I'm really struggling with an exercise in the book Introduction to Dynamical Systems written by Micheal Brin and Garret Stuck (exercise 2.2.6 (a)). Let $\alpha$ be an irrational number and consider the homeomorphism $$f_{\alpha}\colon\mathbb{R}^{2}/\mathbb{Z}^{2}\to\mathbb{R}^{2}/\mathbb{Z}^{2},\qquad f_{\alpha}((x,y)+\mathbb{Z}^{2}):=(x+\alpha,x+y)+\mathbb{Z}^{2}.$$ In the exercise you are asked to show that any non-empty open $f_{\alpha}$-invariant subset of the torus is dense in the torus. (The exercise also says "... i.e., that $f_{\alpha}$ is topologically transitive.", but I do not see why that is equivalent to what they ask.) I have no idea how to prove this.
I'm aware that this question has been asked on this platform more than once, for example, here. In this link someone answered using Fourier theory and ergodic theory, but this has not yet been treated in the course on dynamical systems I'm following. So I think that there must be an easier (or more basic) argument.
Any suggestions are greatly appreciated.