The Inverse Conjecture for The Gowers Norm Over Finite Fields Via The Correspondence Principle

Terence Tao; Tamar Ziegler

doi:10.2140/apde.2010.3.1

The Inverse Conjecture for The Gowers Norm Over Finite Fields Via The Correspondence Principle

Terence Tao^*, Tamar Ziegler

^*Corresponding author for this work

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

44 Scopus citations

Abstract

The inverse conjecture for the Gowers norms U^d(V) for finite-dimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm ∥ f ∥U^d(V)if and only if it correlates with a phase polynomial (Equation Presented) of degree at most d-1, thus P:V ⌜ F is a polynomial of degree at most d-1. In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case char F > d from an ergodic theory counterpart, which was recently established by Bergelson, Tao and Ziegler.

Original language	English
Journal	Analysis and PDE
Volume	3
Issue number	1
DOIs	https://doi.org/10.2140/apde.2010.3.1
State	Published - 2010
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2010 by Mathematical Sciences Publishers

Keywords

Characteristic factor
Furstenberg correspondence principle
Gowers uniformity norm
Polynomials over finite fields

Access to Document

10.2140/apde.2010.3.1

Cite this

@article{13b485f9c84d49379569d5edcd4063ec,

title = "The Inverse Conjecture for The Gowers Norm Over Finite Fields Via The Correspondence Principle",

abstract = "The inverse conjecture for the Gowers norms Ud(V) for finite-dimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm ∥ f ∥Ud(V)if and only if it correlates with a phase polynomial (Equation Presented) of degree at most d-1, thus P:V ⌜ F is a polynomial of degree at most d-1. In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case char F > d from an ergodic theory counterpart, which was recently established by Bergelson, Tao and Ziegler.",

keywords = "Characteristic factor, Furstenberg correspondence principle, Gowers uniformity norm, Polynomials over finite fields",

author = "Terence Tao and Tamar Ziegler",

note = "Publisher Copyright: {\textcopyright} 2010 by Mathematical Sciences Publishers",

year = "2010",

doi = "10.2140/apde.2010.3.1",

language = "אנגלית",

volume = "3",

journal = "Analysis and PDE",

issn = "2157-5045",

publisher = "Mathematical Sciences Publishers",

number = "1",

}

TY - JOUR

T1 - The Inverse Conjecture for The Gowers Norm Over Finite Fields Via The Correspondence Principle

AU - Tao, Terence

AU - Ziegler, Tamar

PY - 2010

Y1 - 2010

N2 - The inverse conjecture for the Gowers norms Ud(V) for finite-dimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm ∥ f ∥Ud(V)if and only if it correlates with a phase polynomial (Equation Presented) of degree at most d-1, thus P:V ⌜ F is a polynomial of degree at most d-1. In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case char F > d from an ergodic theory counterpart, which was recently established by Bergelson, Tao and Ziegler.

AB - The inverse conjecture for the Gowers norms Ud(V) for finite-dimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm ∥ f ∥Ud(V)if and only if it correlates with a phase polynomial (Equation Presented) of degree at most d-1, thus P:V ⌜ F is a polynomial of degree at most d-1. In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case char F > d from an ergodic theory counterpart, which was recently established by Bergelson, Tao and Ziegler.

KW - Characteristic factor

KW - Furstenberg correspondence principle

KW - Gowers uniformity norm

KW - Polynomials over finite fields

UR - http://www.scopus.com/inward/record.url?scp=79952194340&partnerID=8YFLogxK

U2 - 10.2140/apde.2010.3.1

DO - 10.2140/apde.2010.3.1

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:79952194340

SN - 2157-5045

VL - 3

JO - Analysis and PDE

JF - Analysis and PDE

IS - 1

ER -

The Inverse Conjecture for The Gowers Norm Over Finite Fields Via The Correspondence Principle

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this