,$$ how large does $n$ have to be depending on $d$ and $\varepsilon$? We give an elementary proof of the currently best known result, due to Hinrichs, showing that $n \gtrsim d \cdot \varepsilon^{-1}$.
