Let $\alpha$ be an irrational number, and $\left\{ \cdot \right\}$ denotes the fractional part function. We have focused on how $\left\{ n! \alpha \right\}$ distributes in this MO question. And now I am interested in the series $\sum_{n = 1}^\infty \left\{ n! \alpha \right\}$.

The above question mentions that, I quote, there is a subset of $\mathbb{R}$ of Hausdorff dimension $1$, such that the corresponding sequence is bounded away from $0$, which means the series $\sum_{n = 1}^\infty \left\{ n! \alpha \right\}$ must diverge.

Though we cannot tell the exact distribution of $\left\{ n! \alpha \right\}$ yet, it might be reasonable to suspect that the convergence of $\sum_{n = 1}^\infty \left\{ n! \alpha \right\}$ have been fully settled, just as the series $\sum_{n = 1}^\infty \sin( n! \pi \alpha )$.

Sadly, I exhaust my scope of search without finding a similar result. Therefore, my question is if we can completely determine when the series $\sum_{n = 1}^\infty \left\{ n! \alpha \right\}$ converges and if there is a reference to it.

Any advice is welcomed.