I'm working on the following question and would like some hints or solutions
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 $n^\theta \Delta_n\to 0$ in probability as $n\to \infty$, for all $0<\theta<2$. HINT: Divide the circle into small arcs and find the probability that at least one arc contains 2 or more points
So I tried following the hint, and I considered dividing up the circle into $n-1$ pieces that would give with probability 1, that two are in the same section. However, $n^\theta/n-1$ does not go to zero. The other things I tried were $n^2$ pieces and $n$ pieces, but the probability calculations are rather messy for these and I'd be dealing with factorials, which does not seem like it would go well with this problem.