关于高维的Small Ball不等式
On the Small Ball Inequality in All Dimensions
摘要：
令h_R表示L ^{ \infty }归一化的Haar函数，该Haar函数适合于包含在维度d的单位立方体中的并元矩形R。本文建立了双曲型和$$ \sum _{|R|=2 ^{-n}} α(R) h_R (x) $$的L^{ \infty }范数的一个非平凡下界。这个下界的非平凡性在于，我们改进了平均情况下n^η的界，其中η是一个维数为d的函数。据作者所知，这是第一个在4维和更高维度上的此类结果。这个问题与（1）分布的不规则性，（2）逼近论和（3）概率论中的猜想有关。本文的证明方法给出了在四维及四维以上的所有维上这些图的新结果。本文建立在Jozef Beck从1989年开始的先前工作和2006年的前两位作者的基础上。这些结果具有相同的性质，但只是在第三维中。
英文摘要：
Let h_R denote an L ^{\infty} normalized Haar function adapted to a dyadic rectangle R contained in the unit cube in dimension d. We establish a non-trivial lower bound on the L^{\infty} norm of the `hyperbolic' sums $$ \sum _{|R|=2 ^{-n}} α(R) h_R (x) $$ The lower bound is non-trivial in that we improve the average case bound by n^η for some positive η, a function of dimension d. As far as the authors know, this is the first result of this type in dimension 4 and higher. This question is related to Conjectures in (1) Irregularity of Distributions, (2) Approximation Theory and (3) Probability Theory. The method of proof of this paper gives new results on these conjectures in all dimensions 4 and higher. This paper builds upon prior work of Jozef Beck, from 1989, and first two authors from 2006. These results were of the same nature, but only in dimension 3.
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