Properties of multicorrelation sequences and large returns under some ergodicity assumptions

Abstract

<p style='text-indent:20px;'>We prove that given a measure preserving system <inline-formula><tex-math id="M1">\begin{document}$ (X,\mathcal{B},\mu,T_1,\dots, T_d) $\end{document}</tex-math></inline-formula> with commuting, ergodic transformations <inline-formula><tex-math id="M2">\begin{document}$ T_i $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M3">\begin{document}$ T_iT_j^{-1} $\end{document}</tex-math></inline-formula> are ergodic for all <inline-formula><tex-math id="M4">\begin{document}$ i \neq j $\end{document}</tex-math></inline-formula>, the multicorrelation sequence <inline-formula><tex-math id="M5">\begin{document}$ a(n) = \int_X f_0 \cdot T_1^nf_1 \cdot \dotso \cdot T_d^n f_d \ d\mu $\end{document}</tex-math></inline-formula> can be decomposed as <inline-formula><tex-math id="M6">\begin{document}$ a(n) = a_{ \rm{st}}(n)+a_{ \rm{er}}(n) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M7">\begin{document}$ a_{ \rm{st}} $\end{document}</tex-math></inline-formula> is a uniform limit of <inline-formula><tex-math id="M8">\begin{document}$ d $\end{document}</tex-math></inline-formula>-step nilsequences and <inline-formula><tex-math id="M9">\begin{document}$ a_{ \rm{er}} $\end{document}</tex-math></inline-formula> is a nullsequence (that is, <inline-formula><tex-math id="M10">\begin{document}$ \lim_{N-M \to \infty} \frac{1}{N-M} \sum_{n = M}^{N-1} |a_{ \rm{er}}|^2 = 0 $\end{document}</tex-math></inline-formula>). Under some additional ergodicity conditions on <inline-formula><tex-math id="M11">\begin{document}$ T_1,\dots,T_d $\end{document}</tex-math></inline-formula> we also establish a similar decomposition for polynomial multicorrelation sequences of the form <inline-formula><tex-math id="M12">\begin{document}$ 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 $\end{document}</tex-math></inline-formula>, where each <inline-formula><tex-math id="M13">\begin{document}$ p_{i,k}: {\mathbb{Z}} \rightarrow {\mathbb{Z}} $\end{document}</tex-math></inline-formula> is a polynomial map. We also show, for <inline-formula><tex-math id="M14">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula>, that if <inline-formula><tex-math id="M15">\begin{document}$ T_1, T_2, T_1T_2^{-1} $\end{document}</tex-math></inline-formula> are invertible and ergodic, we have large triple intersections: for all <inline-formula><tex-math id="M16">\begin{document}$ \varepsilon&gt;0 $\end{document}</tex-math></inline-formula> and all <inline-formula><tex-math id="M17">\begin{document}$ A \in \mathcal{B} $\end{document}</tex-math></inline-formula>, the set <inline-formula><tex-math id="M18">\begin{document}$ \{n \in {\mathbb{Z}} : \mu(A \cap T_1^{-n}A \cap T_2^{-n}A)&gt;\mu(A)^3-\varepsilon\} $\end{document}</tex-math></inline-formula> is syndetic. Moreover, we show that if <inline-formula><tex-math id="M19">\begin{document}$ T_1, T_2, T_1T_2^{-1} $\end{document}</tex-math></inline-formula> are totally ergodic, and we denote by <inline-formula><tex-math id="M20">\begin{document}$ p_n $\end{document}</tex-math></inline-formula> the <inline-formula><tex-math id="M21">\begin{document}$ n $\end{document}</tex-math></inline-formula>-th prime, the set <inline-formula><tex-math id="M22">\begin{document}$ \{n \in \mathbb{N} : \mu(A \cap T_1^{-(p_n-1)}A \cap T_2^{-(p_n-1)}A)&gt;\mu(A)^3-\varepsilon\} $\end{document}</tex-math></inline-formula> has positive lower density.