Existence of sublattice points in lattice polygons

Abstract

We state the formula for the critical number of vertices of a convex lattice polygon that guarantees that the polygon contains at least one point of a given sublattice and give a partial proof of the formula. We show that the proof can be reduced to finding upper bounds on the number of vertices in certain classes of polygons. To obtain these bounds, we establish inequalities relating the number of edges of a broken line and the coordinates of its endpoints within a suitable class of broken lines.