AbstractWe 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 polynomial (i.e., p has a root modulo m for every $m \in \mathcal{O}_K$ ) with $p(\mathcal{O}_K) \subseteq \mathcal{O}_K$ and $r, s \in \mathcal{O}_K$ are distinct and nonzero, then for any $\varepsilon > 0$ , there is a syndetic set $S \subseteq \mathcal{O}_K$ such that for any $n \in S$ , \begin{align*} d^* \left( \left\{ x \in \mathcal{O}_K \;:\; \{x, x + rp(n), x + sp(n)\} \subseteq E \right\} \right) > \delta^3 - \varepsilon. \end{align*} Moreover, if ${s}/{r} \in \mathbb{Q}$ , then there are syndetically many $n \in \mathcal{O}_K$ such that \begin{align*} d^* \left( \left\{ x \in \mathcal{O}_K \;:\; \{x, x + rp(n), x + sp(n), x + (r+s)p(n)\} \subseteq E \right\} \right) > \delta^4 - \varepsilon; \end{align*} (2)if $\{p_1, \dots, p_k\} \subseteq K[x]$ is a jointly intersective family (i.e., $p_1, \dots, p_k$ have a common root modulo m for every $m \in \mathcal{O}_K$ ) of linearly independent polynomials with $p_i(\mathcal{O}_K) \subseteq \mathcal{O}_K$ , then there are syndetically many $n \in \mathcal{O}_K$ such that \begin{align*} d^* \left( \left\{ x \in \mathcal{O}_K \;:\; \{x, x + p_1(n), \dots, x + p_k(n)\} \subseteq E \right\} \right) > \delta^{k+1} - \varepsilon. \end{align*} These two results generalise and extend previous work of Frantzikinakis and Kra [21] and Franztikinakis [19] on polynomial configurations in $\mathbb{Z}$ and build upon recent work of the authors and Best [2] on linear patterns in general abelian groups. The above combinatorial results follow from multiple recurrence results in ergodic theory via a version of Furstenberg’s correspondence principle. The ergodic-theoretic recurrence theorems 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 generalisation of Weyl’s equidistribution theorem for polynomials of several variables:(3)let $d, k, l \in \mathbb{N}$ . Let $(X, \mathcal{B}, \mu, T_1, \dots, T_l)$ be an ergodic, connected $\mathbb{Z}^l$ -nilsystem. Let $\{p_{i,j} \;:\; 1 \le i \le k, 1 \le j \le l\} \subseteq \mathbb{Q}[x_1, \dots, x_d]$ be a family of polynomials such that $p_{i,j}\left( \mathbb{Z}^d \right) \subseteq \mathbb{Z}$ and $\{1\} \cup \{p_{i,j}\}$ is linearly independent over $\mathbb{Q}$ . Then the $\mathbb{Z}^d$ -sequence $\left( \prod_{j=1}^l{T_j^{p_{1,j}(n)}}x, \dots, \prod_{j=1}^l{T_j^{p_{k,j}(n)}}x \right)_{n \in \mathbb{Z}^d}$ is well-distributed in $X^k$ for every x in a co-meager set of full measure.