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 and Carlos di Fiore.
The exact formulation of this conjecture is as follows:
Let n {\displaystyle n} n be a natural number and S {\displaystyle S} S a set of 4n − 3 lattice points in plane. Then there exists a subset S 1 ⊆ S {\displaystyle S_{1}\subseteq S} S_{1}\subseteq S with n {\displaystyle n} n points such that the centroid of all points from S 1 {\displaystyle S_{1}} 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 2n − 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 4n − 2 lattice points. Then, in 2003, Christian Reiher proved the full conjecture using the Chevalley–Warning theorem.
References
Erdős, P.; Ginzburg, A.; Ziv, A. (1961). "Theorem in additive number theory". Bull. Research Council Israel. 10F: 41–43.
Kemnitz, A. (1983). "On a lattice point problem". Ars Combinatoria. 16b: 151–160.
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: 333–337. arXiv:1603.06161. doi:10.1007/s11139-006-0256-y.
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.
Savchev, S.; Chen, F. (2005). "Kemnitz' conjecture revisited". Discrete Mathematics. 297 (1–3): 196–201. doi:10.1016/j.disc.2005.02.018.
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