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