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:

• (Existence of density-1 UI and OI set) Let (X, ${\mathcal B}$, μ, T) be an invertible probability measure preserving weakly mixing system. Then for any d∈ℕ, any non-constant integer-valued polynomials p₁, p₂,…, p_d such that p_i − p_j are also non-constant for all i ≠ j

  • (i) there is $A \in {\mathcal B}$ such that the set

    $$\left\{n \in \mathbb{N}: \mu\left(A \cap T^{p_{1}(n)} A \cap \cdots \cap T^{p_{d}(n)} A\right)<\mu(A)^{d+1}\right\}$$

    is of density 1.

  • (ii) there is

    $$A \in {\mathcal B}$$

    such that the set $of density 1.$ is of density 1.

• (Existence of Cesàro OI set) Let (X, ${\mathcal B}$, μ, T) be a free, invertible, ergodic probability measure preserving system and M ∈ ℕ. Then there is $A \in {\mathcal B}$ such that

  $$\frac{1}{N} \sum_{n=M}^{N+M-1} \mu\left(A \cap T^{n} A\right)>\mu(A)^{2}$$

  for all N ∈ ℕ.

• (Nonexistence of Cesàro UI set) Let (X, ${\mathcal B}$, 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

  $$\frac{1}{N} \sum_{n=0}^{N-1} \mu\left(A \cap T^{n} A\right)>\mu(A)^{2}.$$