Lattice angles of lattice polygons

TL;DR

This paper extends the classical angle sum theorem to lattice polygons, providing a complete solution for all n-gons using recent advances in geometry of continued fractions.

Contribution

It offers the first complete characterization of lattice angles for polygons with any number of sides, building on previous partial results for triangles.

Findings

Derived a formula for lattice angles of n-gons for all n

Connected lattice angle conditions with continued fractions

Resolved previous open problem for polygons with n > 3

Abstract

This paper is dedicated to a lattice analog to the classical ``sum of interior angles of a polygon theorem''. In 2008, the first formula expressing conditions on the geometric continued fractions for lattice angles of triangles was derived, while the cases of $n$-gons for $n > 3$ remained unresolved. In this paper, we provide the complete solution for all integer $n$. The main results are based on recent advances in geometry of continued fractions.