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. 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.

本文致力于研究单位正方形中设置的斐波那契点的差异类型特征（固定体积差异）。最近观察到，这一新特性使我们能够从数值积分结果中获得最佳分散速度。这种观察促使我们彻底研究这种新版本的差异，这本身似乎很有趣。本文的新内容是在帽子功能的转变中使用平均值，而不是在转变中获得最高值。我们显示，设置的这种变化导致平滑固定体积差异的上限有所提高，这与通常情况下众所周知的结果类似${\mathrm{大号}}_p$-差异。即，上限的对数的幂减小。有趣的是，这表明对于通常的差异，“坏盒子”不能太小。关于平滑差异的已知结果表明，在一定意义上不能改善获得的边界。