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

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?

I don't see much connection between the linked question and mine.Your question seems to differ from that one in two ways: (1) you changed sine to cosine, and (2) you seem to be asking for an algorithm or decision procedure that determines convergence for a given $x$. What is the motivation for changing both of these things at the same time? Re #2, it's not clear to me what it would mean to have a decision procedure that took an "arbitrary irrational" as an input, since only countably many irrationals can even be described. This question seems to lack motivation and research effort. $\endgroup$ – Ben Crowell Dec 26 '16 at 19:204more comments