Kemnitz's conjecture

Short description: Every set of lattice points in the plane has a large subset with centroid 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.

The exact formulation of this conjecture is as follows:

Let [math]\displaystyle{ n }[/math] be a natural number and [math]\displaystyle{ S }[/math] a set of [math]\displaystyle{ 4n-3 }[/math] lattice points in plane. Then there exists a subset [math]\displaystyle{ S_1 \subseteq S }[/math] with [math]\displaystyle{ n }[/math] points such that the centroid of all points from [math]\displaystyle{ S_1 }[/math] 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 [math]\displaystyle{ 2n-1 }[/math] integers have a subset of size [math]\displaystyle{ n }[/math] whose average is an integer. In 2000, Lajos Rónyai proved a weakened form of Kemnitz's conjecture for sets with [math]\displaystyle{ 4n-2 }[/math] 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. doi:10.1016/j.jcta.2004.03.009.