Bounds on the number of vertices of sublattice-free lattice polygons

TL;DR

This paper establishes bounds on the vertices of certain lattice polygons without sublattice points, aiding in understanding the critical number of vertices needed for lattice point existence.

Contribution

It provides new bounds on vertices of sublattice-free lattice polygons, crucial for determining when such polygons contain lattice points.

Findings

Bounds on vertices for specific classes of lattice polygons

Relations between broken line edges and endpoint coordinates

Implications for critical vertex counts in lattice polygons

Abstract

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.