Properties of multicorrelation sequences and large returns under some ergodicity assumptions

We prove that given a measure preserving system $(X,\mathcal{B},\mu ,T_1,\dots ,T_d)$ with commuting, ergodic transformations $T_i$ such that $T_i{T}_j^{-1}$ are ergodic for all $i\mathrm{\ne }j$, the multicorrelation sequence $a(n)=\int_X f_0 \cdot T_1^nf_1 \cdot \dotso \cdot T_d^n f_d \ d\mu$canbedecomposedas$a(n)=a_{$\text{st}$}(n)+a_{$\text{er}$}(n)$,where$a_{$\text{st}$}$ is a uniform limit of $d$-step nilsequences and $a_{\text{er}}$ is a nullsequence (that is, $\lim_{N-M \to \infty} \frac{1}{N-M} \sum_{n=M}^{N-1} |a_{$\text{er}$}|^2=0$)\mathrm{.}Undersomeadditionalergodicityconditionson$T_1,\dots,T_d$ we also establish a similar decomposition for polynomial multicorrelation sequences of the form $a(n)=\int_X f_0 \cdot \prod_{i=1}^dT_i^{p_{i,1}(n)}f_1\cdot\dotso \cdot \prod_{i=1}^dT_i^{p_{i,k}(n)}f_k \ d\mu$,whereeach$p_{i,k}: $\mathbb{Z}$ \rightarrow $\mathbb{Z}$$isapolynomialmap\mathrm{.}Wealsoshow,for$d=2$,thatif$T_1, T_2, T_1T_2^{-1}$ are invertible and ergodic, we have large triple intersections: for all $\epsilon >0$ and all $A\in \mathcal{B}$, the set ${n \in $\mathbb{Z}$ : \mu(A \cap T_1^{-n}A \cap T_2^{-n}A)>\mu(A)^3-\varepsilon}$issyndetic\mathrm{.}Moreover,weshowthatif$T_1, T_2, T_1T_2^{-1}$aretotallyergodic,andwedenoteby$p_n$the$n$-th prime, the set ${n \in $\mathbb{N}$ : \mu(A \cap T_1^{-(p_n-1)}A \cap T_2^{-(p_n-1)}A)>\mu(A)^3-\varepsilon}$ has positive lower density.