A Uniform Product-Difference Theorem for Dense Subsets of ${\mathbb{Z}}^2$

Abstract

We establish a uniform product-difference theorem for dense subsets of ${\mathbb{Z}}^2$, which gives an affirmative answer to Problem~2 of Fish and, as consequences, to both parts of his Problem~1. More precisely, we prove that for every $\delta >0$ there exists an integer $K(\delta )\ge 1$ such that every set $E\subseteq {\mathbb{Z}}^2$ with upper Banach density $d^{\star }(E)\ge \delta$ satisfies \[ K(δ)\mathbb Z \subseteq \{ab:(a,b)\in E-E\}. \] As consequences, we obtain affirmative answers to both parts of Fish's Problem~1: for positive-density sets $E_1,E_2\subseteq \mathbb{Z}$ and $E\subseteq {\mathbb{Z}}^2$, respectively, the sets \[ (E_1-E_1)^2-(E_2-E_2)^2 \quad\text{and}\quad \{x^2-y^2:(x,y)\in E-E\} \] contain nontrivial ideals of $\mathbb{Z}$, with generators depending only on the corresponding density thresholds. In particular, the latter result also settles a conjecture of Davies concerning differences of the indefinite quadratic form $x^2-y^2$ in dense subsets of ${\mathbb{Z}}^2$.

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