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.^[1]

The exact formulation of this conjecture is as follows:

Let ${\displaystyle n}$ be a natural number and ${\displaystyle S}$ a set of ${\displaystyle 4n-3}$ 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^[2] as a generalization of the Erdős–Ginzburg–Ziv theorem, an analogous one-dimensional result stating that every ${\displaystyle 2n-1}$ integers have a subset of size ${\displaystyle n}$ whose average is an integer.^[3] In 2000, Lajos Rónyai proved a weakened form of Kemnitz's conjecture for sets with ${\displaystyle 4n-2}$ lattice points.^[4] Then, in 2003, Christian Reiher proved the full conjecture using the Chevalley–Warning theorem.^[5]