Note on polynomial recurrence

Let $(X,μ,T_1,...,T_l)$ be a measure-preserving system with those $T_i$ are commuting. Suppose that the polynomials $p_1(t),...,p_{l}(t)\in\Z[t]$ with $p_j(0)=0$ have distinct degrees. Then for any $ε>0$ and $A\subseteq X$ with $μ(A)>0$, the set $\{n:\,μ(A\cap T_1^{-p_1(n)}A\cap...\cap T_l^{-p_l(n)}A)\geqμ(A)^{l+1}-ε\}$ has bounded gaps.

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