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Portugaliae Mathematica

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Volume 75, Issue 3/4, 2018, pp. 205–248
DOI: 10.4171/PM/2017

Published online: 2019-06-06

An extremal property of lattice polygons

Nikolai Bliznyakov[1] and Stanislav Kondratyev[2]

(1) Voronezh State University, Russian Federation
(2) Universidade de Coimbra, Portugal

We find the critical number of vertices of a convex lattice polygon that guarantees that the polygon contains at least one point of a given square sublattice. As a tool, we prove Diophantine inequalities relating the number of edges of a broken line and the coordinates of its endpoints.

Keywords: Integral polygons, lattice-free polygons, broken lines

Bliznyakov Nikolai, Kondratyev Stanislav: An extremal property of lattice polygons. Port. Math. 75 (2018), 205-248. doi: 10.4171/PM/2017