A mapping from $\mathbb{R}^1$ to a dense subset of the surface of torus in $\mathbb{R}^3$

Solution 1:

Let $f(s_0, t_0)$ be a point on $K$. Consider $g(s_0 + 2\pi n)$ for $n \in \mathbb N$. By Kronecker's approximation theorem, and since $\lambda$ is irrational, we can choose an integer $m$ and a value for $n$ so that:

$$ \left|\frac{t_0 - \lambda s_0}{4\pi} - n \frac{\lambda}{2} + m\right| < \epsilon $$

We have:

\begin{align} \left|\sin t_0 - \sin \lambda(s_0 + 2\pi n)\right| &\le 2 \left|\sin\frac{t_0 - \lambda s_0 - 2\pi n \lambda}{2}\right| \\ &= 2 \left|\sin2\pi\left(\frac{t_0 - \lambda s_0}{4\pi} - n \frac{\lambda}{2} + m\right)\right| \\ &\le 4\pi\left|\frac{t_0 - \lambda s_0}{4\pi} - n \frac{\lambda}{2} + m\right| < 4\pi\epsilon \end{align}

Similarly, we can show that for the same value of $n$: $$ \left|\cos t_0 - \cos \lambda(s_0 + 2\pi n)\right| < 4\pi\epsilon $$

It follows that $g(s_0 + 2\pi n)$ can get arbitrarily close to $f(s_0, t_0)$ as desired. Thus, the image of $g$ is a dense subset of $K$.