Multiple recurrence and popular differences for polynomial patterns in rings of integers

We demonstrate that the phenomenon of popular differences (aka the phenomenon of large intersections) holds for natural families of polynomial patterns in rings of integers of number fields. If $K$ is a number field with ring of integers ${\mathcal{O}}_K$ and $E\subseteq {\mathcal{O}}_K$ has positive upper Banach density $d^{\ast }(E)=\delta >0$, we show, inter alia:

1. If $p(x)\in K[x]$ is an intersective ${\mathcal{O}}_K$-valued polynomial and $r,s\in {\mathcal{O}}_K$ are distinct and nonzero, then for any $\epsilon >0$, the set of $n\in {\mathcal{O}}_K$ such that $$d^{\ast }(\{x\in {\mathcal{O}}_K:\{x,x+rp(n),x+sp(n)\}\subseteq E\})>{\delta }^3-\epsilon \mathrm{.}$$ is syndetic. Moreover, if $\frac{s}{r} \in $\mathbb{Q}$$,thentherearesyndeticallymany$n \in $\mathcal{O}$_K$ such that $$d^{\ast }(\{x\in {\mathcal{O}}_K:\{x,x+rp(n),x+sp(n),x+(r+s)p(n)\}\subseteq E\})>{\delta }^4-\epsilon \mathrm{.}$$
2. If $\{p_1,\dots ,p_k\}\subseteq K[x]$ is a jointly intersective family of linearly independent ${\mathcal{O}}_K$-valued polynomials, then the set of $n \in $\mathcal{O}$_K$ such that $$d^{\ast }(\{x\in {\mathcal{O}}_K:\{x,x+p_1(n),\dots ,x+p_k(n)\}\subseteq E\})>{\delta }^{k+1}-\epsilon$$ is syndetic. These two results generalize and extend previous work of Frantzikinakis and Kra on polynomial configurations in $\mathbb{Z}$ and build upon recent work of the authors and Best on linear patterns in general abelian groups. The above combinatorial results follow from multiple recurrence results in ergodic theory, which require a sharpening of existing tools for handling polynomial multiple ergodic averages. A key advancement made in this paper is a new result on the equidistribution of polynomial orbits in nilmanifolds, which can be seen as a far-reaching generalization of Weyl’s equidistribution theorem.