I would like to hear opinions regarding this problem of mine.
Given a set of points on an x-y plane, is it possible to construct an equal size grid (of rows and columns)? In other words, if I have a set of points on a transparency paper, I would like to superimpose it on a paper that I would print the grid on such that every point will be exactly on and only on an intersection.



Drawing an equal size grid based on a set of points
If you view $(x,y)$ as the complex number $x+iy$ then you are asking for all differences between pairs of points to be contained in a rank one free $\mathbb Z[i]$ submodule of the complex numbers. This only happens if the ratios between the numbers $x\_k-x\_0+i(y\_j-y\_0)$ are all in $\mathbb Q(i)$, and, for infinite sets, the ideals generated by the ratios holding the denominator fixed are bounded from below. 


There are many finite sets that do not satisfy this, such as $(0,0),(1,0),(0,\sqrt{2})$.


