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] ?

One thought on “Dirichlet’s approximation theorem

  1. Pingback: Constructive proof of Kronecker’s density theorem – Kepler Lounge

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