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

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 -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.