English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We demonstrate that the phenomenon of popular differences (aka the phenomenonof large intersections) holds for natural families of polynomial patterns inrings of integers of number fields. If $K$ is a number field with ring ofintegers $\mathcal{O}_K$ and $E \subseteq \mathcal{O}_K$ has positive upperBanach density $d^*(E) = \delta > 0$, we show, inter alia:  1. If $p(x) \in K[x]$ is an intersective $\mathcal{O}_K$-valued polynomialand $r, s \in \mathcal{O}_K$ 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 familyof 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 andKra on polynomial configurations in $\mathbb{Z}$ and build upon recent work ofthe authors and Best on linear patterns in general abelian groups. The abovecombinatorial results follow from multiple recurrence results in ergodictheory, which require a sharpening of existing tools for handling polynomialmultiple ergodic averages. A key advancement made in this paper is a new resulton the equidistribution of polynomial orbits in nilmanifolds, which can be seenas a far-reaching generalization of Weyl's equidistribution theorem.

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