Rational polygons: Odd compression ratio and odd plane coverings

We show that for any rational polygon P in the plane, and any odd-sized collection of translates of P, the area of the set of points covered by an odd number of these translates is bounded away from 0 by a universal constant depending on P alone. 

The key ingredient of the proof is a construction of an odd cover of the plane by translates of P. That is, we establish a family of translates of P covering (almost) every point in the plane a uniformly bounded odd number of times. 

Joint work with Rom Pinchasi.