On nilspace systems and their morphisms

Candela, Pablo; González-Sánchez, Diego; Szegedy, Balázs

doi:10.1017/etds.2019.24

Cited by 12 publications

(20 citation statements)

References 15 publications

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“…For example, from the existing theory of compact nilspaces it follows that an ergodic nilspace system whose group of translations is finitely generated is an inverse limit of nilsystems. This is proved in [12,Theorem 5.2], and can also be derived from [ Theorem 5.12. Let G be a finitely generated nilpotent group acting ergodically on a Borel probability space Ω, and let G • be a filtration on G. Then for each positive integer k the k-th Host-Kra factor of (Ω, (G, G • )) is isomorphic to an inverse limit of k-step nilsystems.…”

Section: On Characteristic Factors Associated With Nilpotent Group Ac...mentioning

confidence: 64%

Nilspace factors for general uniformity seminorms, cubic exchangeability and limits

Candela¹,

Szegedy²

2018

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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.These are sequences indexed by the infinite discrete cube, such that for every integer k ≥ 0 the joint distribution's marginals on affine subcubes of dimension k are all equal.In particular, our result gives a description, in terms of compact nilspaces, of a related exchangeability property considered by Austin, inspired by a problem of Aldous. Finally, using nilspaces we obtain limit objects for sequences of functions on compact abelian groups (more generally on compact nilspaces) such that the densities of certain patterns in these functions converge. The paper thus proposes a measure-theoretic framework on which the area of higher-order Fourier analysis can be based, and which yields new applications of this area in a unified way in ergodic theory and arithmetic combinatorics.

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Section: On Characteristic Factors Associated With Nilpotent Group Ac...mentioning

confidence: 64%

Nilspace factors for general uniformity seminorms, cubic exchangeability and limits

Candela¹,

Szegedy²

2018

Preprint

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“…We can now obtain the main result of this section. 12 As usual, given a probability space (Ω, A, µ), two sub-σ-algebras B 1 , B 2 of A are independent according to µ if for every…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…More precisely, for a measure of the form ζ X,m to be affine-exchangeable, the underlying nilspace X must have a more specific structure than in the general cubicexchangeable setting. These more specific structures are the so-called 2-homogeneous nilspaces, introduced in [12] (and recalled in more detail below). As a consequence, when we replace cubic exchangeability with affine exchangeability, the representation theorem from [14] can be significantly refined, leading to one of the main results of this paper, Theorem 1.5 below.…”

Section: )mentioning

confidence: 99%

On $\mathbb{F}_2^ω$-affine-exchangeable probability measures

Candela¹,

González-Sánchez²,

Szegedy³

2022

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For any standard Borel space B, let P(B) denote the space of Borel probability measures on B. In relation to a difficult problem of Aldous in exchangeability theory, and in connection with arithmetic combinatorics, Austin raised the question of describing the structure of affine-exchangeable probability measures on product spaces indexed by the vector space F ω 2 , i.e., the measures in P(B F ω 2 ) that are invariant under the coordinate permutations on B F ω 2 induced by all affine automorphisms of F ω 2 . We answer this question by describing the extreme points of the space of such affine-exchangeable measures. We prove that there is a single structure underlying every such measure, namely, a random infinite-dimensional cube (sampled using Haar measure adapted to a specific filtration) on a group that is a countable power of the 2-adic integers. Indeed, every extreme affineexchangeable measure in P(B F ω 2 ) is obtained from a P(B)-valued function on this group, by a vertex-wise composition with this random cube. The consequences of this result include a description of the convex set of affine-exchangeable measures in P(B F ω 2 ) equipped with the vague topology (when B is a compact metric space), showing that this convex set is a Bauer simplex. We also obtain a correspondence between affine-exchangeability and limits of convergent sequences of (compact-metric-space valued) functions on vector spaces F n 2 as n → ∞. Via this correspondence, we establish the above-mentioned group as a general limit domain valid for any such sequence.

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“…Nilspace related topics have now grown into an active research area, including detailed treatments by the first named author [7,8] and by Gutman, Manners and Varjú [22,23,24], as well as further applications in arithmetic combinatorics, ergodic theory, probability theory, and topological dynamics [9,12,20,22,23,24]. Initial applications of nilspaces in higher-order Fourier analysis were obtained in [32],…”

Section: Introductionmentioning

confidence: 99%

“…Such a system can be viewed as a topological dynamical system, and if we equip X with its Haar probability measure then the nilspace system becomes a measure-preserving G-system. Nilspace systems were shown in [9,12] to yield extensions of the Ergodic Structure Theorem (in particular, ergodic nilspace Z-systems are inverse limits of nilsystems [9, Theorem 5.1]).…”

Section: Introductionmentioning

confidence: 99%

On higher-order Fourier analysis in characteristic $p$

Candela¹,

González-Sánchez²,

Szegedy³

2021

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In this paper, the nilspace approach to higher-order Fourier analysis is developed in the setting of vector spaces over a prime field F p , with applications mainly in ergodic theory. A key requisite for this development is to identify a class of nilspaces adequate for this setting. We introduce such a class, whose members we call p-homogeneous nilspaces. One of our main results characterizes these objects in terms of a simple algebraic property. We then prove various further results on these nilspaces, leading to a structure theorem describing every finite p-homogeneous nilspace as the image, under a nilspace fibration, of a member of a simple family of filtered finite abelian p-groups.The applications include a description of the Host-Kra factors of ergodic F ω p -systems as p-homogeneous nilspace systems. This enables the analysis of these factors to be reduced to the study of such nilspace systems, with central questions on the factors thus becoming purely algebraic problems on finite nilspaces. We illustrate this approach by proving that for k ≤ p + 1 the k-th Host-Kra factor is an Abramov system of order ≤ k, extending a result of Bergelson-Tao-Ziegler that holds for k < p. We illustrate the utility of p-homogeneous nilspaces also by showing that the above-mentioned structure theorem yields a new proof of the Tao-Ziegler inverse theorem for Gowers norms on F n p .

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On nilspace systems and their morphisms

Cited by 12 publications

References 15 publications

Nilspace factors for general uniformity seminorms, cubic exchangeability and limits

Nilspace factors for general uniformity seminorms, cubic exchangeability and limits

On $\mathbb{F}_2^ω$-affine-exchangeable probability measures

On higher-order Fourier analysis in characteristic $p$

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