March 1, 2005
The Gowers U^3 norm is one of a sequence of norms used in the study of arithmetic progressions. If G is an abelian group and A is a subset of G then the U^3(G) of the characteristic function 1_A is useful in the study of progressions of length 4 in A. We give a comprehensive study of the U^3(G) norm, obtaining a reasonably complete description of functions f : G -> C for which ||f||_{U^3} is large and providing links to recent results of Host, Kra and Ziegler in ergodic theory. As an application we generalise a result of Gowers on Szemeredi's theorem. Writing r_4(G) for the size of the largest set A not containing four distinct elements in arithmetic progression, we show that r_4(G) << |G|(loglog|G|)^{-c} for some absolute constant c. In future papers we will develop these ideas further, obtaining an asymptotic for the number of 4-term progressions p_1 < p_2 < p_3 < p_4 < N of primes as well as superior bounds for r_4(G). Update, December 2023. Proposition 3.2 in the paper, which is stated without detailed proof, is incorrect. For a counterexample, see Candela, Gonzalez-Sanchez and Szegedy arXiv:2311.13899, Remark 4.3. Proposition 3.2 is invoked twice in the paper. First, it is used immediately after its statement to deduce the second part of Theorem 2.3. However, that theorem concerns only vector spaces over finite fields, and in this setting Proposition 3.2 is correct by standard linear algebra. The remark at the end of Section 3 that the argument works for arbitrary $G$ should, however, be deleted. The second application is in the proof of Lemma 10.6. It may well be possible to salvage this lemma, particularly if $P$ is assumed proper, but in any case it is only applied once, in the proof of Proposition 10.8. There, $P$ is proper and, more importantly, $H = \{0\}$ is trivial; in this setting Lemma 10.6 and its proof remain valid.
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November 30, 2009
We prove the so-called inverse conjecture for the Gowers U^{s+1}-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we establish that if f : [N] -> C is a function with |f(n)| <= 1 for all n and || f ||_{U^4} >= \delta then there is a bounded complexity 3-step nilsequence F(g(n)\Gamma) 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 conce...
June 1, 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 precisely we outline a proof (details of which will appear in a forthcoming paper) that if f : [N] -> [-1,1] is a function with || f ||_{U^{s+1}[N]} => \delta then there is a bounded-complexity s-step nilsequence F(g(n)\Gamma) which correlates with f, where the bounds on the complexity ...
December 27, 2021
We state and prove a quantitative inverse theorem for the Gowers uniformity norm $U^3(G)$ on an arbitrary finite abelian group $G$; the cases when $G$ was of odd order or a vector space over ${\mathbf F}_2$ had previously been established by Green and the second author and by Samorodnitsky respectively by Fourier-analytic methods, which we also employ here. We also prove a qualitative version of this inverse theorem using a structure theorem of Host--Kra type for ergodic ${\m...
December 16, 2010
The \emph{Gowers uniformity norms} $\|f\|_{U^k(G)}$ of a function $f: G \to \C$ on a finite additive group $G$, together with the slight variant $\|f\|_{U^k([N])}$ defined for functions on a discrete interval $[N] := \{1,...,N\}$, are of importance in the modern theory of counting additive patterns (such as arithmetic progressions) inside large sets. Closely related to these norms are the \emph{Gowers-Host-Kra seminorms} $\|f\|_{U^k(X)}$ of a measurable function $f: X \to \C$...
March 10, 2021
Let $G$ be a finite-dimensional vector space over a prime field $\mathbb{F}_p$ with some subspaces $H_1, \dots, H_k$. Let $f \colon G \to \mathbb{C}$ be a function. Generalizing the notion of Gowers uniformity norms, Austin introduced directional Gowers uniformity norms of $f$ over $(H_1, \dots, H_k)$ as \[\|f\|_{\mathsf{U}(H_1, \dots, H_k)}^{2^k} = \mathbb{E}_{x \in G,h_1 \in H_1, \dots, h_k \in H_k} \partial_{h_1} \dots \partial_{h_k} f(x)\] where $\partial_u f(x) \colon= f...
May 7, 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 claimed in a previous paper by the authors and published in Proc. London Math. Society. Unfortunately the proof had a gap, and we issue an erratum for that paper here. Our new argument is different and significantly shorter. In fact we prove a stronger result, which can be viewed as a ...
November 23, 2023
In recent work, Jamneshan, Shalom and Tao proved an inverse theorem for the Gowers $U^{k+1}$-norm on finite abelian groups of fixed torsion $m$, where the final correlating harmonic is a polynomial phase function of degree at most $C(k,m)$. They also posed a related central question, namely, whether the bound $C$ can be reduced to the optimal value $k$ for every $m$. We make progress on this question using nilspace theory. First we connect the question to the study of finite ...
September 21, 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<3 have also been previously established. More precisely, we establish that if f : [N] -> [-1,1] is a function with || f ||_{U^{s+1}[N]} > \delta then there is a bounded-complexity s-step nilsequence F(g(n)\Gamma) which correlates with f, where the bounds on the complexity and correlation depend only on s and \delta. From previous results, this conjecture impli...
February 11, 2010
Using the density-increment strategy of Roth and Gowers, we derive Szemeredi's theorem on arithmetic progressions from the inverse conjectures GI(s) for the Gowers norms, recently established by the authors and Ziegler.
October 19, 2006
Define $r_4(N)$ to be the largest cardinality of a set $A$ in $\{1,\dots,N\}$ which does not contain four elements in arithmetic progression. In 1998 Gowers proved that $r_4(N) \ll N(\log \log N)^{-c}$ for some absolute constant $c> 0$. In this paper (part II of a series) we improve this to $r_4(N) \ll N e^{-c\sqrt{\log \log N}}$. In part III of the series we will use a more elaborate argument to improve this to $r_4(N) \ll N(\log N)^{-c}$.