非交换变换沿有理无关多项式的多重递推

两个可换变换的多重递归性

Multiple recurrence for two commuting transformations

本文研究了两个交换变换的多重递归性。我们得到了一个结果，这是类似的，但不完全相同的一个单一的转换所建立的Bergelson，主机和Kra。我们将使用B.主持人证明。

非交换确定性变换的双重回复性的反例

Counterexamples to double recurrence for non-commuting deterministic transformations

本文构造反例证明：对于在原点为零的单射整数多项式，若两者次数同为一次或同为二次及以上，则非交换混合零熵变换不满足双重回复性，从而完全解决了Frantzikinakis和Host提出的问题。

A good universal weight for multiple recurrence averages with commuting transformations in norm

We will show that the sequences appearing in Bourgain's double recurrence result are good universal weights to the multiple recurrence averages with commuting measure-preserving transformations in norm. This will extend the pointwise converge result of Bourgain, the norm convergence result of Tao, and the authors' previous work on the single measure-preserving transformation.

Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

A set $R\subset \mathbb{N}$ is called rational if it is well-approximable by finite unions of arithmetic progressions. Examples of rational sets include many classical sets of number-theoretical origin such as the set of squarefree numbers, the set of abundant numbers, or sets of the form $Φ_x:=\{n\in\mathbb{N}: \frac{\boldsymbolφ(n)}{n}<x\}$, where $x\in[0,1]$ and $\boldsymbolφ$ is Euler's totient function. We investigate the combinatorial and dynamical properties of rational sets and obtain new results in ergodic Ramsey theory. We show that if $R$ is a rational set with $\overline{d}(R)>0$, …

交换保测变换的点态递推

Pointwise recurrence for commuting measure preserving transformations

设$(X,\mathcal{A}, μ)$是概率测度空间，$T_i,$ $1\leq i\leq H,$是该测度空间上的交换可逆保测变换.我们证明了如下逐点结果：平均值$$\frac{1}{N}\sum_{n=1}^N f_1(T_1^nx)f_2(T_2^nx)\cdots f_H(T_H^nx)$$收敛a.e.为每个函数$f_i \in L^{\infty}(μ)$。\\因此，如果$T_i = T^i$对于$1\leq i \leq H$，其中$T$是$(X, \mathcal{A}, μ)$上的可逆测度保持变换，则平均值$$\frac{1}{N}\sum_{n=1}^N f_1(T^nx)f_2(T^{2n}x)...f_H(T^{Hn}x)$$收敛a.e.这解决了一个长期开放的问题上的逐点收敛的非常规遍历平均。对于$H=2$，它提供了J. Bourgain a.e.双递归定理

两个和三个变换的定量多重递归

Quantitative multiple recurrence for two and three transformations

我们提供了各种反例定量多个递归问题的系统与一个以上的转换。我们证明了$\bullet$存在具有两个交换变换的遍历系统$(X,\mathcal{X},μ,T_1,T_2)$使得对于每个$0<\ell< 4$，存在$A\in\mathcal{X}$使得$$μ(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A)<μ(A)^{\ell} \text{ for every } n\neq 0;$$ $\bullet$存在具有三个交换变换的遍历系统$(X,\mathcal{X},μ,T_1,T_2, T_{3})$使得对于每个$\ell>0$，存在$A\in\mathcal{X}$使得$$μ(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A\cap T_{3}^{-n}A)<μ(A)^{\ell} \text{ for every } n\neq 0;$$ $\bullet$存在遍历系统$(X,\mathcal{X},μ,T_1,T_2)$，其具有两个生成2-阶幂零群使得对于每个$\ell>0$，存在$A\in\mathcal{X}$使得$$μ(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A)<μ(A)^{\ell} \text{ for every } n\neq 0.$$