On Hilbert cubes and primitive roots in finite fields

Ali Alsetri; Xuancheng Shao

doi:10.1007/s00013-021-01672-3

On Hilbert cubes and primitive roots in finite fields

Ali Alsetri, Xuancheng Shao

Mathematics

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

2 Scopus citations

Abstract

We consider the problem of bounding the dimension of Hilbert cubes in a finite field F_p that does not contain any primitive roots. We show that the dimension of such Hilbert cubes is O_ε(p¹^/⁸⁺^ε) for any ε> 0 , matching what can be deduced from the classical Burgess estimate in the special case when the Hilbert cube is an arithmetic progression. We also consider the dual problem of bounding the dimension of multiplicative Hilbert cubes avoiding an interval.

Original language	English
Pages (from-to)	49-56
Number of pages	8
Journal	Archiv der Mathematik
Volume	118
Issue number	1
DOIs	https://doi.org/10.1007/s00013-021-01672-3
State	Published - Jan 2022

Bibliographical note

Publisher Copyright:
© 2021, Springer Nature Switzerland AG.

Keywords

Character sums
Finite field
General arithmetic progression
Hilbert cubes
Multiplicative Hilbert cube
Primitive roots
Quadratic residues

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1007/s00013-021-01672-3

Cite this

@article{54cab54da7d1479e8f28ace1130e5b10,

title = "On Hilbert cubes and primitive roots in finite fields",

abstract = "We consider the problem of bounding the dimension of Hilbert cubes in a finite field Fp that does not contain any primitive roots. We show that the dimension of such Hilbert cubes is Oε(p1/8+ε) for any ε> 0 , matching what can be deduced from the classical Burgess estimate in the special case when the Hilbert cube is an arithmetic progression. We also consider the dual problem of bounding the dimension of multiplicative Hilbert cubes avoiding an interval.",

keywords = "Character sums, Finite field, General arithmetic progression, Hilbert cubes, Multiplicative Hilbert cube, Primitive roots, Quadratic residues",

author = "Ali Alsetri and Xuancheng Shao",

note = "Publisher Copyright: {\textcopyright} 2021, Springer Nature Switzerland AG.",

year = "2022",

month = jan,

doi = "10.1007/s00013-021-01672-3",

language = "English",

volume = "118",

pages = "49--56",

number = "1",

}

TY - JOUR

T1 - On Hilbert cubes and primitive roots in finite fields

AU - Alsetri, Ali

AU - Shao, Xuancheng

PY - 2022/1

Y1 - 2022/1

N2 - We consider the problem of bounding the dimension of Hilbert cubes in a finite field Fp that does not contain any primitive roots. We show that the dimension of such Hilbert cubes is Oε(p1/8+ε) for any ε> 0 , matching what can be deduced from the classical Burgess estimate in the special case when the Hilbert cube is an arithmetic progression. We also consider the dual problem of bounding the dimension of multiplicative Hilbert cubes avoiding an interval.

AB - We consider the problem of bounding the dimension of Hilbert cubes in a finite field Fp that does not contain any primitive roots. We show that the dimension of such Hilbert cubes is Oε(p1/8+ε) for any ε> 0 , matching what can be deduced from the classical Burgess estimate in the special case when the Hilbert cube is an arithmetic progression. We also consider the dual problem of bounding the dimension of multiplicative Hilbert cubes avoiding an interval.

KW - Character sums

KW - Finite field

KW - General arithmetic progression

KW - Hilbert cubes

KW - Multiplicative Hilbert cube

KW - Primitive roots

KW - Quadratic residues

UR - http://www.scopus.com/inward/record.url?scp=85118303742&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85118303742&partnerID=8YFLogxK

U2 - 10.1007/s00013-021-01672-3

DO - 10.1007/s00013-021-01672-3

M3 - Article

AN - SCOPUS:85118303742

SN - 0003-889X

VL - 118

SP - 49

EP - 56

JO - Archiv der Mathematik

JF - Archiv der Mathematik

IS - 1

ER -

On Hilbert cubes and primitive roots in finite fields

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this