  Let P be a polygon with rational vertices in the plane. We show that for any
finite odd-sized collection of translates of P, the area of the set of points
lying in an odd number of these translates is bounded away from 0 by a 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 F of
translates of P covering (almost) every point in the plane a uniformly bounded
odd number of times.
