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



Are square tiled surfaces dense in the moduli space of translation surfaces?

> 
> 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.


**EDIT**: There is some nonsense (due to me) in the comments below. Caveat emptor!


