TITLE: Are square tiled surfaces dense in the moduli space of translation surfaces?
QUESTION [7 upvotes]: I'm reading the survey "An introduction to Veech surfaces" by Pascal Hubert and Thomas Schmidt. 
At page 19 they state "In any fixed stratum, the set of square-tiled surfaces of that stratum is dense.". The reason should be that in the coordinates for the moduli space of translation surfaces given by the period map (integration of the 1-form on relative periods) the coordinates of the translation surfaces are exactly $\mathbb{Q}+i\mathbb{Q}$. So the assertion follows because of the density of $\mathbb{Q}$ in $\mathbb{R}$.
Well, it's hard for me to believe in Hubert and Schmidt's assertion.
Isn't it true that the square tiled surfaces are the "integer points" of the moduli space of translation surfaces? If I have a translation surface tiled by squares with the side of length one, isn't it true that the relative periods are contained in $\mathbb{Z}+i\mathbb{Z}$?
So my questions are:
Is Hubert and Schmidt's assertion wrong?
If so, which are the translation surfaces corresponding to rational points?
Thank you

REPLY [3 votes]: Is Hubert and Schmidt's assertion wrong?

No, they are correct.  They allow squares where the sidelength is not equal to one. 
If you like, we can say that the two translation surfaces $(X, \omega)$ and $(X, r\omega)$ (for $r$ positive and real) are "scalar multiples" of each other.  Then under any definitions, the scalar multiples of the square-tiled surfaces are dense in the space of translation surfaces.