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_iT_j^{-1} $ are ergodic for all $ i \neq 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 $ can be decomposed as $ a(n) = a_{ \rm{st}}(n)+a_{ \rm{er}}(n) $, where $ a_{ \rm{st}} $ is a uniform limit of $ d $-step nilsequences and $ a_{ \rm{er}} $ is a nullsequence (that is, $ \lim_{N-M \to \infty} \frac{1}{N-M} \sum_{n = M}^{N-1} |a_{ \rm{er}}|^2 = 0 $). Under some additional ergodicity conditions on $ 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 $, where each $ p_{i,k}: {\mathbb{Z}} \rightarrow {\mathbb{Z}} $ is a polynomial map. We also show, for $ d = 2 $, that if $ T_1, T_2, T_1T_2^{-1} $ are invertible and ergodic, we have large triple intersections: for all $ \varepsilon>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\} $ is syndetic. Moreover, we show that if $ T_1, T_2, T_1T_2^{-1} $ are totally ergodic, and we denote by $ 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.