Gowers U^3 范数是用于研究等差数列的一系列范数之一。若 G 是一个 Abel 群，且 A 是 G 的一个子集，则特征函数 1_A 的 U^3(G) 范数在研究 A 中长度为 4 的等差数列时十分有用。我们对 U^3(G) 范数进行了全面研究，对满足 ||f||_{U^3} 较大的函数 f : G -> C 给出了相当完整的描述，并将其与 Host、Kra 和 Ziegler 近期在遍历理论中的结果联系起来。作为应用，我们推广了 Gowers 关于 Szemerédi 定理的一项结果。记 r_4(G) 为不含四个互异元素构成等差数列的最大集合 A 的大小，我们证明对于某个绝对常数 c，有 r_4(G) << |G|(loglog|G|)^{-c}。在后续论文中，我们将进一步发展这些思想，得到素数中满足 p_1 < p_2 < p_3 < p_4 < N 的 4 项等差数列数量的渐近公式，并得到关于 r_4(G) 的更优界。2023 年 12 月更新：本文中未给出详细证明而陈述的命题 3.2 是不正确的。反例见 Candela、Gonzalez-Sanchez 和 Szegedy 的 arXiv:2311.13899，注记 4.3。命题 3.2 在本文中被引用两次。首先，它在陈述之后立即被用于推导定理 2.3 的第二部分。然而，该定理只涉及有限域上的向量空间；在此情形下，由标准线性代数可知命题 3.2 是正确的。不过，第 3 节末尾关于该论证适用于任意 G 的注记应当删除。第二次应用是在引理 10.6 的证明中。该引理很有可能可以被挽救，尤其是在假设 P 为真（proper）的情况下；但无论如何，它只在命题 10.8 的证明中被应用一次。在那里，P 是真的，而且更重要的是，H = {0} 是平凡的；在此情形下，引理 10.6 及其证明仍然有效。
英文摘要：
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.
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…
Let $μ$ be the Möbius function and let $k \geq 1$. We prove that the Gowers $U^k$-norm of $μ$ restricted to progressions $\{n \leq X: n\equiv a_q\pmod{q}\}$ is $o(1)$ on average over $q\leq X^{1/2-σ}$ for any $σ> 0$, where $a_q\pmod{q}$ is an arbitrary residue class with $(a_q,q) = 1$. This generalizes the Bombieri-Vinogradov inequality for $μ$, which corresponds to the special case $k=1$.
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 $+1$.
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. …
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…
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…
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.
Let $P_1,\dots,P_k \colon {\bf Z} \to {\bf Z}$ be polynomials of degree at most $d$ for some $d \geq 1$, with the degree $d$ coefficients all distinct, and admissible in the sense that for every prime $p$, there exists integers $n,m$ such that $n+P_1(m),\dots,n+P_k(m)$ are all not divisible by $p$. We show that there exist infinitely many natural numbers $n,m$ such that $n+P_1(m),\dots,n+P_k(m)$ are simultaneously prime, generalizing a previous result of the authors, which was restricted to the special case $P_1(0)=\dots=P_k(0)=0$ (though it allowed for the top degree coefficients to coincide)…
We obtain a polynomial upper bound in the finite-field version of the multidimensional polynomial Szemerédi theorem for distinct-degree polynomials. That is, if $P_1, ..., P_t$ are nonconstant integer polynomials of distinct degrees and $v_1, ..., v_t$ are nonzero vectors in $\mathbb{F}_p^D$, we show that each subset of $\mathbb{F}_p^D$ lacking a nontrivial configuration of the form $$ x, x + v_1 P_1(y), ..., x + v_t P_t(y)$$ has at most $O(p^{D-c})$ elements. In doing so, we apply the notion of Gowers norms along a vector adapted from ergodic theory, which extends the classical concept of Gow…
We obtain a double exponential bound in Brauer's generalisation of van der Waerden's theorem, which concerns progressions with the same colour as their common difference. Such a result has been obtained independently and in much greater generality by Sanders. Using Gowers' local inverse theorem, our bound is quintuple exponential in the length of the progression. We refine this bound in the colour aspect for three-term progressions, and combine our arguments with an insight of Lefmann to obtain analogous bounds for the Ramsey numbers of certain nonlinear quadratic equations.
We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the $L^q$-dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of $\times p$ and $\times q$-invariant sets. Among several other applications, we also show that Bernoulli convolutions have a…
