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

TL;DR

This paper proves that large intersections of polynomial patterns occur in rings of integers of number fields with positive density, extending classical results to a broader algebraic setting using advanced ergodic theory and equidistribution techniques.

Contribution

It introduces new results on polynomial multiple recurrence and equidistribution in rings of integers of number fields, generalizing prior work from integers to algebraic number rings.

Findings

Large intersections of polynomial patterns are syndetic in rings of integers.

Generalization of polynomial recurrence results to algebraic number rings.

New equidistribution results for polynomial orbits in nilmanifolds.

Abstract

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.