Article Contents
Article Contents

# 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.

Mathematics Subject Classification: Primary: 37A15, 37A30.

 Citation:

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