Every set of lattice points in the plane has a large subset whose centroid is also a lattice point
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 be a natural number and a set of lattice points in plane. Then there exists a subset with points such that the centroid of all points from 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 integers have a subset of size whose average is an integer.[3] In 2000, Lajos Rónyai proved a weakened form of Kemnitz's conjecture for sets with lattice points.[4] Then, in 2003, Christian Reiher proved the full conjecture using the Chevalley–Warning theorem.[5]
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