Furstenberg--S\'{a}rk\"{o}zy theorem over number fields

Abstract

We introduce the notion of intersective polynomials having coefficients in the ring of integers $\mathscr{O}_K$ of a number field $K$, and define a notion of upper density of subsets of $\mathscr{O}_K$. We prove that given any intersective polynomial $p(x)$ over $\mathscr{O}_K$, every subset $A$ of $\mathscr{O}_K$ of positive upper density contains two distinct elements whose difference is equal to $p(x)$ for some element $x$ in $\mathscr{O}_K$. 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--S\'{a}rk\"{o}zy theorem over the integers by Green, Tao and Ziegler.