An exact formula for the $L_2$ discrepancy of the symmetrized Hammersley point set

TL;DR

This paper derives an exact formula for the $L_2$ discrepancy of the symmetrized Hammersley point set, demonstrating its independence from the shift and confirming optimal discrepancy bounds.

Contribution

It provides the first exact formula for the $L_2$ discrepancy of the symmetrized Hammersley point set, highlighting shift invariance.

Findings

Exact $L_2$ discrepancy formula derived

Discrepancy is independent of shift choice

Confirms optimal discrepancy order

Abstract

The process of symmetrization is often used to construct point sets with low $L_p$ discrepancy. In the current work we apply this method to the shifted Hammersley point set. It is known that for every shift this symmetrized point set achieves an $L_p$ discrepancy of order $\mathcal{O}\left(\sqrt{\log{N}}/N\right)$ for $p\in [1,\infty)$, which is best possible in the sense of results by Roth, Schmidt and Hal\'asz. In this paper we present an exact formula for the $L_2$ discrepancy of the symmetrized Hammersley point set, which shows in particular that it is independent of the choice for the shift.