A subset of $[0,1]\times[0,1]$ containing at most one point from each horizontal and vertical section whose boundary is $[0,1]\times[0,1]$

Let $\alpha, \beta$ be two irrationals such that $\alpha/\beta$ is also irrational. What can one say about the set $$A=\{(n\alpha \mod 1,n\beta\mod 1):n\in \Bbb Z\}?$$

Edit: probably I should give some more hints:

1. The fact $A$ contains at most one point in each vertical/horizontal stripe follows by irrationality of $\alpha$ and $\beta$.
2. The set $\{(n\alpha,n\beta):n\in \Bbb Z\}$ is dense in the line $\{(t\alpha,t\beta):t\in \Bbb R\}$. Now wrap that line around the unit square $[0,1]\times [0,1]$, using irrationality of $\alpha/\beta$.

Edit 2: As Thomas pointed out in the comment, Statement 2. above is not correct unless $n\in \Bbb Q$. One then can modify the set to $$A'=\{(r\alpha\mod 1,r\beta\mod 1):\color{red}{r\in\Bbb Q}\}.$$ Note that the original $A$ still works, just not with the given argument.