David Conlon:
Difference sets in higher dimensions

Let $d \geq 2$ be a natural number. We determine the minimum possible size of the difference set $A-A$ in terms of $|A|$ 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. If time permits, we will also discuss some recent related results.