Lower Bounds for $L_1$ Discrepancy

TL;DR

This paper establishes optimal asymptotic lower bounds for the $L_1$ discrepancy in two dimensions using advanced mathematical techniques, improving understanding of discrepancy measures in numerical integration.

Contribution

It provides the best possible asymptotic lower bounds for the $L_1$ discrepancy coefficient using Roth's orthogonal function method within a broad class of test functions.

Findings

Established the tight asymptotic lower bounds for $L_1$ discrepancy

Applied combinatorics, probability, and harmonic analysis techniques

Enhanced the theoretical understanding of discrepancy measures

Abstract

We find the best asymptotic lower bounds for the coefficient of the leading term of the $L_1$ norm of the two-dimensional (axis-parallel) discrepancy that can be obtained by K.Roth's orthogonal function method among a large class of test functions. We use methods of combinatorics, probability, complex and harmonic analysis.