Under- and over-independence in measure preserving systems

Abstract

We introduce the notions of over- and under-independence for weakly mixing and (free) ergodic measure preserving actions and establish new results which complement and extend the theorems obtained in [BoFW] and [A]. Here is a sample of results obtained in this paper: $\cdot$ (Existence of density-1 UI and OI set) Let $(X,\mathcal{B},\mu ,T)$ be an invertible probability measure preserving weakly mixing system. Then for any $d\in \mathbb{N}$, any non-constant integer-valued polynomials $p_1,p_2,\dots ,p_d$ such that $p_i-p_j$ are also non-constant for all $i\ne j$, (i) there is $A\in \mathcal{B}$ such that the set $$\{n\in \mathbb{N}:\mu (A\cap T^{p_1(n)}A\cap \cdots \cap T^{p_d(n)}A)<\mu (A{)}^{d+1}\}$$ is of density 1. (ii) there is $A\in \mathcal{B}$ such that the set $$\{n\in \mathbb{N}:\mu (A\cap T^{p_1(n)}A\cap \cdots \cap T^{p_d(n)}A)>\mu (A{)}^{d+1}\}$$ is of density 1. $\cdot$ (Existence of Cesàro OI set) Let $(X,\mathcal{B},\mu ,T)$ be a free, invertible, ergodic probability measure preserving system and $M\in \mathbb{N}$. %Suppose that $X$ contains an ergodic component which is aperiodic. Then there is $A\in \mathcal{B}$ such that $$\frac{1}{N}\sum _{n=M}^{N+M-1}\mu (A\cap T^{n}A)>\mu (A{)}^2$$ for all $N\in \mathbb{N}$. $\cdot$ (Nonexistence of Cesàro UI set) Let $(X,\mathcal{B},\mu ,T)$ be an invertible probability measure preserving system. For any measurable set $A$ satisfying $\mu (A)\in (0,1)$, there exist infinitely many $N\in \mathbb{N}$ such that $$\frac{1}{N}\sum _{n=0}^{N-1}\mu (A\cap T^{n}A)>\mu (A{)}^2.$$

25 pages