Motivated by the definition of the Gowers uniformity norms, we introduce and study a wide class of norms. Our aim is to establish them as a natural generalization of the $L_p$ norms. We shall prove that these normed spaces share many of the nice properties of the $L_p$ spaces. Some examples of these norms are $L_p$ norms, trace norms $S_p$ when $p$ is an even integer, and Gowers uniformity norms.
Every such norm is defined through a pair of weighted hypergraphs. In regard to a question of Laszlo Lovasz, we prove several results in the direction of characterizing all hypergraph pairs that correspond to norms.
In this note, we prove a certain hypergraph generalization of the Balog-Szemeredi-Gowers Theorem. Our result shares some features in common with a similar such generalizsation due to Sudakov, Szemeredi and Vu, though the conclusion of our result is somewhat different.
In his proof of Szemeredi's Theorem, Gowers introduced certain norms that are defined on a parallelepiped structure. A natural question is on which sets a parallelepiped structure (and thus a Gowers norm) can be defined. We focus on dimensions 2 and 3 and show when this possible, and describe a correspondence between the parallelepiped structures nilpotent 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 nilspaces. As an application we prove inverse theorems for the Gowers norms on bounded exponent abelian groups. It says roughly speaking that if a function on A has non negligible U(k+1)-norm then it correlates with a phase polynomial of degree k when lifted to some abelian group extension of A. This result is closely related to a conjecture by Tao and Ziegler. In prticular we obtain a new proof for the Tao…
We consider a class of two-dimensional functions f(x,y) with the property that the smallness of its rectangular norm implies the smallness of rectangular norm for f(x,x+y). Also we study a family of functions f(x,y) having a similar property for higher Gowers norms. The method based on a transference principle for a class of sums over special systems of linear equations.
This is a companion note to our paper 'A relative Szemerédi theorem', elaborating on a concluding remark. In that paper, we showed how to prove a relative Szemerédi theorem for $(r+1)$-term arithmetic progressions assuming a linear forms condition. Here we show how to replace this condition with an assumption about the Gowers uniformity norm $U^r$.
We estimate Gowers uniformity norms for some classical automatic sequences, such as the Thue-Morse and Rudin-Shapiro sequences. The methods can also be extended to other automatic sequences. As an application, we asymptotically count arithmetic progressions in the set of integers $\leq N$ where the Thue-Morse (resp. Rudin-Shapiro) sequence takes the value $+1$.
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 $f:\F_p^N \to \F_p$ is non-negligible, that is larger than a constant independent of $N$, then $f$ can be non-trivially approximated by 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 any polynomial of degree 3 is expon…
Gowers norms have been studied extensively both in the direct sense, starting with a function and understanding the associated norm, and in the inverse sense, starting with the norm and deducing properties of the function. Instead of focusing on the norms themselves, we study associated dual norms and dual functions. Combining this study with a variant of the Szemeredi Regularity Lemma, we give a decomposition theorem for dual functions, linking the dual norms to classical norms and indicating that the dual norm is easier to understand than the norm itself. Using the dual functions, we introdu…
We prove a generalization of Gowers' theorem for $\mathrm{FIN}_{k}$ where, instead of the single tetris operation $T:\mathrm{FIN}_{k}\rightarrow \mathrm{FIN}_{k-1}$, one considers all maps from $\mathrm{FIN}_{k}$ to $\mathrm{FIN}_{j}$ for $0\leq j\leq k$ arising from nondecreasing surjections $f:\left\{ 0,1,\ldots ,k+1\right\} \rightarrow \left\{ 0,1,\ldots ,j+1\right\} $. This answers a question of Bartošová and Kwiatkowska. We also prove a common generalization of such a result and the Galvin--Glazer--Hindman theorem on finite products, in the setting of layered partial semigroups introduced…
We establish the \emph{inverse conjecture for the Gowers norm over finite fields}, which asserts (roughly speaking) that if a bounded function $f: V \to \C$ on a finite-dimensional vector space $V$ over a finite field $\F$ has large Gowers uniformity norm $\|f\|_{U^{s+1}(V)}$, then there exists a (non-classical) polynomial $P: V \to \T$ of degree at most $s$ such that $f$ correlates with the phase $e(P) = e^{2πi P}$. This conjecture had already been established in the "high characteristic case", when the characteristic of $\F$ is at least as large as $s$. Our proof relies on the weak form of t…
We prove quantitative bounds for the inverse theorem for Gowers uniformity norms $\mathsf{U}^5$ and $\mathsf{U}^6$ in $\mathbb{F}_2^n$. The proof starts from an earlier partial result of Gowers and the author which reduces the inverse problem to a study of algebraic properties of certain multilinear forms. The bulk of the work in this paper is a study of the relationship between the natural actions of $\operatorname{Sym}_4$ and $\operatorname{Sym}_5$ on the space of multilinear forms and the partition rank, using an algebraic version of regularity method. Along the way, we give a positive answ…
A central tool in the study of systems of linear equations with integer coefficients is the Generalised von Neumann Theorem of Green and Tao. This theorem reduces the task of counting the weighted solutions of these equations to that of counting the weighted solutions for a particular family of forms, the Gowers norms $\Vert f \Vert_{U^{s+1}[N]}$ of the weight $f$. In this paper we consider systems of linear inequalities with real coefficients, and show that the number of solutions to such weighted diophantine inequalities may also be bounded by Gowers norms. Furthermore, we provide a necessar…
In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our 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 we say that P has small rank. We give several applications of this result, paying particular attention to consequences for the theory of the so-called Gowers norms. We establish an inverse result for the Gowers U^{d+1}-norm of functions of the form f(x)= e_F(P(x)), where P : F…
