The problem:
A rectangle R can be tiled with smaller rectangles such that
- The sides of the smaller rectanges are parallel the sides of R.
- At least one side of each of the smaller rectangle is integral.
Show that at least one side of R is integral.
The proof:
Consider $\iint_{R} e^{2 \pi i (x+y)} dxdy$
This is zero, by adding up along each small rectangle. The result follows.
