TITLE: Accumulation points of $ \{x_n \in \mathbb{R}, n \in \mathbb{N} \ \ | \ x_n = n\sin(n) \}$?
QUESTION [14 upvotes]: A younger student asked me:
What are accumulation points of the following set?
$$ \{x_n \in \mathbb{R}, n \in \mathbb{N} \ \ | \ x_n = n\sin(n) \}$$
I really can't answer this question, could anyone help me?

REPLY [5 votes]: The irrationality measure $\mu(\pi)$ of $\pi$ is not known better than $2\le\mu(\pi)\le7.6063$, it seems. If we knew $\mu(\pi)>2$ then there are infinitely many fractions with $\left|\frac ab-\pi\right|<\frac1{b^\kappa}$ for some $\kappa>2$. Then $|\sin a\|<|a-b\pi|<\frac 1{b^{\kappa-1}}$ and so $|a\sin a|$ becomes arbitrarily small, making $0$ an accumulation point.
Unfortunately, I can only confirm that $\left|\frac ab-\pi\right|<\frac c{b^2}$ for infinitely many fractions, which results in infinitely many $n$ with  $|n\sin n|<c\pi$. So at least some accumulation point exists.