Motivated by a recent MSE question about the sequence of function $\cos(n!\pi x)$, I have read related several related questions:

• On the behaviour of $\sin(n!\pi x)$ when $x$ is irrational.
• Is there a limit of cos (n!)?

Is the existence of $\lim_{n\to\infty}\cos(n!\pi x)$ for given arbitrary irrational $x$ still an open problem? Are there any known related references/results for this question?