论文标题
三分配置的Khintchine型复发
Khintchine-type recurrence for 3-point configurations
论文作者
论文摘要
本文的目的是概括，完善和改善大型交叉点的结果。我们表明，如果$ g $是一个可计数的阿贝利安群体，$φψ：g \ t to g $是同构的，以至于三个亚组$φ（g）$，$ψ（g）$和$（ψ-φ）（g）（g）（g）$中的至少有两个$ g $，则$ g $，inters $ \ \ c $ \ c。 财产}。也就是说，对于任何Ergodic度量保留系统$ x =（x，x，\ nathcal {x}，μ，（t_g）_ {g \ in G}）$，任何$ a \ in \ in \ nathcal {x} $，以及任何$ \ varepsilon> 0 $ 0 $，set $ \ \ v $ \ \ \ \ \ f：g： t_ {φ（g）}^{ - 1} a \ cap t_ {ψ（g）}^{ - 1} a）>μ（a）^3- \ varepsilon \} $ 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.一个特别有趣的情况是$ g =（\ mathbb {q} _ {> 0}，\ cdot）$和$φ（g）= g $，$ψ（g）= g^2 $，这导致了Bergelson-Host-Host-kra的大相交结果的多词版本。我们还完全表征同构$φ，ψ$对具有较大相交属性的对成对。我们的主要结果的证明取决于对多个ergodic平均值的\ emph {通用特征因子}的结构的分析，$$ \ frac {1} {|φ_n|} \ sum_ {g \ inφ_n} t_ {φ（g）} f_1} f_1 \ cdot t_ $ c $ c $ g）有限生成的，这种平均值的特征因素是\ emph {kronecker因子}。在本文中，我们研究了不一定有限生成的群体的动作，特别表明，通过传递$ x $的扩展，人们可以用\ emph {conze-lesigne factor}和$σ$ -Slgebras $ $φ（g）$和$ψ（g）和$ψ（g）$ novariant函数来描述特征因素。
The goal of this paper is to generalize, refine, and improve results on large intersections. We show that if $G$ is a countable abelian group and $φ, ψ: G \to G$ are homomorphisms such that at least two of the three subgroups $φ(G)$, $ψ(G)$, and $(ψ-φ)(G)$ have finite index in $G$, then $\{φ, ψ\}$ has the \emph{large intersections property}. That is, for any ergodic measure preserving system $X=(X,\mathcal{X},μ,(T_g)_{g\in G})$, any $A\in\mathcal{X}$, and any $\varepsilon>0$, the set $$\{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.
