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 and Carlos di Fiore.

The exact formulation of this conjecture is as follows:

Let ${\displaystyle n}$ be a natural number and ${\displaystyle S}$ a set of 4n − 3 lattice points in plane. Then there exists a subset ${\displaystyle S_{1}\subseteq S}$ with ${\displaystyle n}$ points such that the centroid of all points from ${\displaystyle S_{1}}$ 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.