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

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 $p\in (1,\infty)$.