Polynomial Szemerédi characterization over rings of integers of global fields

Let $K$ be a global field with ring of integers $\mathcal{O}_K$. Let $P_1[x],\dots,P_k[x]\in\mathcal{O}_K[x]$ be nonconstant polynomials, and let $d^*(A)$ denote the upper Banach density of a subset $A\subseteq\mathcal{O}_K$. Polynomial Szemerédi characterization. The following are equivalent: (i) for every nonzero ideal $I\subseteq\mathcal{O}_K$, there exists $n\in\mathcal{O}_K$ such that

$$\{P_1(n),\dots,P_k(n)\}\subseteq I;$$

(ii) for every subset $A\subseteq\mathcal{O}_K$ with $d^*(A)>0$, there exist $x,n\in\mathcal{O}_K$ such that

$$\{x,x+P_1(n),\dots,x+P_k(n)\}\subseteq A.$$

This conjecture seeks a necessary and sufficient algebraic condition for the polynomial Szemerédi theorem over rings of integers of global fields. The analogous Furstenberg–Sárközy characterization is established in the paper, while the equivalence asserted here is posed as an open problem.