# Dirichlet’s approximation theorem

Last summer I was attempting to show that $sin(\mathbb{N})$ was dense in $[-1,1]$. After more reflection I realised that this was an interesting corollary of Dirichlet’s approximation theorem which is stated as follows:

Let $\alpha \in \mathbb{R_+}$ and $n \in \mathbb{N}$. Dirichlet’s approximation theorem says that $\exists k, b \in \mathbb{N}, |k\alpha -b | < \frac{1}{n k}$.

Proof:

Each of the $n+1$ numbers $a_i = i \alpha -\left\lfloor i \alpha \right\rfloor, i \in [0,n]$ lies in $0 \leq a_i < 1$ so by the pigeon-hole principle there’s at least one semi-closed interval of the form $[\frac{r}{n},\frac{r+1}{n}), 0 \leq r < n$ which contains two of them.

It follows that there exists $m, j \in \mathbb{N}$ such that:

\begin{aligned} |(m-j)\alpha -(\left\lfloor m \alpha \right\rfloor) -\left\lfloor j \alpha \right\rfloor) | < \frac{1}{n} \end{aligned}

Now if we set $k= m-j$ and

\begin{aligned} b = \left\lfloor m \alpha \right\rfloor -\left\lfloor j \alpha \right\rfloor, i \in [0,n] \end{aligned}

we have the result we desire.

The above proof can be found anywhere on the internet but using this I managed to deduce that integer angles are dense in the unit circle. I also tried to show that $sin(\mathbb{P})$ is dense in $[-1,1]$ where $\mathbb{P}$ is the set of prime numbers but this appears to be a more difficult result.

An interesting further question I wondered about is the following: what’s the necessary and sufficient condition on $S \subset \mathbb{N}$ such that $sin(S)$ is dense in $[-1,1]$?