The number of configurations in lattice point counting II

A convex plane set S is discretized by first mapping the centre of S to a point (u, v), preserving orientation, enlarging by a factor t to obtain the image S(t, u, v) and then taking the discrete set J(t, u, v) of integer points in S(t, u, v). Let N(t, u, v) be the size of the 'configuration' J(t, u, v). Let L(N) be the number of different configurations (up to equivalence by translation) of size N(t, u, v) = N and let M(N) be the number of different configurations with 1 < N(t, u, v) < N. Then L(N) < 2N-1, M(N) < N-2, with equality if S satisfies the Quadrangle Condition, that no image S(t, u, v) has four or more integer points on the boundary. For the circle, which does not satisfy the Quadrangle Condition, we expect that L(N) should be asymptotic to 2N, despite the numerical evidence.