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 .
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 ?