Bounds on the number of vertices of sublattice-free lattice polygons
  In this paper we establish bounds on the number of vertices for a few classes of convex sublattice-free lattice polygons. The bounds are essential for proving the formula for the critical number of vertices of a lattice polygon that ensures the existence of a sublattice point in the polygon. To obtain the bounds, we use relations between the number of edges of lattice broken lines and the coordinates of their endpoints. 
Nikolai Bliznyakov and Stanislav Kondratyev
math.NT