三点构形的Khintchine型递推式
Khintchine-type recurrence for 3-point configurations
摘要：
本文的目标是概括、细化和改进大型交叉口的结果。证明了若$G$是可数交换群，且$φ, ψ: G \to G$是同态，使得$φ(G)$，$ψ(G)$，$(ψ-φ)(G)$三个子群中至少有两个子群在$G$中有有限指数，则$\{φ, ψ\}$有\emph{large intersections property} .也就是说，对于任何遍历测度保持系统$X=(X,\mathcal{X},μ,(T_g)_{g\in G})$、任何$A\in\mathcal{X}$和任何$\varepsilon>0$，集合$$\{g\in G : μ(A\cap T_{φ(g)}^{-1} A\cap T_{ψ(g)}^{-1}A)>μ(A)^3-\varepsilon\}$$是联合的。此外，在特殊情况下，$φ(g)=ag$和$ψ(g)=bg$为$a,b\in\mathbb{Z}$，我们表明，我们只需要一个组$aG$，$bG$，或$(b-a)G$是有限的指数在$G$，我们表明，该性质失败，在一般情况下，如果所有三个组是无限的指数。一个特别有趣的情况是，其中$G=(\mathbb{Q}_{>0},\cdot)$和$φ(g)=g$，$ψ(g)=g^2$，这导致Bergelson-Host-Kra的大交集结果的乘法版本。我们还完全刻画了当$G=\mathbb{Z}^2$时具有大交性质的同态对$φ,ψ$。我们的主要结果的证明依赖于分析的结构的\emph{universal characteristic factor}的多遍历平均$$\frac{1}{|Φ_N|} \sum_{g\in Φ_N}T_{φ(g)}f_1\cdot T_{ψ(g)} f_2.$$的情况下，$G$是有限生成的，这种平均的特征因子是\emph{Kronecker factor}。在本文中，我们研究行动的群体，不一定是有限生成的，特别是表明，通过通过扩展$X$，可以描述的特征因子的\emph{Conze--Lesigne factor}和$σ$ -代数的$φ(G)$和$ψ(G)$不变函数。
英文摘要：
The goal of this paper is to generalize, refine, and improve results on large intersections. We show that if $G$ is a countable abelian group and $φ, ψ: G \to G$ 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 \emph{large intersections property}. That is, for any ergodic measure preserving system $X=(X,\mathcal{X},μ,(T_g)_{g\in G})$, any $A\in\mathcal{X}$, and any $\varepsilon>0$, the set $$\{g\in G : μ(A\cap T_{φ(g)}^{-1} A\cap T_{ψ(g)}^{-1}A)>μ(A)^3-\varepsilon\}$$ is syndetic. Moreover, in the special case where $φ(g)=ag$ and $ψ(g)=bg$ 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 $G$, and we show that the property fails in general if all three groups are of infinite index. One particularly interesting case is where $G=(\mathbb{Q}_{>0},\cdot)$ and $φ(g)=g$, $ψ(g)=g^2$, which leads to a multiplicative version for the large intersection result of Bergelson-Host-Kra. We also completely characterize the pairs of homomorphisms $φ,ψ$ 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 \emph{universal characteristic factor} for the multiple ergodic averages $$\frac{1}{|Φ_N|} \sum_{g\in Φ_N}T_{φ(g)}f_1\cdot T_{ψ(g)} f_2.$$ In the case where $G$ is finitely-generated, the characteristic factor for such averages is the \emph{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 \emph{Conze--Lesigne factor} and the $σ$-algebras of $φ(G)$ and $ψ(G)$ invariant functions.
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