Dichotomy results for the L1 norm of the discrepancy function

Gagik Amirkhanyan; Dmitriy Bilyk; Michael T. Lacey

doi:10.1016/j.jmaa.2013.08.002

Dichotomy results for the L¹ norm of the discrepancy function

Gagik Amirkhanyan, Dmitriy Bilyk, Michael T. Lacey

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

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 L² norm. In dimension d=2 this fact has been established by Halász, 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 L¹ 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.

Original language	English (US)
Pages (from-to)	1-6
Number of pages	6
Journal	Journal of Mathematical Analysis and Applications
Volume	410
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2013.08.002
State	Published - Feb 1 2014

Bibliographical note

Funding Information:
The authors gratefully acknowledge that this research was supported in part by NSF grants DMS-1101519 (D. Bilyk), DMS-0968499 (M. Lacey), a grant from the Simons Foundation # 229596 (M. Lacey), and the Australian Research Council through grant ARC-DP120100399 (M. Lacey).

Keywords

Discrepancy function
Function spaces
Irregularities of distribution

Access

10.1016/j.jmaa.2013.08.002

OpenUrl availability

Full text

Cite this

@article{4b5dad7dae7040678469728b45a06020,

title = "Dichotomy results for the L1 norm of the discrepancy function",

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 L2 norm. In dimension d=2 this fact has been established by Hal{\'a}sz, 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 L1 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.",

keywords = "Discrepancy function, Function spaces, Irregularities of distribution",

author = "Gagik Amirkhanyan and Dmitriy Bilyk and Lacey, {Michael T.}",

note = "Funding Information: The authors gratefully acknowledge that this research was supported in part by NSF grants DMS-1101519 (D. Bilyk), DMS-0968499 (M. Lacey), a grant from the Simons Foundation # 229596 (M. Lacey), and the Australian Research Council through grant ARC-DP120100399 (M. Lacey).",

year = "2014",

month = feb,

day = "1",

doi = "10.1016/j.jmaa.2013.08.002",

language = "English (US)",

volume = "410",

pages = "1--6",

journal = "Journal of Mathematical Analysis and Applications",

issn = "0022-247X",

publisher = "Academic Press Inc.",

number = "1",

}

TY - JOUR

T1 - Dichotomy results for the L1 norm of the discrepancy function

AU - Amirkhanyan, Gagik

AU - Bilyk, Dmitriy

AU - Lacey, Michael T.

N1 - Funding Information: The authors gratefully acknowledge that this research was supported in part by NSF grants DMS-1101519 (D. Bilyk), DMS-0968499 (M. Lacey), a grant from the Simons Foundation # 229596 (M. Lacey), and the Australian Research Council through grant ARC-DP120100399 (M. Lacey).

PY - 2014/2/1

Y1 - 2014/2/1

N2 - 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 L2 norm. In dimension d=2 this fact has been established by Halász, 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 L1 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.

AB - 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 L2 norm. In dimension d=2 this fact has been established by Halász, 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 L1 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.

KW - Discrepancy function

KW - Function spaces

KW - Irregularities of distribution

UR - http://www.scopus.com/inward/record.url?scp=84884983576&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884983576&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2013.08.002

DO - 10.1016/j.jmaa.2013.08.002

M3 - Article

AN - SCOPUS:84884983576

SN - 0022-247X

VL - 410

SP - 1

EP - 6

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -