Uniform syndeticity in multiple recurrence
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929675453857792 |
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| author | Jamneshan, Asgar Pan, Minghao |
| author_facet | Jamneshan, Asgar Pan, Minghao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_02833 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Uniform syndeticity in multiple recurrence Jamneshan, Asgar Pan, Minghao Dynamical Systems 37A30 (Primary), 37A15 (Secondary) 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. |
| title | Uniform syndeticity in multiple recurrence |
| topic | Dynamical Systems 37A30 (Primary), 37A15 (Secondary) |
| url | https://arxiv.org/abs/2208.02833 |