Dense sets in $[-1,1]$

Question

For any fixed real number $c\in (0,\pi)$ with $\frac{c}{\pi}$ is irrational, the set $\{\sin(nc): n \in \mathbb N\}$ is dense in $[-1,1]$.

P.S. I know the proof for $c=1$, but how to proceed for other real numbers. Any hint would be helpful.

Answer 1

Let $\alpha$ and $\beta$ be any positive reals such that their ratio is irrational, and let $A=\{n\alpha+m\beta\in\mathbb{R}|n\in\mathbb{Z^+},m\in\mathbb{Z}\}$.

Claim: (For all distinct $(n,m),\ (n',m')\in\mathbb{Z}^2$, $n\alpha+m\beta\neq n'\alpha+m'\beta$.)
Suppose they are distinct and $n\alpha+m\beta= n'\alpha+m'\beta$. Then we must have both $n\neq n'$ and $m\neq m'$, and $\frac{\alpha}{\beta}=\frac{m'-m}{n-n'}\in\mathbb{Q}$, a contradiction.

For all $n\in\mathbb{Z^+}$, there is some $m\in\mathbb{Z}$ such that $n\alpha+m\beta\in[0,\beta)$, because $[0,\beta)$ is an interval of length $\beta$. Thus, for any $N\in\mathbb{Z^+}$, we must have some $n, n'\in\mathbb{Z^+}$ and $m, m'\in\mathbb{Z}$ such that $n<n'$ and $n\alpha+m\beta$, $n'\alpha+m'\beta$ are within the same $\frac{\beta}{N}$ length interval for some $[\frac{(k-1)\beta}{N},\frac{k\beta}{N})$ by the pigeonhole principle. Then, $(n'-n)\alpha+(m'-m)\beta$ is a nonzero element of $A$ within $\frac{\beta}{N}$ of $0$, so $A$ has a limit point at $0$.

Now, we reinterpret $\alpha=c$ and $\beta=2\pi$.
Fix any $x\in[-1,1]$ and any $\epsilon >0$. Now, using the facts that $\sin$ surjects onto $[-1,1]$ (continuity and IVT), is periodic, and is uniformly continuous, we may choose $y<0<z$ such that $\sin y=\sin z=x$ and let $\delta>0$ such that for all $a,b$ such that $|a-b|<\delta$, $|\sin a-\sin b|<\epsilon$. Then, fix any nonzero $d\in A$ of absolute value less than $\delta$. $A$ is closed over positive integer multiplication, so we may find some $d'=nc+m2\pi\in A$ within $\delta$ of either $y$ or $z$ (so that we do not have to specify if $d$ is positive or negative), so that $$|\sin(nc)-x|=|\sin(d')-\sin y|=|\sin (d')-\sin z|<\epsilon.$$Thus, $x$ is in the closure of the set $\{\sin (nc)|n\in\,\mathbb{N}\}$, so this set is dense in [-1,1].

Also, see the answer from Brian M. Scott here.