关于Fibonacci集在积分范数下的定容差
On the fixed volume discrepancy of the Fibonacci sets in the integral norms
摘要：
本文研究单位正方形内Fibonacci点集的一个离散型特征--固定体积离散。最近观察到，这种新的特性使我们能够从数值积分结果中获得最佳的色散率。这一观察促使我们深入研究这个新版本的差异，它本身似乎很有趣。本文的新成分是使用平均的帽子函数的移位，而不是采取的上确界的移位。我们表明，这种变化的设置结果在改善的上界为平滑固定体积的差异，类似于众所周知的结果通常的$L_p$ -差异。有趣的是，这表明通常差异的“坏盒”不可能"太小“。已有的关于光滑偏差的结果表明，所得到的界在一定意义上不能得到改进。
英文摘要：
This paper is devoted to the study of a discrepancy-type characteristic -- the fixed volume discrepancy -- of the Fibonacci point set in the unit square. It was observed recently that this new characteristic allows us to obtain optimal rate of dispersion from numerical integration results. This observation motivates us to thoroughly study this new version of discrepancy, which seems to be interesting by itself. The new ingredient of this paper is the use of the average over the shifts of hat functions instead of taking the supremum over the shifts. We show that this change in the setting results in an improvement of the upper bound for the smooth fixed volume discrepancy, similarly to the well-known results for the usual $L_p$-discrepancy. Interestingly, this shows that ``bad boxes'' for the usual discrepancy cannot be ``too small''. The known results on smooth discrepancy show that the obtained bounds cannot be improved in a certain sense.
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