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 $\bullet$ There exists an ergodic system $(X,\mathcal{X},\mu,T_1,T_2)$ with two commuting transformations such that for every $0<\ell< 4$, there exists $A\in\mathcal{X}$ such that $$\mu(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A)<\mu(A)^{\ell} \text{ for every } n\neq 0;$$ $\bullet$ There exists an ergodic system $(X,\mathcal{X},\mu,T_1,T_2, T_{3})$ with three commuting transformations such that for every $\ell>0$, there exists $A\in\mathcal{X}$ such that $$\mu(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A\cap T_{3}^{-n}A)<\mu(A)^{\ell} \text{ for every } n\neq 0;$$ $\bullet$ There exists an ergodic system $(X,\mathcal{X},\mu,T_1,T_2)$ with two transformations generating a 2-step nilpotent group such that for every $\ell>0$, there exists $A\in\mathcal{X}$ such that $$\mu(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A)<\mu(A)^{\ell} \text{ for every } n\neq 0.$$