阿贝尔群中的辛钦型二重递归
Khintchine-type double recurrence in abelian groups
AI总结：
该研究证明了可数离散阿贝尔群自同态对的辛钦型二重递归定理，推广了相关结果并回答了遗留问题，还推导了$\boldsymbol{Z}^d$群的特殊情形，此前仅$d=2$已知。
摘要：
我们证明了可数离散阿贝尔群的自同态对的辛钦型递归定理。作为主要结果的特例，若$Γ$是可数离散阿贝尔群，$φ, ψ∈End(Γ)$，且$ψ- φ$是像具有有限指数的单射自同态，则对任意遍历保测$Γ$-系统$\left( X, \boldsymbol{X}, μ, (T_g)_{g ∈ Γ} \right)$、任意可测集$A ∈ \boldsymbol{X}$及任意$\boldsymbol{ε} > 0$，满足$$μ(A ∩ T_{φ(g)}^{-1}A ∩ T_{ψ(g)}^{-1}A) > μ(A)^3 - ε$$的$g ∈ Γ$构成的集合是 syndetic 集。这推广了（Ackelsberg--Bergelson--Shalom, 2022）的主要结果，且本质上回答了该论文中遗留的问题1.12。对于群$Γ= \boldsymbol{Z}^d$，我们推导出：若任意矩阵$M_1, M_2 ∈ M_{d×d}(\boldsymbol{Z})$的差$M_2 - M_1$是非奇异的，则对任意遍历保测$\boldsymbol{Z}^d$-系统$\left( X, \boldsymbol{X}, μ, (T_{\boxed{n}})_{\boxed{n} ∈ \boldsymbol{Z}^d} \right)$、任意可测集$A ∈ \boldsymbol{X}$及任意$\boldsymbol{ε} > 0$，满足$$μ(A ∩ T_{M_1 \boxed{n}}^{-1}A ∩ T_{M_2 \boxed{n}}^{-1}A) > μ(A)^3 - ε$$的$\boxed{n} ∈ \boldsymbol{Z}^d$构成的集合是 syndetic 集，该结果此前仅在$d=2$时已知。证明的关键要素包括：(1) 与Bergelson和Shalom合作得到的近期结果，即相关遍历平均由与拟仿射（或Conze--Lesigne）因子密切相关的特征因子控制；(2) 一种约化技巧，将问题约化到具有（关于$φ$和$ψ$）良态离散谱的系统；(3) 对具有良态离散谱的旋转系统上拟仿射余圈相关的Mackey群的描述。
英文摘要：
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.
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