Difference sets in ${\mathbb{R}}^d$

Abstract

Let $d\ge 2$ be a natural number. We show that $$|A-A|\ge \left(2d-2+\frac{1}{d-1}\right)|A|-(2d^2-4d+3)$$ for any sufficiently large finite subset $A$ of ${\mathbb{R}}^d$ that is not contained in a translate of a hyperplane. By a construction of Stanchescu, this is best possible and thus resolves an old question first raised by Uhrin.

15 pages