Set $$ \Omega=\left\{x \in \mathbb{R} \mid \sum_{n=1}^{\infty} \sin (n!\pi x)\text{ converges}\right\}. $$
Claim: $\Omega$ is dense in $\mathbb{R}$, and $\Omega \cap(\mathbb{R} \backslash \mathbb{Q}) \neq \varnothing$.
Remark:
It is obvious that the $\Omega$ is dense in $\mathbb{R}$, and as long as we take $x$ as a rational number, the general term of the series equals zero when $n$ is sufficiently large. Thus $\mathbb{Q}$ is a subset of $\Omega$, and $\Omega$ is dense in $\mathbb{R}$.
But what puzzles me is what irrational number can make this sequence converge? According to the method of irrational number approximation of Roth's theorem, I choose such numbers as algebraic numbers with degree greater than 2, and try to prove that it can make the series converge, but I have no effective method to prove it. Do you have an idea?
