What is the condition for ({$n\alpha$},{$n\beta$}) being dense in [0,1]$\times$[0,1]? (Here {a} represents the fraction part of a).
If the qustion changes to ({$n\alpha$},{$m\beta$}), then it is fairly easy: $\alpha$ and $\beta$ are irrationals. However, if the question is ({$n\alpha$},{$n\beta$}), it seems quite complex. For instance, if $\alpha$ and $\beta$ are both irrational but they are equal, then ({$n\alpha$},{$n\beta$}) lies on y=x, so it's not dense.
I guess the condition is that $\alpha$ and $\beta$ are irrational while their division is also irrational. But I've got no idea about how to prove\disaprove my conjecture.
