For an irrational number $\alpha$, let $e(k\alpha):=\exp(2k\pi i\alpha)$. It was indicated in this thread that

$$\limsup_{n \to \infty} \prod_{k=1}^n |1-e(k\alpha)|=\infty$$

(actually a weaker result was shown and the author of that answer claimed the general thing)

Some further questions can be asked from the aspects of dynamics. Given an integer $l$, how frequently will this product lie above $l$.

Specifically, let

$$\overline{p}_l:=\limsup_{N\to \infty} \frac{\{n=1,\dots,N:\prod_{k=1}^n |1-e(k\alpha)|\ge l \}}{N}$$ and $$\underline{p}_l:=\liminf_{N\to \infty} \frac{\{n=1,\dots,N:\prod_{k=1}^n |1-e(k\alpha)|\ge l \}}{N}. $$

I have two questions:

(1) Is $\overline{p}_l= \underline{p}_l$? (If not, for what $l$?)

(2) What are the limits $\lim_{l \to \infty} \overline{p}_l$ and $\lim_{l \to \infty} \underline{p}_l$. (I specular that the limit is zero... then one could further ask about the rate of convergence).