This is a qual problem。
Let $n$ points be iid uniformly distributed on the unit circle. Let $\Delta_n$ be the smallest distance between any two of these points. Show that $n^{\theta}\Delta_n\rightarrow 0$ in probability as $n\rightarrow \infty$ for all $0<\theta<2$.
I started by denoting $\Delta_n$ and it got related to the angle, which got really messy and didn't think that's the right way. So I 'm wondering whether I can find some hints or solutions here. Thanks.