torus filling curve

Question

I'm trying solve this problem but I didn't many ideas how to do it. So, if someone can give me a hint or the step of a solution I would greatly appreciate it. This is the problem:

"Let $\gamma:\mathbb{R}\rightarrow\mathbb{T^2}$ be the following curve: $$\gamma(t)=(e^{2\pi it},e^{2\pi ict})\qquad \textrm{where $c$ is any irrational number}$$ Show that the image set $\gamma(\mathbb{R})$ is dense in $\mathbb{T^2}$.

I know that $\mathbb{T^2}=\mathbb{S^1}\times\mathbb{S^1}\subset\mathbb{C^2}$, $\gamma$ is a immersion (this is clear because $\gamma'(t)$ never vanishes). I know also that $\gamma$ is injective.

Thanks!

Answer 1

First try to prove the following fact (or solve the problem and prove this later):

If $a$ is irrational, then the set $\{e^{2\pi ina}\}_{n\in \mathbb{Z}}$ is dense in $S^1$.

Now pick a point $(e^{2\pi is},e^{2\pi it}) \in \mathbb{T}^2$. Finding a sequence in $\gamma(\mathbb{R})$ that converges to this point now amounts to using this fact and a little algebra.

I hope that this gives you enough to get started.