LAUSR

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: (Existence of density-1 UI and OI set) Let ( X , $${\mathcal B}$$ ℬ , μ, T ) be an invertible probability measure preserving weakly mixing system. Then for any d ∈ℕ, any non-constant integer-valued polynomials p 1 , p 2 ,…, p d such that p i − p j are also non-constant for all i ≠ j (i) there is $$A \in {\mathcal B}$$ A ∈ ℬ such that the set $$\left\{n \in \mathbb{N}: \mu\left(A \cap T^{p_{1}(n)} A \cap \cdots \cap T^{p_{d}(n)} A\right)<\mu(A)^{d+1}\right\}$$ { n ∈ ℕ : μ ( A ∩ T p 1 ( n ) A ∩ ⋯ ∩ T p d ( n ) A ) < μ ( A ) d + 1 } is of density 1. (ii) there is $$A \in {\mathcal B}$$ A ∈ ℬ such that the set $$of density 1.$$ { n ∈ ℕ : μ ( A ∩ T p 1 ( n ) A ∩ ⋯ ∩ T p d ( n ) A ) > μ ( A ) d + 1 } is of density 1. (Existence of Cesàro OI set) Let ( X , $${\mathcal B}$$ ℬ , μ , T ) be a free, invertible, ergodic probability measure preserving system and M ∈ ℕ. Then there is $$A \in {\mathcal B}$$ A ∈ ℬ such that $$\frac{1}{N} \sum_{n=M}^{N+M-1} \mu\left(A \cap T^{n} A\right)>\mu(A)^{2}$$ 1 N ∑ n = M N + M − 1 μ ( A ∩ T n A ) > μ ( A ) 2 for all N ∈ ℕ. (Nonexistence of Cesàro UI set) Let ( X , $${\mathcal B}$$ ℬ , μ, T ) be an invertible probability measure preserving system. For any measurable set A satisfying μ ( A ) ∈ (0, 1), there exist infinitely many N ∈ ℕ such that $$\frac{1}{N} \sum_{n=0}^{N-1} \mu\left(A \cap T^{n} A\right)>\mu(A)^{2}.$$ 1 N ∑ n = 0 N − 1 μ ( A ∩ T n A ) > μ ( A ) 2 .