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 $d,l\ge 1$ and any $\epsilon >0$, we prove the existence of $\delta >0$ and $K\ge 1$ (dependent only on $d$, $l$, and $\epsilon$) such that the following holds: Consider a solvable group $\mathrm{\Gamma }$ of derived length $l$, a probability space $(X,\mu )$, and $d$ pairwise commuting measure-preserving $\mathrm{\Gamma }$-actions $T_1,\dots ,T_d$ on $(X,\mu )$. Let $E$ be a measurable set in $X$ with $\mu (E)\ge \epsilon$. 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 $\mathrm{\Gamma }$. This result extends and refines uniformity results by Furstenberg and Katznelson. As a combinatorial application, we obtain the following uniformity result. For any integers $d,l\ge 1$ and any $\epsilon >0$, there are $\delta >0$ and $K\ge 1$ (dependent only on $d$, $l$, and $\epsilon$) such that for all finite solvable groups $G$ of derived length $l$ and any subset $E\subset G^d$ with $m^{\otimes d}(E)\ge \epsilon$ (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.