  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
$i\neq j$,
  (i) there is $A\in\mathcal{B}$ such that the set
  $$\{n\in\mathbb{N}\colon\mu(A\cap T^{p_{1}(n)}A\cap\dots\cap
T^{p_{d}(n)}A)<\mu(A)^{d+1}\}$$
  is of density 1.
  (ii) there is $A\in\mathcal{B}$ such that the set
  $$\{n\in\mathbb{N}\colon\mu(A\cap T^{p_{1}(n)}A\cap\dots\cap
T^{p_{d}(n)}A)>\mu(A)^{d+1}\}$$
  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 $M\in\mathbb{N}$.
%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 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 $$\frac{1}{N} \sum_{n=0}^{N-1} \mu ( A \cap T^{n}A) > \mu(A)^2.$$
