A Lissajous curve, or a Bowditch curve, is given by the parametric equations
$x(t)=Asin(ω_{x}t + \phi)$
$y(t)=Bsin(ω_{y}t+δ)$,
Now if $\frac{\omega_{x}}{\omega_{y}}$ is irrational, and $\phi$ and $\delta$ are fixed, then the set $\mathcal{L} = (x(t), y(t) | -\infty < t < \infty)$ is supposedly dense in the rectangle $R =[-A,A]$x$[-B,B]$
I have seen:
Show that a Lissajous curve has incommesurate frequencies iff it isdense in a rectangle
But a completed proof was not given. This problem is frequently mentioned in the literature, but a reference to a full proof is never given. Does anyone have such a reference, or could someone provide a proof here?