Lattice Points on Similar Figures and Conics

Let us say that a plane figure F satisfies Steinhaus’ condition if for any positive integer n, there exists a figure F _n similar to F which satisfies the condition |F_n?mathbb Z²|=n{|F_ncap{mathbb Z}^2|=n}. For example, the circular disc satisfies Steinhaus’ condition. We prove that every compact convex region in the plane mathbb R²{mathbb R^2} satisfies Steinhaus’ condition. As for plane curves, it is known that the circle satisfies Steinhaus’ condition. We consider Steinhaus’ condition for other conics, and present several results.