An inverse theorem for the uniformity seminorms associated with the action of $F^ω$

Uniformity seminorms on $\ell^\infty$ and applications

A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on $\Z/N\Z$ introduced by Gowers in his proof of Szemerédi's Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg's proof of Szemerédi's Theorem) defined by the authors. For each integer $k\geq 1$, we define seminorms on $\ell^\infty(\Z)$ analogous to these norms and seminorms. We study t…

Nilspace factors for general uniformity seminorms, cubic exchangeability and limits

We study a class of measure-theoretic objects that we call cubic couplings, on which there is a common generalization of the Gowers norms and the Host-Kra seminorms. Our main result yields a complete structural description of cubic couplings, using nilspaces. We give three applications. Firstly, we describe the characteristic factors of Host-Kra type seminorms for measure-preserving actions of countable nilpotent groups. This yields an extension of the structure theorem of Host and Kra. Secondly, we characterize sequences of random variables with a property that we call cubic exchangeability. …

Large values of the Gowers-Host-Kra seminorms

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…

完全不连通的Host–Kra–Ziegler因子的结构，以及有界挠有限阿贝尔群上$U^k$ Gowers均匀性范数的逆定理

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

本文研究完全不连通的Host–Kra–Ziegler因子的结构，证明其为Weyl型$\boldsymbol{\text{Z^ω}}$系统的广义因子，进而利用对应原理得到有界挠有限阿贝尔群上$U^k$ Gowers均匀性范数的弱逆定理。

沿算术级数的遍历平均的次数降低

Degree lowering for ergodic averages along arithmetic progressions

本文研究差取自固定整数序列的算术级数相关多重遍历平均的极限行为，给出其与常规ℓ项算术级数平均同极限的充要条件，推导整数稠密集含受限差ℓ+1长算术级数的充分条件，并通过Gowers-Host-Kra半范数相关逆定理与新估计证明了次数降低的高阶变体定理。

A point of view on Gowers uniformity norms

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…