Kemnitz's conjecture explained

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 4 n -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

2 n -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 4 n -2 lattice points. Then, in 2003, Christian Reiher proved the full conjecture using the Chevalley–Warning theorem.

Further reading

• Gao . W. D. . Thangadurai . R. . 2004 . A variant of Kemnitz Conjecture . Journal of Combinatorial Theory. Series A . 107 . 1 . 69–86 . 10.1016/j.jcta.2004.03.009 .