Difference sets in higher dimensions

Abstract

Let $d\ge 3$ 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 \[ |A-A| \geq (2d-2)|A| - O_d(|A|^{1- δ}),\] where $\delta >0$ 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.

14 pages

References in corpus (2)

• A generalization of the discrete version of Minkowski's fundamental theorem
• Sum of dilates in vector spaces