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

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 Kemnitz^[2] 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.^[3] In 2000, Lajos Rónyai proved a weakened form of Kemnitz's conjecture for sets with $4n-2$ lattice points.^[4] Then, in 2003, Christian Reiher proved the full conjecture using the Chevalley–Warning theorem.^[5]

References

^ Savchev, S.; Chen, F. (2005). "Kemnitz' conjecture revisited". Discrete Mathematics. 297 (1–3): 196–201. doi:10.1016/j.disc.2005.02.018.
^ Kemnitz, A. (1983). "On a lattice point problem". Ars Combinatoria. 16b: 151–160.
^ Erdős, P.; Ginzburg, A.; Ziv, A. (1961). "Theorem in additive number theory". Bull. Research Council Israel. 10F: 41–43.
^ Rónyai, L. (2000). "On a conjecture of Kemnitz". Combinatorica. 20 (4): 569–573. doi:10.1007/s004930070008.
^ 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.

References

Further reading