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 ) ⩽ 2 N −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 2 N , despite the numerical evidence.