Inverse conjecture for the gowers norm is false

Shachar Lovett*, Roy Meshulam, Alex Samorodnitsky

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

24 Scopus citations

Abstract

Let p be a fixed prime number and N be a large integer. The "Inverse Conjecture for the Gowers Norm" states that if the "d-th Gowers norm" of a function f : double strok F signpN → double strok F signp is non-negligible, that is larger than a constant independent of N, then f has a non-trivial correlation with a degree d - 1 polynomial. The conjecture is known to hold for d = 2, 3 and for any prime p. In this paper we show the conjecture to be false for p = 2 and for d = 4, by presenting an explicit function whose 4-th Gowers norm is non-negligible, but whose correlation with any polynomial of degree 3 is exponentially small. Essentially the same result, with different bounds for correlation, was independently obtained by Green and Tao [8]. Their analysis uses a modification of a Ramsey-type argument of Alon and Beigel [1] to show inapproximability of certain functions by low-degree polynomials. We observe that a combination of our results with the argument of Alon and Beigel implies the inverse conjecture to be false for any prime p, for d = p2.

Original languageAmerican English
Title of host publicationSTOC'08
Subtitle of host publicationProceedings of the 2008 ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery (ACM)
Pages547-556
Number of pages10
ISBN (Print)9781605580470
DOIs
StatePublished - 2008
Event40th Annual ACM Symposium on Theory of Computing, STOC 2008 - Victoria, BC, Canada
Duration: 17 May 200820 May 2008

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference40th Annual ACM Symposium on Theory of Computing, STOC 2008
Country/TerritoryCanada
CityVictoria, BC
Period17/05/0820/05/08

Keywords

  • Gowers norm
  • Low degree tests
  • Multivariate polynomials

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