Properties of multicorrelation sequences and large returns under some ergodicity assumptions

Abstract

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 $i\ne j$, the multicorrelation sequence $a(n)={\int }_X{f}_0\cdot T_1^n{f}_1\cdot \dots \cdot T_d^n{f}_{d}d\mu$ can be decomposed as $a(n)=a_{\text{st}}(n)+a_{\text{er}}(n)$, where $a_{\text{st}}$ is a uniform limit of $d$-step nilsequences and $a_{\text{er}}$ is a nullsequence (that is, ${\lim }_{N-M\to \infty }\frac{1}{N-M}{\sum }_{n=M}^{N-1}|a_{\text{er}}{|}^2=0$). Under some additional ergodicity conditions on $T_1,\dots ,T_d$ we also establish a similar decomposition for polynomial multicorrelation sequences of the form $a(n)={\int }_X{f}_0\cdot {\prod }_{i=1}^d{T}_i^{p_{i,1}(n)}{f}_1\cdot \dots \cdot {\prod }_{i=1}^d{T}_i^{p_{i,k}(n)}{f}_{k}d\mu$, where each $p_{i,k}:\mathbb{Z}\to \mathbb{Z}$ is a polynomial map. We also show, for $d=2$, that if $T_1,T_2,T_1{T}_2^{-1}$ are invertible and ergodic, we have large triple intersections: for all $\epsilon >0$ and all $A\in \mathcal{B}$, the set $\{n\in \mathbb{Z}:\mu (A\cap T_1^{-n}A\cap T_2^{-n}A)>\mu (A{)}^3-\epsilon \}$ is syndetic. Moreover, we show that if $T_1,T_2,T_1{T}_2^{-1}$ are totally ergodic, and we denote by $p_n$ the $n$-th prime, the set $\{n\in \mathbb{N}:\mu (A\cap T_1^{-(p_n-1)}A\cap T_2^{-(p_n-1)}A)>\mu (A{)}^3-\epsilon \}$ has positive lower density.

21 pages