Kemnitz's conjecture

In additive number theory, Kemnitz's conjecture states that every set of lattice points in the plane has a large subset whose centroid is also a lattice point. It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student.

The exact formulation of this conjecture is as follows:


 * Let $$n$$ be a natural number and $$S$$ a set of $$4n-3$$ lattice points in plane. Then there exists a subset $$S_1 \subseteq S$$ with $$n$$ points such that the centroid of all points from $$S_1$$ is also a lattice point.

Kemnitz's conjecture was formulated in 1983 by Arnfried Kemnitz as a generalization of the Erdős–Ginzburg–Ziv theorem, an analogous one-dimensional result stating that every $$2n-1$$ integers have a subset of size $$n$$ whose average is an integer. In 2000, Lajos Rónyai proved a weakened form of Kemnitz's conjecture for sets with $$4n-2$$ lattice points. Then, in 2003, Christian Reiher proved the full conjecture using the Chevalley–Warning theorem.