(This is the corrected version of the question I asked here: Ratio of maximal to minimal jump in the set of angle multiples.)
Let $S^1$ be the unit circle in the complex plain. Let $d:S^1\times S^1\to\mathbb{R}$ be the distance function given by the arc length. Let $\theta\in S^1$ be an element of infinite order, that is $\theta^n\neq 1$ for any $n\neq 0$. Define $J_N$ to be the set of those pairs $(i,j)$, such that $1\leq i,j\leq N$ and the smaller open arc between $\theta^i$ and $\theta^j$ does not contain any other $\theta^k$, $1\leq k\leq N$.
I would like to know what is the behavior of the ratio \begin{equation} \frac{\max_{(i,j)\in J_N} d(\theta^i,\theta^j)}{\min_{(i,j)\in J_N}d(\theta^i,\theta^j)} \end{equation}
when $N$ goes to $\infty$. I would like it to be bounded above and below, but any information would be useful.