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

2019/08/13 by Vladimir Temlyakov, Temlyakov, Vladimir, Mario Ullrich +1
Mathematics · #FOS: Mathematics #Mathematical Approximation and Integration #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1908.04658

openalex publication_date 2019/08/13 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

This paper is devoted to the study of a discrepancy-type characteristic --\nthe fixed volume discrepancy -- of the Fibonacci point set in the unit square.\nIt was observed recently that this new characteristic allows us to obtain\noptimal rate of dispersion from numerical integration results. This observation\nmotivates us to thoroughly study this new version of discrepancy, which seems\nto be interesting by itself. The new ingredient of this paper is the use of the\naverage over the shifts of hat functions instead of taking the supremum over\nthe shifts. We show that this change in the setting results in an improvement\nof the upper bound for the smooth fixed volume discrepancy, similarly to the\nwell-known results for the usual L_p-discrepancy. Interestingly, this shows\nthat ``bad boxes'' for the usual discrepancy cannot be ``too small''. The known\nresults on smooth discrepancy show that the obtained bounds cannot be improved\nin a certain sense.\n

Related

• Dispersion of the Fibonacci and the Frolov point sets
• Connections between numerical integration, discrepancy, dispersion, and universal discretization
• Fibonacci lattices have minimal dispersion on the two-dimensional torus
• Fixed volume discrepancy in the periodic case
• Low-discrepancy point sets for non-uniform measures