Polynomial maps which are not good for nice recurrence and applications

Let $\mathbb F_2$ be the finite field with two elements and let $\mathbb F_2^\omega$ denote the countably infinite-dimensional vector space over $\mathbb F_2$. We show that, unlike the case of polynomial maps $p:\mathbb Z\rightarrow\mathbb Z$ vanishing at zero which are always good for nice recurrence, there are polynomials $p:\mathbb F_2^\omega\rightarrow \mathbb F_2^\omega$ with $p(0_{\mathbb F_2^\omega})=0_{\mathbb F_2^\omega}$ which fail to have this property. This disproves a conjecture of Bergelson and McCutcheon (c. 2000), which predicted that for any countably infinite abelian groups $H$ and $G$, every polynomial map $p:H\to G$ with $p(0_H)=0_G$ is good for nice recurrence. Moreover, we develop a dynamical mechanism which shows that the magnitude of intersections along polynomial paths is degree-sensitive (even when one considers only weakly mixing systems). Among other things, we also show that the Furstenberg-Sarkozy theorem for $\mathbb F_2^\omega$-valued polynomials of degree at most $d$ vanishing at zero is equivalent to a $d$-dimensional symmetric-difference weakening of the density polynomial Hales-Jewett conjecture. Thus, as we explain in detail in this paper, our observations not only shed new light on the phenomenon of polynomial recurrence but also constrain possible strategies for proving or disproving the density polynomial Hales-Jewett conjecture.