(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 additive combinatorics, and play an important role in both Gowers' proof of Szemerédi's theorem and the Green-Tao theorem. The inverse theorem states that if a function has a large uniformity norm, which is a robust combinatorial measure of structure, then it must correlate with a nilsequence, which is a highly structured algebraic object. This was proved in a qualitative sense by Green, Tao, and Ziegler…
We establish a number of "concatenation theorems" that assert, roughly speaking, that if a function exhibits "polynomial" (or "Gowers anti-uniform", "uniformly almost periodic", or "nilsequence") behaviour in two different directions separately, then it also exhibits the same behavior (but at higher degree) in both directions jointly. Among other things, this allows one to control averaged local Gowers uniformity norms by global Gowers uniformity norms. In a sequel to this paper, we will apply such control to obtain asymptotics for "polynomial progressions" $n+P_1(r),\dots,n+P_k(r)$ in various…
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 $f: X \to \C$ on a measure-preserving system $X = (X, {\mathcal X}, μ, T)$. Much recent effort has been devoted to the question of ob…
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 $ϕ= e_\F(P)$ of degree at most $d-1$, thus $P: V \to \F$ 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 $\charac(F) \geq d$ from an ergodic theory counterpart, which was recently est…
It is shown that there is an oscillating sequence of higher order which is not orthogonal to the class of dynamical flow with topological entropy zero. We further establish that any oscillating sequence of order $d$ is orthogonal to any $d$-nilsequence arising from the skew product on the $d$-dimensional torus $\mathbb{T}^d$. The proof yields that any oscillating sequence of higher order is orthogonal to any dynamical sequence arising from topological dynamical systems with quasi-discrete spectrum. however, we provide an example of oscillating sequence of higher order with large Gowers norms. …
We show that if A is a subset of Z/pZ (p a prime) of density bounded away from 0 and 1 then the A(Z/pZ)-norm (that is the l^1-norm of the Fourier transform) of the characterstic function of A is bounded below by an absolute constant times (log p)^{1/2 - ε} as p tends to infinity. This improves on the exponent 1/3 in recent work of Green and Konyagin.
For every natural number k we prove a decomposition theorem for bounded measurable functions on compact abelian groups into a structured part, a quasi random part and a small error term. In this theorem quasi randomness is measured with the Gowers norm U(k+1) and the structured part is a bounded complexity ``nilspace-polynomial'' of degree k. This statement implies a general inverse theorem for the U(k+1) norm. (We discuss some consequences in special families of groups such as bounded exponent groups, zero characteristic groups and the circle group.) Along these lines we introduce a convergen…
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 $\partial_u f(x) \colon= f(x + u) \overline{f(x)}$ is the discrete multiplicative derivative. Suppose that $G$ is a direct sum of subspaces $G = U…
We show that, under some mild hypotheses, the Gowers uniformity norms (both in the additive and in the hypergraph setting) are essentially equivalent to certain weaker norms which are easier to understand. We present two applications of this equivalence: a variant of the Koopman--von Neumann decomposition, and a proof of the relative inverse theorem for the Gowers $U^s[N]$-norm using a norm-type pseudorandomness condition.
The structure of totally disconnected Host--Kra--Ziegler factors, and the inverse theorem for the $U^k$ Gowers uniformity norms on finite abelian groups of bounded torsion
University of Bonn（波恩大学）；Bar Ilan University（巴伊兰大学）；University of California Los Angeles（加州大学洛杉矶分校）
We consider some variants of the Gowers box norms, introduced by Hatami, and show their relevance in the context of sparse hypergraphs. Our main results are the following. Firstly, we prove a generalized von Neumann theorem for $L_p$ graphons. Secondly, we give natural examples of pseudorandom families, that is, sparse weighted uniform hypergraphs which satisfy relative versions of the counting and removal lemmas.
In this paper we study the structure of polynomials of degree three and four that have high bias or high Gowers norm, over arbitrary prime fields. In particular we obtain the following results. 1. We give a canonical representation for degree three or four polynomials that have a significant bias (i.e. they are not equidistributed). This result generalizes the corresponding results from the theory of quadratic forms. It also significantly improves the results of Green and Tao and Kaufman and Lovett for such polynomials. 2. For the case of degree four polynomials with high Gowers norm we show t…
We consider the problem of testing if a given function f : F_2^n -> F_2 is close to any degree d polynomial in n variables, also known as the Reed-Muller testing problem. The Gowers norm is based on a natural 2^{d+1}-query test for this property. Alon et al. [AKKLR05] rediscovered this test and showed that it accepts every degree d polynomial with probability 1, while it rejects functions that are Omega(1)-far with probability Omega(1/(d 2^{d})). We give an asymptotically optimal analysis of this test, and show that it rejects functions that are (even only) Omega(2^{-d})-far with Omega(1)-prob…
Let $\F$ a finite field. We show that the universal characteristic factor for the Gowers-Host-Kra uniformity seminorm $U^k(\X)$ for an ergodic action $(T_g)_{g \in \F^ω}$ of the infinite abelian group $\F^ω$ on a probability space $X = (X,\B,μ)$ is generated by phase polynomials $ϕ: X \to S^1$ of degree less than $C(k)$ on $X$, where $C(k)$ depends only on $k$. In the case where $k \leq \charac(\F)$ we obtain the sharp result $C(k)=k$. This is a finite field counterpart of an analogous result for $\Z$ by Host and Kra. In a companion paper to this paper, we shall combine this result with a corr…
