Gowers introduced, for d\geq 1, the notion of dimension-d uniformity U^d(f) of a function f: G -> \C, where G is a finite abelian group and \C are the complex numbers. Roughly speaking, if U^d(f) is small, then f has certain "pseudorandomness" properties.
We prove the following property of functions with large U^d(f). Write G=G_1 x >... x G_n as a product of groups. If a bounded balanced function f:G_1 x ... x G_n -> \C is such that U^{d} (f) > epsilon, then one of the coordinates of f has influence at least epsilon/2^{O(d)}.
The Gowers inner product of a collection of functions is a related notion of pseudorandomness. We prove that if a collection of bounded functions has large Gowers inner product, and at least one function in the collection is balanced, then there is a variable that has high influence for at least four of the functions in the collection.
Finally, we relate the acceptance probability of the "hypergraph long-code test" proposed by Samorodnitsky and Trevisan to the Gowers inner product of the functions being tested and we deduce applications to the construction of Probabilistically Checkable Proofs and to hardness of approximation.
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 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…
(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…
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…
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 corre…
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.
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…
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…
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 demonstrate that $$\|μ\|_{U^3([N])} \ll_{A}^{\text{ineff}} \log^{-A}(N)$$ $$\|Λ- Λ_Q\|_{U^3([N])} \ll_{A}^{\text{ineff}} \log^{-A}(N)$$ for any $A > 0$ where $Λ_Q$ is an approximant to the von Mangoldt function and will be defined below, improving upon a bound of Tao-Teräväinen (2021). As a consequence, among other things, we have the following: $$\mathbb{E}_{x, y \in [N], x + 3y \in [N]} Λ(x)Λ(x + y)Λ(x + 2y)Λ(x + 3y) = \mathfrak{S} + O_A(\log^{-A}(N))$$ where $\mathfrak{S}$ is the singular series for the configuration $(x, x + y, x + 2y, x + 3y)$. In fact, we show that $$\|μ- μ_{Siegel}\|…
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…
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…
Since the theorems of Schur and van der Waerden, numerous partition regularity results have been proved for linear equations, but progress has been scarce for non-linear ones, the hardest case being equations in three variables. We prove partition regularity for certain equations involving forms in three variables, showing for example that the equations $16x^2+9y^2=n^2$ and $x^2+y^2-xy=n^2$ are partition regular, where $n$ is allowed to vary freely in $\mathbb{N}$. For each such problem we establish a density analogue that can be formulated in ergodic terms as a recurrence property for actions…
A bracket polynomial on the integers is a function formed using the operations of addition, multiplication and taking fractional parts. For a fairly large class of bracket polynomials we show that if p is a bracket polynomial of degree k-1 on [N] then the function f defined by f(n) = e(p(n)) has Gowers U^k[N]-norm bounded away from zero, uniformly in N. We establish this result by first reducing it to a certain recurrence property of sets of bracket polynomials. Specifically, for a fairly large class of bracket polynomials we show that if p_1, ..., p_r are bracket polynomials then their values…
