Multiple ergodic averages in abelian groups and Khintchine type recurrence

Or, Shalom,

doi:10.1090/tran/8558

Cited by 10 publications

(17 citation statements)

References 24 publications

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“…Under certain assumptions on a and b , two different (but related) formulas were obtained previously in [2] and in [27] (see Theorems 5.1 and 5.2 below). Neither of the previously obtained formulas is sufficient for our purposes, so we prove a new one in this section.…”

Section: A Limit Formula Formentioning

confidence: 99%

“…We will use the above extension theorem in order to express these characteristic factors in terms of and the invariant -algebras, and . Then, using a result of Tao and Ziegler [29] (see Theorem 4.8 below), we will reduce matters further to studying the Conze–Lesigne factor with respect to the action of G , which is already well understood for arbitrary countable discrete abelian groups (see [2], [27]).…”

Section: Extensionsmentioning

confidence: 99%

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Khintchine-type recurrence for 3-point configurations

Ackelsberg

Bergelson

2022

Forum of Mathematics, Sigma

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The goal of this paper is to generalise, refine and improve results on large intersections from [2, 8]. We show that if G is a countable discrete abelian group and $\varphi , \psi : G \to G$ are homomorphisms, such that at least two of the three subgroups $\varphi (G)$ , $\psi (G)$ and $(\psi -\varphi )(G)$ have finite index in G, then $\{\varphi , \psi \}$ has the large intersections property. That is, for any ergodic measure preserving system $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$ , any $A\in \mathcal {X}$ and any $\varepsilon>0$ , the set $$ \begin{align*} \{g\in G : \mu(A\cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A)>\mu(A)^3-\varepsilon\} \end{align*} $$ is syndetic (Theorem 1.11). Moreover, in the special case where $\varphi (g)=ag$ and $\psi (g)=bg$ for $a,b\in \mathbb {Z}$ , we show that we only need one of the groups $aG$ , $bG$ or $(b-a)G$ to be of finite index in G (Theorem 1.13), and we show that the property fails, in general, if all three groups are of infinite index (Theorem 1.14). One particularly interesting case is where $G=(\mathbb {Q}_{>0},\cdot )$ and $\varphi (g)=g$ , $\psi (g)=g^2$ , which leads to a multiplicative version of the Khintchine-type recurrence result in [8]. We also completely characterise the pairs of homomorphisms $\varphi ,\psi $ that have the large intersections property when $G = {{\mathbb Z}}^2$ . The proofs of our main results rely on analysis of the structure of the universal characteristic factor for the multiple ergodic averages $$ \begin{align*} \frac{1}{|\Phi_N|} \sum_{g\in \Phi_N}T_{\varphi(g)}f_1\cdot T_{\psi(g)} f_2. \end{align*} $$ In the case where G is finitely generated, the characteristic factor for such averages is the Kronecker factor. In this paper, we study actions of groups that are not necessarily finitely generated, showing, in particular, that, by passing to an extension of $\textbf {X}$ , one can describe the characteristic factor in terms of the Conze–Lesigne factor and the $\sigma $ -algebras of $\varphi (G)$ and $\psi (G)$ invariant functions (Theorem 4.10).

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Section: A Limit Formula Formentioning

confidence: 99%

Section: Extensionsmentioning

confidence: 99%

Khintchine-type recurrence for 3-point configurations

Ackelsberg

Bergelson

2022

Forum of Mathematics, Sigma

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“…Let us mention that there are other descriptions of the characteristic factors of ergodic G-systems. There is for instance the concept of nilpotent system introduced in [33, Definition 1.29] for the 2-step case with G = p∈P F p (where P is a multiset of primes), and more generally, for any countable abelian group G in the 2-step case, there is a description of the 2-factor as a double coset space [34,Theorem 1.21]. 6 Here Z ∞ p denotes the direct product Z N p .…”

Section: Introductionmentioning

confidence: 99%

On higher-order Fourier analysis in characteristic p

Candela

González-Sánchez²,

Szegedy³

2023

Ergod. Th. Dynam. Sys.

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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 $\mathbb {F}_p$ , 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 $\mathbb {F}_p^\omega $ -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 $k\leq p+1$ the kth Host–Kra factor is an Abramov system of order at most 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 structure theorem yields a new proof of the Tao–Ziegler inverse theorem for Gowers norms on $\mathbb {F}_p^n$ .

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“…Austin's machinery will be also useful in our paper. See [8,30,1,14] for some other Khintchine-type results for multiple recurrence for various countable abelian group actions.…”

Section: Introductionmentioning

confidence: 99%

Uniform syndeticity in multiple recurrence

Jamneshan¹,

Pan²

2022

Preprint

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We establish uniform lower bounds on the size of the return set and the lower Banach density of return times, respectively, for d-term multiple recurrence for arbitrary d, depending only on the measure of the measurable set and d but otherwise independent of the system for any fixed uniformly amenable group. Our proof method relies on a multiple recurrence theorem for ultraproducts of measure-preserving systems, which we obtain as a special case of an uncountable extension of Austin's amenable multiple recurrence theorem.

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Multiple ergodic averages in abelian groups and Khintchine type recurrence

Cited by 10 publications

References 24 publications

Khintchine-type recurrence for 3-point configurations

Khintchine-type recurrence for 3-point configurations

On higher-order Fourier analysis in characteristic p

Uniform syndeticity in multiple recurrence

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