Fixed volume discrepancy in the periodic case

TL;DR

This paper investigates the fixed volume discrepancy in the periodic setting, demonstrating that Frolov point sets achieve optimal decay rates for smooth periodic discrepancy and establishing upper bounds for these measures.

Contribution

It proves the optimal decay order of the smooth periodic discrepancy for Frolov point sets and provides upper bounds for the r-smooth fixed volume discrepancy.

Findings

Frolov point sets have optimal decay rates for smooth periodic discrepancy.

Upper bounds are established for the r-smooth fixed volume discrepancy.

The results improve understanding of discrepancy behavior in the periodic case.

Abstract

The smooth fixed volume discrepancy in the periodic case is studied here. It is proved that the Frolov point sets adjusted to the periodic case have optimal in a certain sense order of decay of the smooth periodic discrepancy. The upper bounds for the $r$-smooth fixed volume periodic discrepancy for these sets are established.