Khintchine-type double recurrence in abelian groups
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| Format: | Preprint |
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2023
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| _version_ | 1866915056445292544 |
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| author | Ackelsberg, Ethan |
| author_facet | Ackelsberg, Ethan |
| contents | We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if $Γ$ is a countable discrete abelian group, $φ, ψ\in End(Γ)$, and $ψ- φ$ is an injective endomorphism with finite index image, then for any ergodic measure-preserving $Γ$-system $\left( X, \mathcal{X}, μ, (T_g)_{g \in Γ} \right)$, any measurable set $A \in \mathcal{X}$, and any $\varepsilon > 0$, the set of $g \in Γ$ for which $$μ\left( A \cap T_{φ(g)}^{-1} A \cap T_{ψ(g)}^{-1} A \right) > μ(A)^3 - \varepsilon$$ is syndetic. This generalizes the main results of (Ackelsberg--Bergelson--Shalom, 2022) and essentially answers a question left open in that paper (Question 1.12).
For the group $Γ= \mathbb{Z}^d$, we deduce that for any matrices $M_1, M_2 \in M_{d \times d}(\mathbb{Z})$ whose difference $M_2 - M_1$ is nonsingular, any ergodic measure-preserving $\mathbb{Z}^d$-system $\left( X, \mathcal{X}, μ, (T_{\vec{n}})_{\vec{n} \in \mathbb{Z}^d} \right)$, any measurable set $A \in \mathcal{X}$, and any $\varepsilon > 0$, the set of $\vec{n} \in \mathbb{Z}^d$ for which $$μ\left( A \cap T_{M_1 \vec{n}}^{-1} A \cap T_{M_2 \vec{n}}^{-1} A \right) > μ(A)^3 - \varepsilon$$ is syndetic, a result that was previously known only in the case $d = 2$.
The key ingredients in the proof are: (1) a recent result obtained jointly with Bergelson and Shalom that says that the relevant ergodic averages are controlled by a characteristic factor closely related to the quasi-affine (or Conze--Lesigne) factor; (2) an extension trick to reduce to systems with well-behaved (with respect to $φ$ and $ψ$) discrete spectrum; and (3) a description of Mackey groups associated to quasi-affine cocycles over rotational systems with well-behaved discrete spectrum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_04698 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Khintchine-type double recurrence in abelian groups Ackelsberg, Ethan Dynamical Systems Combinatorics 37A15 (Primary) 37A30, 05D10 (Secondary) We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if $Γ$ is a countable discrete abelian group, $φ, ψ\in End(Γ)$, and $ψ- φ$ is an injective endomorphism with finite index image, then for any ergodic measure-preserving $Γ$-system $\left( X, \mathcal{X}, μ, (T_g)_{g \in Γ} \right)$, any measurable set $A \in \mathcal{X}$, and any $\varepsilon > 0$, the set of $g \in Γ$ for which $$μ\left( A \cap T_{φ(g)}^{-1} A \cap T_{ψ(g)}^{-1} A \right) > μ(A)^3 - \varepsilon$$ is syndetic. This generalizes the main results of (Ackelsberg--Bergelson--Shalom, 2022) and essentially answers a question left open in that paper (Question 1.12). For the group $Γ= \mathbb{Z}^d$, we deduce that for any matrices $M_1, M_2 \in M_{d \times d}(\mathbb{Z})$ whose difference $M_2 - M_1$ is nonsingular, any ergodic measure-preserving $\mathbb{Z}^d$-system $\left( X, \mathcal{X}, μ, (T_{\vec{n}})_{\vec{n} \in \mathbb{Z}^d} \right)$, any measurable set $A \in \mathcal{X}$, and any $\varepsilon > 0$, the set of $\vec{n} \in \mathbb{Z}^d$ for which $$μ\left( A \cap T_{M_1 \vec{n}}^{-1} A \cap T_{M_2 \vec{n}}^{-1} A \right) > μ(A)^3 - \varepsilon$$ is syndetic, a result that was previously known only in the case $d = 2$. The key ingredients in the proof are: (1) a recent result obtained jointly with Bergelson and Shalom that says that the relevant ergodic averages are controlled by a characteristic factor closely related to the quasi-affine (or Conze--Lesigne) factor; (2) an extension trick to reduce to systems with well-behaved (with respect to $φ$ and $ψ$) discrete spectrum; and (3) a description of Mackey groups associated to quasi-affine cocycles over rotational systems with well-behaved discrete spectrum. |
| title | Khintchine-type double recurrence in abelian groups |
| topic | Dynamical Systems Combinatorics 37A15 (Primary) 37A30, 05D10 (Secondary) |
| url | https://arxiv.org/abs/2307.04698 |