# AN INVERSE THEOREM FOR THE GOWERS U4-NORM

@article{Green2010ANIT,
title={AN INVERSE THEOREM FOR THE GOWERS U4-NORM},
author={Ben Green and Terence Tao and Tamar D. Ziegler},
journal={Glasgow Mathematical Journal},
year={2010},
volume={53},
pages={1 - 50}
}
• Published 1 March 2005
• Mathematics
• Glasgow Mathematical Journal
Abstract We prove the so-called inverse conjecture for the Gowers Us+1-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we show that if f : [N] → ℂ is a function with |f(n)| ≤ 1 for all n and ‖f‖U4 ≥ δ then there is a bounded complexity 3-step nilsequence F(g(n)Γ) which correlates with f. The approach seems to generalise so as to prove the inverse conjecture for s ≥ 4 as well, and a longer paper will follow concerning this. By combining the main result…
157 Citations
An inverse theorem for the Gowers U^{s+1}[N]-norm (announcement)
• Mathematics
• 2010
In this note we announce the proof of the inverse conjecture for the Gowers U^{s+1}[N]-norm for all s => 3; this is new for s => 4, the cases s = 1,2,3 having been previously established. More
An inverse theorem for the Gowers U^{s+1}[N]-norm
• Mathematics
• 2010
We prove the inverse conjecture for the Gowers U^{s+1}[N]-norm for all s >= 3; this is new for s > 3, and the cases s [-1,1] is a function with || f ||_{U^{s+1}[N]} > \delta then there is a
The Littlewood-Gowers problem
AbstractThe paper has two main parts. To begin with, suppose that G is a compact abelian group. Chang’s Theorem can be viewed as a structural refinement of Bessel’s inequality for functions ƒ ∈
LINEAR FORMS AND QUADRATIC UNIFORMITY FOR FUNCTIONS ON n p
• Mathematics
• 2011
We give improved bounds for our theorem in [W. T. Gowers and J. Wolf, The true complexity of a system of linear equations. Proc. London Math. Soc.  (3) 100 (2010), 155–176], which shows that a system
Gowers norms control Diophantine inequalities
The classical 'Generalised von Neumann Theorem' is a central tool in the study of systems of linear equations with integer coefficients. This theorem reduces the task of counting weighted solutions
Quantitative inverse theory of Gowers uniformity norms
(This text is a survey written for the Bourbaki seminar on the work of F. Manners.) Gowers uniformity norms are the central objects of higher order Fourier analysis, one of the cornerstones of
New bounds for Szemeredi's theorem, Ia: Progressions of length 4 in finite field geometries revisited
• Mathematics
• 2012
Let p > 4 be a prime. We show that the largest subset of F_p^n with no 4-term arithmetic progressions has cardinality << N(log N)^{-c}, where c = 2^{-22} and N := p^n. A result of this type was
RECURRENCE AND NON-UNIFORMITY OF BRACKET POLYNOMIALS
In his celebrated proof of Szemerédi’s theorem that a set of integers of positive density contains arbitrarily long arithmetic progressions, W. T. Gowers introduced a certain sequence of norms ‖ · ‖U
A refinement of the Cameron–Erdős conjecture
• Mathematics
• 2012
In this paper, we study sum‐free subsets of the set {1, …, n}, that is, subsets of the first n positive integers which contain no solution to the equation x+y=z. Cameron and Erdős conjectured in 1990
A NOTE ON THE FREIMAN AND BALOG–SZEMERÉDI–GOWERS THEOREMS IN FINITE FIELDS
• Mathematics
Journal of the Australian Mathematical Society
• 2009
Abstract We prove quantitative versions of the Balog–Szemerédi–Gowers and Freiman theorems in the model case of a finite field geometry 𝔽2n, improving the previously known bounds in such theorems.

## References

SHOWING 1-10 OF 122 REFERENCES
The inverse conjecture for the Gowers norm over finite fields via the correspondence principle
• Mathematics
• 2010
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 normk fkU d.V/
The Mobius function is strongly orthogonal to nilsequences
• Mathematics
• 2012
We show that the Mobius function (n) is strongly asymptotically or- thogonal to any polynomial nilsequence (F (g(n))) n2N. Here, G is a sim- ply-connected nilpotent Lie group with a discrete and
AN INVERSE THEOREM FOR THE GOWERS $U^3(G)$ NORM
• Mathematics
Proceedings of the Edinburgh Mathematical Society
• 2008
Abstract There has been much recent progress in the study of arithmetic progressions in various sets, such as dense subsets of the integers or of the primes. One key tool in these developments has
New bounds for Szemerédi's theorem, I: progressions of length 4 in finite field geometries
• Mathematics
• 2009
Let k ⩾ 3 be an integer, and let G be a finite abelian group with |G| = N, where (N, (k − 1)!) = 1. We write rk(G) for the largest cardinality |A| of a set A ⊆ G which does not contain k distinct
A polynomial bound in Freiman's theorem
.Earlier bounds involved exponential dependence in αin the second estimate. Ourargument combines I. Ruzsa’s method, which we improve in several places, as well asY. Bilu’s proof of Freiman’s
Regularity properties for triple systems
• Mathematics
Random Struct. Algorithms
• 2003
The aim of this paper is to establish the analogous statement for 3-uniform hypergraphs, called The Counting Lemma, together with Theorem 3.5 of P. Frankl and V. Rodl, which can be applied in various situations as Szemeredi's Regularity Lemma is for graphs.
Fourier Analysis and Szemerédi's Theorem
The famous theorem of Szemer edi asserts that for every positive integer k and every positive real number > 0 there is a positive integer N such that every subset of f1; 2; : : :; Ng of cardinality
Coherence measures for heralded single-photon sources
• Mathematics
• 2009
We show that the Mobius function mu(n) is strongly asymptotically orthogonal to any polynomial nilsequence n -> F(g(n)L). Here, G is a simply-connected nilpotent Lie group with a discrete and
Polynomial extensions of van der Waerden’s and Szemerédi’s theorems
• Mathematics
• 1996
An extension of the classical van der Waerden and Szemeredi the- orems is proved for commuting operators whose exponents are polynomials. As a consequence, for example, one obtains the following
Linear equations in primes
• Mathematics
• 2006
Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation