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

Abstract

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 $\hat{I}¦$, and every $\epsilon >0$, the set $$\left\{n\in \mathbb{N}:d_{\hat{I}}¦\left(E\cap (E+p_1(n))\cap \cdots \cap (E+p_k(n))\right)>d_{\hat{I}}¦(E{)}^{k+1}-\epsilon \right\}$$ 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.

23 pages