That's an interesting number set. The set of all points that can be produced from a straight-edge and the grid of integers.
I'm pretty sure this is just the grid of rational numbers. Interestingly, this is a subset of the 'constructible' numbers. That is, the numbers that can be produced from a compass and straight-edge (without even needing a grid).
It’s not so surprising, since once you have a compass, straightedge, and a line segment you call “unit length”, you can construct all of the integer valued points in the plane.
So he has to place points with integer coordinates on the graph paper, and draw lines between them.