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

3 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.

Funding

XS was supported by NSF Grant DMS-1802224.

Funders	Funder number
Center for Hierarchical Manufacturing, National Science Foundation
Center for Selective C-H Functionalization, National Science Foundation
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	DMS-1802224

Keywords

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

ASJC Scopus subject areas

General Mathematics

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10.1007/s00013-021-01672-3

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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

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On Hilbert cubes and primitive roots in finite fields

Abstract

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Funding

Keywords

ASJC Scopus subject areas

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