Existence of sublattice points in lattice polygons

TL;DR

This paper presents a formula for the minimum number of vertices in convex lattice polygons needed to ensure they contain a point from a specified sublattice, along with partial proof and bounding techniques.

Contribution

It introduces a new formula for sublattice point existence in convex lattice polygons and reduces the proof to bounding vertices via inequalities involving broken lines.

Findings

Derived a formula for critical vertices guaranteeing sublattice points

Reduced proof to bounding vertices through inequalities

Established bounds on broken line edges and endpoints

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.