Integer chords and configurations of lattice points

The area within a closed convex plane curve  CC may be estimated by enlarging CC by a factor  RR, translating, counting the set  JJ of integer points inside, and scaling back to the original size. This estimate is accurate when CC  is three times continuously differentiable in a certain sense. The set  JJ is very sensitive to translations of the curve. We show that as RR  tends to infinity, the domains in which each set  JJ occurs tend to uniform distribution modulo the integer lattice; this was only known for the special case of the circle.