%0 Journal Article
%T Dichotomy Results for the L1 Norm of the Discrepancy Function
%A Gagik Amirkhanyan
%A Dmitriy Bilyk
%A Michael T Lacey
%J Mathematics
%D 2013
%I arXiv
%R 10.1016/j.jmaa.2013.08.002
%X 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.
%U http://arxiv.org/abs/1306.1761v2