Properties of multicorrelation sequences and large returns under some ergodicity assumptions

TL;DR

This paper studies the structure of multicorrelation sequences in ergodic systems, proving their decomposition into structured and negligible parts, and establishes results on large intersections and recurrence properties under ergodicity assumptions.

Contribution

It introduces a decomposition of multicorrelation sequences into nilsequence and nullsequence components under ergodicity, extending to polynomial multicorrelations and large recurrence results.

Findings

Decomposition of multicorrelation sequences into structured and null parts.

Existence of large triple intersections in ergodic systems.

Positive density of recurrence sets involving prime numbers.

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 $T_iT_j^{-1}$ 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 $a(n)=a_{\textrm{st}}(n)+a_{\textrm{er}}(n)$, where $a_{\textrm{st}}$ is a uniform limit of $d$-step nilsequences and $a_{\textrm{er}}$ is a nullsequence (that is, $\lim_{N-M \to \infty} \frac{1}{N-M} \sum_{n=M}^{N-1} |a_{\textrm{er}}|^2=0$). 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.