Furstenberg--Sárközy theorem over number fields

Abstract

We introduce the notion of intersective polynomials having coefficients in the ring of integers ${\mathcal{O}}_K$ of a number field $K$, and define a notion of upper density of subsets of ${\mathcal{O}}_K$. We prove that given any intersective polynomial $p(x)$ over ${\mathcal{O}}_K$, every subset $A$ of ${\mathcal{O}}_K$ of positive upper density contains two distinct elements whose difference is equal to $p(x)$ for some element $x$ in ${\mathcal{O}}_K$. Moreover, we obtain a quantitative version of this result. The proof is motivated by an argument due to Lucier, and the Fourier-free proof of the Furstenberg--Sárközy theorem over the integers by Green, Tao and Ziegler.