Dichotomy results for the L1 norm of the discrepancy function

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:410

Article

Amirkhanyan, Gagik¹  Bilyk, Dmitriy²  Lacey, Michael T.¹ 

[1] Georgia Inst Technol, Sch Math, Atlanta, GA 30332 USA

[2] Univ Minnesota, Sch Math, Minneapolis, MN 55455 USA

关键词: Discrepancy function;    Irregularities of distribution;    Function spaces;   

DOI  :  10.1016/j.jmaa.2013.08.002

来源: Elsevier

【 摘 要 】

It is a well-known conjecture in the theory of irregularities of distribution that the L-1 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. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free