Difference sets in higher dimensions

Abstract

AbstractLet d ≥ 3 be a natural number. We show that for all finite, non-empty sets $A \subseteq \mathbb{R}^d$ that are not contained in a translate of a hyperplane, we have$$\begin{equation*} |A-A| \geq (2d-2)|A| - O_d(|A|^{1- \delta}),\end{equation*}$$where δ > 0 is an absolute constant only depending on d. This improves upon an earlier result of Freiman, Heppes and Uhrin, and makes progress towards a conjecture of Stanchescu.