Quantitative multiple recurrence for two and three transformations

Abstract

We provide various counter examples for quantitative multiple recurrence problems for systems with more than one transformation. We show that:

• There exists an ergodic system (X,X, μ,T₁, T₂) with two commuting transformations such that, for every 0 < ℓ < 4, there exists A ∈ X such that

  $$\mu \left( {A \cap T_1^{ - n}A \cap T_2^{ - n}A} \right) < \mu {\left( A \right)^\ell }foreveryn \ne 0$$

• There exists an ergodic system (X,X, μ,T₂, T₃) with three commuting transformations such that, for every ℓ > 0, there exists A ∈ X such that

  $$\mu \left( {A \cap T_1^{ - n}A \cap T_2^{ - n}A \cap T_3^{ - n}A} \right) < \mu {\left( A \right)^\ell }foreveryn \ne 0$$

• There exists an ergodic system (X,X, μ,T₁, T₂) with two transformations generating a 2-step nilpotent group such that, for every ℓ > 0, there exists A ∈ X such that

  $$\mu \left( {A \cap T_1^{ - n}A \cap T_2^{ - n}A} \right) < \mu {\left( A \right)^\ell }foreveryn \ne 0$$