Popular differences for polynomial patterns in rings of integers

Ackelsberg, Ethan; Bergelson, Vitaly

doi:10.48550/arxiv.2107.07626

Cited by 2 publications

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“…A notion of upper Banach density can be defined in the semigroup (N, •) by the formula in (1), where the supremum is now taken over Følner sequences in (N, •). Examples of Følner sequences in (N, •) include sequences of the form…”

Section: Patterns In (N •)mentioning

confidence: 99%

“…Remark 3.2. In the example above, the factor Z 1 (𝑋) is isomorphic to 𝑗 ∈N 𝐶 4 equipped with the action 𝑇 (1) 𝑔 𝑥 = (𝑖 𝑔 𝑗 • 𝑥 𝑗 ) 𝑗 ∈N , while Z 2 (𝑋) = X. On the other hand, for the 2𝐺-system 𝑋, (𝑇 𝑔 ) 𝑔 ∈2𝐺 , we have…”

Section: Theorem 111mentioning

confidence: 99%

“…By Theorem 2.5(ii), the Kronecker factor of X, Z 1 (𝑋) is isomorphic to an ergodic rotation. Therefore, it is convenient to identify the Kronecker factor with the system Z = (𝑍, 𝛼), where Z is a compact abelian group and 𝛼 : 𝐺 → 𝑍 is a homomorphism, such that 𝑇 1 𝑔 𝑧 = 𝑧 + 𝛼 𝑔 , where 𝑇 1 is the G-action on Z. The following corollary of Proposition 3.5 will be useful later on in this paper.…”

Section: Uc -Limmentioning

confidence: 99%

“…Now suppose that there exists some constant 𝑐 ∈ 𝑆 1 , such that 𝑇 𝑓 (𝑥, 𝑦) = 𝑐 • 𝑓 (𝑥, 𝑦) for 𝜇-a.e. (𝑥, 𝑦) ∈ 𝑆 1 × 𝑆 1 . By the uniqueness of the Fourier series, we deduce that…”

Section: Relative Orthonormal Basismentioning

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: Patterns In (N •)mentioning

confidence: 99%

Section: Theorem 111mentioning

confidence: 99%

Section: Uc -Limmentioning

confidence: 99%

Section: Relative Orthonormal Basismentioning

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 large intersections property is closely related to the phenomenon of popular differences in combinatorics; see, e.g., [SSZ21,Ber21,M21,BSST21,AB21].…”

Section: Introductionmentioning

confidence: 99%

Khintchine-type recurrence for 3-point configurations

Ackelsberg,

Bergelson,

Shalom

2022

Preprint

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The goal of this paper is to generalize, refine, and improve results on large intersections from [BHK05, ABB21]. We show that if G is a countable abelian group and ϕ, ψ : G → G are homomorphisms such that at least two of the three subgroups ϕ(G), ψ(G), and (ψ − ϕ)(G) have finite index in G, then {ϕ, ψ} has the large intersections property. That is, for any ergodic measure preserving system X = (X, X , µ, (T g ) g∈G ), any A ∈ X , and any ε > 0, the set (Theorem 1.11). Moreover, in the special case where ϕ(g) = ag and ψ(g) = bg for a, b ∈ 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 = (Q >0 , •) and ϕ(g) = g, ψ(g) = g 2 , which leads to a multiplicative version for the Khintchine-type recurrence result in [BHK05]. We also completely characterize the pairs of homomorphisms ϕ, ψ that have the large intersections property when G = Z 2 .The proofs of our main results rely on analysis of the structure of the universal characteristic factor for the multiple ergodic averages 1In 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 X, one can describe the characteristic factor in terms of the Conze-Lesigne factor and the σ-algebras of ϕ(G) and ψ(G) invariant functions (Theorem 4.10). ContentsE. ACKELSBERG, V. BERGELSON, AND O. SHALOM 1.6. Structure of the paper 2. Preliminaries 2.1. Uniform Cesàro limits 2.2. Factors 2.3. Factor of invariant sets 2.4. Host-Kra factors 2.5. Joins and meets of factors 2.6. Ziegler factors 2.7. Seminorms for multiplicative configurations 3. Theorem 1.11 3.1. Characteristic factors 3.2. Relative orthonormal basis 3.3. Proof of Theorem 1.11 4. Extensions 4.1. Characteristic factors related to Theorem 1.13 5. A limit formula for {ag, bg} 5.1. Previous limit formulas 5.2. Mackey group 5.3. Cocycle identities 5.4. Proof of Proposition 5.3 5.5. Limit formula 5.6. Proof of Theorem 1.13 6. Proof of Theorem 1.14 7. 3-point configurations in Z 2 7.1. Ergodic popular difference densities when r(M 1 , M 2 ) = (2, 1, 1) 7.2. Ergodic popular difference densities when r(M 1 , M 2 ) = (1, 1, 1) 7.3. Finitary combinatorial consequences and open questions 8. Khintchine-type recurrence for actions of semigroups 8.1. The group generated by a cancellative abelian semigroup 8.2. Notions of largeness 8.3. Extending main results to actions of cancellative abelian semigroups 8.4. Two combinatorial questions Appendix A. Proof of Lemma 3.6 References

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Popular differences for polynomial patterns in rings of integers

Cited by 2 publications

References 0 publications

Khintchine-type recurrence for 3-point configurations

Khintchine-type recurrence for 3-point configurations

Khintchine-type recurrence for 3-point configurations

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