The number of configurations in lattice point counting I

When a strictly convex plane set S moves by translation, the set J of points of the integer lattice that lie in S changes. The number K of equivalence classes of sets J under lattice translations (configurations) is bounded in terms of the area of the Brunn-Minkowski difference set of S. If S satisfies the Triangle Condition, that no translate of S has three distinct lattice points in the boundary, thenK is asymptotically equal to the area of the difference set, with an error term like that in the corresponding lattice point problem. If S satisfies a Smoothness Condition but not the Triangle Condition, then we obtain a lower bound for K , but not of the right order of magnitude. The case when S is a circle was treated in our earlier paper by a more complicated method. The Triangle Condition was removed by considerations of norms of Gaussian integers, which are special to the circle. 2000 Mathematics Subject Classification: 11P21; 11B75.