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]

References

1. Savchev, S.; Chen, F. (2005). "Kemnitz' conjecture revisited". Discrete Mathematics. 297 (1–3): 196–201. doi:10.1016/j.disc.2005.02.018.
2. Kemnitz, A. (1983). "On a lattice point problem". Ars Combinatoria. 16b: 151–160.
3. Erdős, P.; Ginzburg, A.; Ziv, A. (1961). "Theorem in additive number theory". Bull. Research Council Israel. 10F: 41–43.
4. Rónyai, L. (2000). "On a conjecture of Kemnitz". Combinatorica. 20 (4): 569–573. doi:10.1007/s004930070008.
5. Reiher, Ch. (2007). "On Kemnitz' conjecture concerning lattice-points in the plane". The Ramanujan Journal. 13 (1–3): 333–337. arXiv:1603.06161. doi:10.1007/s11139-006-0256-y.

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.