$\mathrm{IP}_{\mathrm{rat}}$-polynomial recurrence and large intersections

Let $p_1,...,p_k$ be a rationally independent sequence of integer valued nonlinear polynomials. We show that for all $E\subseteq \mathbb{N}$, every Folner sequence $\Phi$, and every $\varepsilon>0$, the set $$\left\{n\in \mathbb{N} : d_{\Phi}\left(E\cap(E+p_1(n))\cap\cdots\cap (E+p_k(n))\right) > d_{\Phi}(E)^{k+1}-\varepsilon\right\}$$ intersects every IP generated by a sequence with rational spectrum. Our methods involve the study of the characteristic factors for multiple ergodic polynomial averages along IPs. In particular, we also prove a pointwise convergence theorem for polynomial averages along IPs with rational spectrum, generalizing a well known result of Leibman.