Rotation of the torus $T^2$ by irrational numbers linearly dependent over $\mathbb Z$

Question

It is known that the rotation $x \to x + \alpha$ of $S^1 = \mathbb R / \mathbb Z$ with irrational $\alpha$ is ergodic and, in particular, $\alpha n$, $n = 1, 2,\dots$, are dense in $S^1$. In two dimensions, the corresponding result is that if $\alpha_1$ and $\alpha_2$ are irrational numbers, linearly independent over $\mathbb Z$, then the rotation $(x_1,x_2) \to (x_1+\alpha_1,x_2+\alpha_2)$ of $T^2 = \mathbb R^2 / \mathbb Z^2$ is also ergodic. In particular, the points $(n\alpha_1,n\alpha_2)$, $n = 1, 2,\dots$, are dense in $T^2$. My question is whether it is possible to find an explicit example of such irrational numbers $\alpha_1$ and $\alpha_2$, which are linearly dependent over $\mathbb Z$, and for which the distance from the set $\{(n\alpha_1,n\alpha_2), \; n = 1,2,\dots \} \subset T^2$ to zero is strictly positive?

Answer 1

No, that is not possible.

If $\alpha_1,\alpha_2$ are linearly dependent irrational numbers over $\mathbb{Z}$ then there exists another irrational number $\alpha'$, which is a $\mathbb{Z}$-linear combination of $\alpha_1$ and $\alpha_2$, such that your original set can be written as multiples of $(m_1\alpha',m_2\alpha')$ for some $m_1,m_2 \in \{1,2,\ldots\}$. That is: $$\{(n \alpha_1, n \alpha_2) \,\bigm|\, n=1,2,...\} = \{(nm_1\alpha',nm_2\alpha') \,\bigm| \, n=1,2,...\} $$ Furthermore, this subset is contained densely in $$C = \{(m_1 t, m_2 t) \,\bigm|\, t \in \mathbb{R}\} $$ which is a circle embedded in $T^2$ that contains zero. Therefore, the original set $A$ has points arbitrarily close to zero.