Dichotomy Results for the L1 Norm of the Discrepancy Function

TL;DR

This paper investigates the L1 norm of the discrepancy function in irregularities of distribution, establishing dichotomy results that link small L1 norms to large norms in other spaces, advancing understanding of discrepancy bounds.

Contribution

It introduces dichotomy results showing that small L1 discrepancy implies large discrepancy in other function spaces, providing new insights into discrepancy theory.

Findings

If L1 norm is too small, discrepancy is large in some other space

Supports the conjecture relating L1 and L2 norms in discrepancy theory

Advances understanding of the structure of discrepancy functions

Abstract

It is a well-known conjecture in the theory of irregularities of distribution that the L1 norm of the discrepancy function of an N-point set satisfies the same asymptotic lower bounds as its L^2 norm. In dimension d=2 this fact has been established by Halasz, while in higher dimensions the problem is wide open. In this note, we establish a series of dichotomy-type results which state that if the L^1 norm of the discrepancy function is too small (smaller than the conjectural bound), then the discrepancy function has to be large in some other function space.