Quantitative multiple recurrence for two and three transformations

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 $Ainmathcal{X}$ such that $$mu(Acap T_{1}^{-n}Acap T_{2}^{-n}A)<mu(A)^{ell} text{ for every } n eq 0;$$ $bullet$ 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 $Ainmathcal{X}$ such that $$mu(Acap T_{1}^{-n}Acap T_{2}^{-n}Acap T_{3}^{-n}A)<mu(A)^{ell} text{ for every } n eq 0;$$ $bullet$ 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 $Ainmathcal{X}$ such that $$mu(Acap T_{1}^{-n}Acap T_{2}^{-n}A)<mu(A)^{ell} text{ for every } n eq 0.$$