  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 $d^*(E) = \delta > 0$, 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
$\varepsilon > 0$, the set of $n \in \mathcal{O}_K$ such that \[ d^* \left( \{
x \in \mathcal{O}_K : \{x, x + rp(n), x + sp(n)\} \subseteq E \} \right) >
\delta^3 - \varepsilon. \] is syndetic. Moreover, if $\frac{s}{r} \in
\mathbb{Q}$, then there are syndetically many $n \in \mathcal{O}_K$ such that
\[ d^* \left( \{ x \in \mathcal{O}_K : \{x, x + rp(n), x + sp(n), x +
(r+s)p(n)\} \subseteq E \} \right) > \delta^4 - \varepsilon. \]
  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 \[ d^* \left( \{ x \in \mathcal{O}_K : \{x, x +
p_1(n), \dots, x + p_k(n)\} \subseteq E \} \right)> \delta^{k+1} - \varepsilon
\] 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.
