Last summer I was attempting to show that was dense in . After more reflection I realised that this was an interesting corollary of Dirichlet’s approximation theorem which is stated as follows:

Let and . Dirichlet’s approximation theorem says that .

__Proof:__

Each of the numbers lies in so by the pigeon-hole principle there’s at least one semi-closed interval of the form which contains two of them.

It follows that there exists such that:

Now if we set and

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 is dense in where 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 such that is dense in ?