Properties of multicorrelation sequences and large returns under some ergodicity assumptions

We prove that given a measure preserving system \begin{document}$ (X,\mathcal{B},\mu,T_1,\dots, T_d) $\end{document} with commuting, ergodic transformations id="M2">\begin{document}$ T_i such id="M3">\begin{document}$ T_iT_j^{-1} are for all id="M4">\begin{document}$ i \neq j $\end{document}, the multicorrelation sequence id="M5">\begin{document}$ a(n) = \int_X f_0 \cdot T_1^nf_1 \dotso T_d^n f_d \ d\mu can be decomposed as id="M6">\begin{document}$ a_{ \rm{st}}(n)+a_{ \rm{er}}(n) where id="M7">\begin{document}$ \rm{st}} is uniform limit of id="M8">\begin{document}$ d $\end{document}-step nilsequences and id="M9">\begin{document}$ \rm{er}} nullsequence (that is, id="M10">\begin{document}$ \lim_{N-M \to \infty} \frac{1}{N-M} \sum_{n M}^{N-1} |a_{ \rm{er}}|^2 0 $\end{document}). Under some additional ergodicity conditions on id="M11">\begin{document}$ T_1,\dots,T_d we also establish similar decomposition polynomial sequences form id="M12">\begin{document}$ \prod_{i 1}^dT_i^{p_{i,1}(n)}f_1\cdot\dotso 1}^dT_i^{p_{i,k}(n)}f_k each id="M13">\begin{document}$ p_{i,k}: {\mathbb{Z}} \rightarrow map. We show, id="M14">\begin{document}$ 2 if id="M15">\begin{document}$ T_1, T_2, T_1T_2^{-1} invertible ergodic, have large triple intersections: id="M16">\begin{document}$ \varepsilon>0 id="M17">\begin{document}$ A \in \mathcal{B} set id="M18">\begin{document}$ \{n : \mu(A \cap T_1^{-n}A T_2^{-n}A)>\mu(A)^3-\varepsilon\} syndetic. Moreover, show id="M19">\begin{document}$ totally denote by id="M20">\begin{document}$ p_n id="M21">\begin{document}$ n $\end{document}-th prime, id="M22">\begin{document}$ \mathbb{N} T_1^{-(p_n-1)}A T_2^{-(p_n-1)}A)>\mu(A)^3-\varepsilon\} has positive lower density.