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On the fixed volume discrepancy of the Fibonacci sets in the integral norms

机译：关于Fibonacci集积分规范的固定卷差异

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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 4-discrepancy. That is, the power of the logarithm in the upper bound decreases. 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. (C) 2020 Elsevier Inc. All rights reserved.
机译：本文致力于研究差异型特征 - 固定体积差异 - 单位广场中设定的斐波纳契点的固定体积差异。最近观察到这一新特征使我们能够从数值积分结果获得最佳分散率。这种观察激励我们彻底研究了这一新版本的差异，这似乎本身很有趣。本文的新成分是在帽子函数的换档方面使用平均值而不是在班次上占用。我们表明，该设置的这种变化导致了平稳固定体积差异的上限的提高，类似于通常的4个差异的众所周知的结果。也就是说，上限在上限中的对数的功率降低。有趣的是，这表明通常差异的“坏箱子”不能“太小”。在平滑差异上的已知结果表明，所获得的界限不能在某种意义上得到改善。 （c）2020 Elsevier Inc.保留所有权利。

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