Uniform syndeticity in multiple recurrence

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 \begin{equation*} \left\{\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 \ldots \circ T^{\gamma^{-1}}_1(E))\geq \delta \right\} \end{equation*} cover $\Gamma$. 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 \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 \delta \} \end{multline*} cover $G$. The proof of our main result is a consequence of an ultralimit version of Austin's amenable ergodic Szem\'eredi theorem.