On difference sets of dense subsets of $\mathbb{Z}^2$

Abstract

In this article, we study the structure of the difference set $E - E$ for subsets $E \subseteq \mathbb{Z}^2$ of positive upper Banach density. Fish asked in [Proc. Amer. Math. Soc. 146 (2018), 3449-3453] whether, for every such set $E$, there exists a nonzero integer $k$ such that $k \cdot \mathbb{Z} \subseteq \{\, xy : (x,y) \in E - E \,\}.$ Although this question remains open, we establish a relatively weaker form of this conjecture. Specifically, we prove that if $\langle a_j\rangle_{j=1}^m$ is any finite sequence in $\mathbb{N},$ then there exist infinitely many integers $k \in \mathbb{Z}$ and a sequence $\langle x_n \rangle_{n \in \mathbb{N}}$ in $\mathbb{Z}$ such that $k \cdot MT\left(\langle a_j \rangle_{j=1}^m, \langle x_n\rangle_{n}\right) \subseteq \{\, xy : (x,y) \in E - E \,\},$ where $MT\left(\langle a_j \rangle_{j=1}^m, \langle x_n\rangle_{n}\right)$ denotes the milliken-Taylor configuration generated by the sequences $\langle a_j\rangle_{j=1}^m$ and $\langle x_n \rangle_{n \in \mathbb{N}}$.