Dichotomy Results for the L1 Norm of the Discrepancy Function
Number Theory
2015-09-02 v2
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.
Cite
@article{arxiv.1306.1761,
title = {Dichotomy Results for the L1 Norm of the Discrepancy Function},
author = {Gagik Amirkhanyan and Dmitriy Bilyk and Michael T Lacey},
journal= {arXiv preprint arXiv:1306.1761},
year = {2015}
}
Comments
8 pages. Accepted to JMAA