Failure of Khintchine-type results along the polynomial image of IP$_0$ sets

Abstract

In "IP-sets and polynomial recurrence", Bergelson, Furstenberg, and McCutcheon established the following far reaching extension of Khintchine's recurrence theorem: For any invertible probability preserving system $(X,\mathcal A,\mu,T)$, any non-constant polynomial $p\in\mathbb Z[x]$ with $p(0)=0$, any $A\in\mathcal A$, and any $\epsilon>0$, the set $$R_\epsilon^p(A)=\{n\in\mathbb N\,|\,\mu(A\cap T^{-p(n)}A)>\mu^2(A)-\epsilon\}$$ is IP$^*$, meaning that for any increasing sequence $(n_k)_{k\in\mathbb N}$ in $\mathbb N$, $$\text{FS}((n_k)_{k\in\mathbb N})\cap R_\epsilon^p(A)\neq \emptyset,$$ where $$\text{FS}((n_k)_{k\in\mathbb N})=\{\sum_{j\in F}n_j\,|\,F\subseteq \mathbb N\,\text{ is finite}\text{ and }F\neq\emptyset\}=\{n_{k_1}+\cdots+n_{k_t}\,|\,k_1<\cdots 1$ and $p(0)=0$ there is an invertible probability preserving system $(X,\mathcal A,\mu,T)$, a set $A\in\mathcal A$, and an $\epsilon>0$ for which the set $R_\epsilon^p(A)$ is not IP$_0^*$.