Furstenberg--Sárközy theorem over number fields
Dev Ranjan Pandey, Jyoti Prakash Saha
TL;DR
The paper extends the Furstenberg–Sárközy paradigm to number fields by introducing intersective polynomials over the ring of integers $\\mathcal{O}_K$ and a corresponding upper density on $S(N)$. It proves that for any intersective $q(x) \\in \\mathcal{O}_K[x]$ of degree $\\kappa \\ge 2$, any subset of $S(N)$ with positive density contains two distinct elements whose difference equals $q(\\xi)$ for some $\\xi \\in \\mathcal{O}_K$, and provides a quantitative bound on the density decay in terms of $\\log\\log N$ with exponent $\\varepsilon = 1/(2^{\\kappa-1}+1)$. The proof synthesizes Lucier’s degree-lowering strategy with a Fourier-free approach à la Green–Tao–Ziegler, constructing an auxiliary intersective polynomial $q_\\alpha$ to propagate density increments along linearized structures and iterating until a density surpasses 1 would occur, which yields the stated bound. This work broadens the scope of additive combinatorics techniques to arithmetic in number fields and highlights a robust method for obtaining quantitative density results without Fourier-analytic machinery in this setting.
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--Sárközy theorem over the integers by Green, Tao and Ziegler.