# The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic

@article{Tao2011TheIC,
title={The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic},
author={Terence Tao and Tamar D. Ziegler},
journal={Annals of Combinatorics},
year={2011},
volume={16},
pages={121-188}
}
• Published 7 January 2011
• Mathematics
• Annals of Combinatorics
We establish the inverse conjecture for the Gowers norm over finite fields, which asserts (roughly speaking) that if a bounded function $${f : V \rightarrow \mathbb{C}}$$ on a finite-dimensional vector space V over a finite field $${\mathbb{F}}$$ has large Gowers uniformity norm $${{\parallel{f}\parallel_{U^{s+1}(V)}}}$$ , then there exists a (non-classical) polynomial $${P: V \rightarrow \mathbb{T}}$$ of degree at most s such that f correlates with the phase e(P) = e2πiP. This conjecture had…
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## References

SHOWING 1-10 OF 54 REFERENCES
An Inverse Theorem for the Uniformity Seminorms Associated with the Action of $${{\mathbb {F}^{\infty}_{p}}}$$
• Mathematics
• 2009
Let $${\mathbb {F}}$$ a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm Uk(X) for an ergodic action $${(T_{g})_{{g} \in \mathbb Linear Forms and Higher-Degree Uniformity for Functions On$${\mathbb{F}^{n}_{p}}$$• Mathematics • 2010 In [GW1] we began an investigation of the following general question. Let L1, . . . , Lm be a system of linear forms in d variables on$${F^n_p}$$, and let A be a subset of$${F^n_p} of positive
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
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 quantitative behaviour of polynomial orbits on nilmanifolds
• Mathematics
• 2007
A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\Gamma$. In this paper
The distribution of polynomials over finite fields, with applications to the Gowers norms
• Mathematics
Contributions Discret. Math.
• 2009
The main result is that a polynomial P : F^n -> F is poorly-distributed only if P is determined by the values of a few polynomials of lower degree, in which case it is said that P has small rank.
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
On the norm convergence of non-conventional ergodic averages
• Tim Austin
• Mathematics
Ergodic Theory and Dynamical Systems
• 2009
Abstract We offer a proof of the following non-conventional ergodic theorem: If Ti:ℤr↷(X,Σ,μ) for i=1,2,…,d are commuting probability-preserving ℤr-actions, (IN)N≥1 is a Følner sequence of subsets of
Structure of finite nilspaces and inverse theorems for the Gowers norms in bounded exponent groups
A result of the author shows that the behavior of Gowers norms on bounded exponent abelian groups is connected to finite nilspaces. Motivated by this, we investigate the structure of finite
Universal characteristic factors and Furstenberg averages
Let X=(X^0,\mu,T) be an ergodic measure preserving system. For a natural number k we consider the averages (*) 1/N \sum_{n=1}^N \prod_{j=1}^k f_j(T^{n a_j}x) where the functions f_j are bounded, and