Kemnitz's conjecture

Template:Short description 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.Template:R

The exact formulation of this conjecture is as follows:

Let $n$ be a natural number and $S$ a set of $4n-3$ 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 KemnitzTemplate:R 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.Template:R In 2000, Lajos Rónyai proved a weakened form of Kemnitz's conjecture for sets with $4n-2$ lattice points.Template:R Then, in 2003, Christian Reiher proved the full conjecture using the Chevalley–Warning theorem.Template:R

Further reading

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