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 $⟨a_j{⟩}_{j=1}^m$ is any finite sequence in $\mathbb{N},$ then there exist infinitely many integers $k\in \mathbb{Z}$ and a sequence $⟨x_n{⟩}_{n\in \mathbb{N}}$ in $\mathbb{Z}$ such that $k\cdot MT\left(⟨a_j{⟩}_{j=1}^m,⟨x_n{⟩}_n\right)\subseteq \{xy:(x,y)\in E-E\},$ where $MT\left(⟨a_j{⟩}_{j=1}^m,⟨x_n{⟩}_n\right)$ denotes the milliken-Taylor configuration generated by the sequences $⟨a_j{⟩}_{j=1}^m$ and $⟨x_n{⟩}_{n\in \mathbb{N}}$.

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