论文标题
多次复发的统一联合性
Uniform syndeticity in multiple recurrence
论文作者
论文摘要
本文的主要定理建立了统一的联合性结果，该结果是对概率空间的量度更具衡量作用的复发。更确切地说，对于任何整数$ d，l \ geq 1 $以及任何$ \ varepsilon> 0 $，我们证明存在$δ> 0 $和$ k \ geq 1 $（仅取决于$ d $，$ l $和$ \ \ varepsilon $），以便以下以下内容。 考虑一个可解决的组$γ$的派生长度$ l $，概率空间$（x，μ）$和$ d $ d $ pairwise Ponsinging Menature-temucting Menature vermuting umporting $γ$ - actions $ t_1，\ ldots，t_d $，t_d $ on $（x，μ）$。让$ e $是$ x $的可测量设置，带有$μ（e）\ geq \ varepsilon $。然后，$ k $许多（左）翻译为\ begin {equation*} \ left \ {γ\inγ\colonμ（t_1^{γ^{γ^{ - 1}}}（e）\ cap t_2^{γ^{γ^{ - 1}}}}}}}}}}}} \ circt t^{circt t^{-1 cyp^{-1} cd {-1}^cd {-1}^c cd {-1}^c cd {-1}^c cd {-1}^{-1} t^{γ^{ - 1}} _ d \ circt t^{γ^{ - 1}} _ {d-1} \ circ \ ldots \ circ t^{γ^{γ^{ - 1}}} _ 1（e））该结果扩展并通过Furstenberg和Katznelson进行了统一的结果。 作为组合应用，我们获得以下均匀性结果。对于任何整数$ d，l \ geq 1 $和任何$ \ varepsilon> 0 $，$δ> 0 $和$ k \ geq 1 $（仅取决于$ d $，$ l $和$ \ \ varepsilon $），以便所有有限的可解决长度$ l $ $ g $ c $ e \ e^$ e^$^dimime $^dimie d}（e）\ geq \ varepsilon $（其中$ m $是$ g $上的统一度量），我们有$ k $ -many（左）翻译\ begin {multline*} \{g\in G\colon m^{\otimes d}(\{(a_1,\ldots,a_n)\in G^d\colon (a_1,\ldots,a_n),(ga_1,a_2,\ldots,a_n),\ldots,(ga_1,ga_2,\ldots, ga_n）\在e \}中）\geqΔ\} \ end {multline*}覆盖$ g $。 我们主要结果的证明是奥斯汀正式埃戈迪克szeméredi定理的超富版版本的结果。
The main theorem of this paper establishes a uniform syndeticity result concerning the multiple recurrence of measure-preserving actions on probability spaces. More precisely, for any integers $d,l\geq 1$ and any $\varepsilon > 0$, we prove the existence of $δ>0$ and $K\geq 1$ (dependent only on $d$, $l$, and $\varepsilon$) such that the following holds: Consider a solvable group $Γ$ of derived length $l$, a probability space $(X, μ)$, and $d$ pairwise commuting measure-preserving $Γ$-actions $T_1, \ldots, T_d$ on $(X, μ)$. Let $E$ be a measurable set in $X$ with $μ(E) \geq \varepsilon$. Then, $K$ many (left) translates of \begin{equation*} \left\{γ\inΓ\colon μ(T_1^{γ^{-1}}(E)\cap T_2^{γ^{-1}} \circ T^{γ^{-1}}_1(E)\cap \cdots \cap T^{γ^{-1}}_d\circ T^{γ^{-1}}_{d-1}\circ \ldots \circ T^{γ^{-1}}_1(E))\geq δ\right\} \end{equation*} cover $Γ$. This result extends and refines uniformity results by Furstenberg and Katznelson. As a combinatorial application, we obtain the following uniformity result. For any integers $d,l\geq 1$ and any $\varepsilon > 0$, there are $δ>0$ and $K\geq 1$ (dependent only on $d$, $l$, and $\varepsilon$) such that for all finite solvable groups $G$ of derived length $l$ and any subset $E\subset G^d$ with $m^{\otimes d}(E)\geq \varepsilon$ (where $m$ is the uniform measure on $G$), we have that $K$-many (left) translates of \begin{multline*} \{g\in G\colon m^{\otimes d}(\{(a_1,\ldots,a_n)\in G^d\colon (a_1,\ldots,a_n),(ga_1,a_2,\ldots,a_n),\ldots,(ga_1,ga_2,\ldots, ga_n)\in E\})\geq δ\} \end{multline*} cover $G$. The proof of our main result is a consequence of an ultralimit version of Austin's amenable ergodic Szeméredi theorem.
