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Uniform syndeticity in multiple recurrence
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- Journal:
- Ergodic Theory and Dynamical Systems , First View
- Published online by Cambridge University Press:
- 28 May 2024, pp. 1-22
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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 
$\delta>0$ and 
$K\geq 1$ (dependent only on d, l, and 
$\varepsilon $) such that the following holds: Consider a solvable group 
$\Gamma $ of derived length l, a probability space 
$(X, \mu )$, and d pairwise commuting measure-preserving 
$\Gamma $-actions 
$T_1, \ldots , T_d$ on 
$(X, \mu )$. Let E be a measurable set in X with 
$\mu (E) \geq \varepsilon $. Then, K many (left) translates of cover
$$ \begin{align*} \big\{\gamma\in\Gamma\colon \mu(T_1^{\gamma^{-1}}(E)\cap T_2^{\gamma^{-1}} \circ T^{\gamma^{-1}}_1(E)\cap \cdots \cap T^{\gamma^{-1}}_d\circ T^{\gamma^{-1}}_{d-1}\circ \cdots \circ T^{\gamma^{-1}}_1(E))\geq \delta \big\} \end{align*} $$
$\Gamma $. 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 
$\delta>0$ 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 cover G. The proof of our main result is a consequence of an ultralimit version of Austin’s amenable ergodic Szeméredi theorem.
$$ \begin{align*} \{g\in G\colon &m^{\otimes d}(\{(a_1,\ldots,a_n)\in G^d\colon \\ & (a_1,\ldots,a_n),(ga_1,a_2,\ldots,a_n),\ldots,(ga_1,ga_2,\ldots, ga_n)\in E\})\geq \delta \} \end{align*} $$
