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 $p_1(t),...,p_{l}(t)\in\Z[t]$ with $p_j(0)=0$ have distinct degrees. Then for any $\epsilon>0$ and $A\subseteq X$ with $\mu(A)>0$, the set $$ \{n:\,\mu(A\cap T_1^{-p_1(n)}A\cap...\cap T_l^{-p_l(n)}A)\geq\mu(A)^{l+1}-\epsilon\} $$ has bounded gaps.