Finding exact formulas for the $L_2$ discrepancy of digital $(0,n,2)$-nets via Haar functions

TL;DR

This paper derives exact formulas for the $L_2$ discrepancy of digital $(0,n,2)$-nets using Haar functions, providing insights into their irregularity of distribution and extending results to $L_p$ discrepancy.

Contribution

It introduces a method to compute exact $L_2$ discrepancy formulas for digital nets via Haar coefficients, including shifted and symmetrized variants, and explores discrepancy bounds.

Findings

Exact formulas for $L_2$ discrepancy of digital nets

Analysis of discrepancy behavior for shifted and symmetrized nets

Extension of discrepancy results to all $p eq 1$ using Littlewood-Paley inequality

Abstract

We use the Haar function system in order to study the $L_2$ discrepancy of a class of digital $(0,n,2)$-nets. Our approach yields exact formulas for this quantity, which measures the irregularities of distribution of a set of points in the unit interval. We will obtain such formulas not only for the classical digital nets, but also for shifted and symmetrized versions thereof. The basic idea of our proofs is to calculate all Haar coefficents of the discrepancy function exactly and insert them into Parseval's identity. We will also discuss reasons why certain (symmetrized) digital nets fail to achieve the optimal order of $L_2$ discrepancy and use the Littlewood-Paley inequality in order to obtain results on the $L_p$ discrepancy for all $p\in (1,\infty)$.