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 $dinmathbb{N}$, 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 $i eq j$, (i) there is $Ainmathcal{B}$ such that the set $${ninmathbb{N}colonmu(Acap T^{p_{1}(n)}Acapdotscap T^{p_{d}(n)}A)<mu(A)^{d+1}}$$ is of density 1. (ii) there is $Ainmathcal{B}$ such that the set $${ninmathbb{N}colonmu(Acap T^{p_{1}(n)}Acapdotscap T^{p_{d}(n)}A)>mu(A)^{d+1}}$$ is of density 1. $cdot$ (Existence of Ces`aro OI set) Let $(X,mathcal{B},mu,T)$ be a free, invertible, ergodic probability measure preserving system and $Minmathbb{N}$. %Suppose that $X$ contains an ergodic component which is aperiodic. Then there is $Ainmathcal{B}$ such that $$frac{1}{N}sum_{n=M}^{N+M-1}mu(Acap T^{n}A)>mu(A)^{2}$$ for all $Ninmathbb{N}$. $cdot$ (Nonexistence of Ces`aro 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 $$frac{1}{N} sum_{n=0}^{N-1} mu ( A cap T^{n}A) > mu(A)^2.$$
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