Note on polynomial recurrence

Abstract

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

This is a very very preliminary draft, which maybe contains some mistakes