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 $0<\mathrm{\ell }<4$, there exists $A\in \mathcal{X}$ such that $$\mu (A\cap T_1^{-n}A\cap T_2^{-n}A)<\mu (A{)}^{\mathrm{\ell }}\text{ for every }n\mathrm{\ne }0;$$ $\bullet$ There exists an ergodic system $(X,\mathcal{X},\mu ,T_1,T_2,T_3)$ with three commuting transformations such that for every $\mathrm{\ell }>0$, there exists $A\in \mathcal{X}$ such that $$\mu (A\cap T_1^{-n}A\cap T_2^{-n}A\cap T_3^{-n}A)<\mu (A{)}^{\mathrm{\ell }}\text{ for every }n\mathrm{\ne }0;$$ $\bullet$ 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 $\mathrm{\ell }>0$, there exists $A\in \mathcal{X}$ such that $$\mu (A\cap T_1^{-n}A\cap T_2^{-n}A)<\mu (A{)}^{\mathrm{\ell }}\text{ for every }n\mathrm{\ne }0.$$