  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.

Number Theory;Combinatorics