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On Kemnitz’ conjecture concerning lattice-points in the plane. (English) Zbl 1126.11011
Let $f\left(n,k\right)$ be the minimal number $f$ such that each set of $f$ lattice-points in the $k$-dimensional Euclidean space contains a subset of cardinality $n$ whose centroid is a lattice-point as well. It was proved by P. Erdős, A. Ginzburg and A. Ziv [Bull. Res. Council Israel 10F, 41–43 (1961; Zbl 0063.00009)] that $f\left(n,1\right)=2n-1$ and conjectured by A. Kemnitz [Ars Comb. 16-B, 151–160 (1983; Zbl 0539.05008)] that $f\left(n,2\right)=4n-3$ who gave some partial results concerning this conjecture. In this note the author proves Kemnitz’ conjecture in general.

##### MSC:
 11B50 Sequences (mod $m$) 11B75 Combinatorial number theory
##### Keywords:
zero-sum-subsets; Kemnitz’ conjecture
##### References:
 [1] Alon, N., Dubiner, D.: A lattice point problem and additive number theory. Combinatorica 15, 301–309 (1995) · Zbl 0838.11020 · doi:10.1007/BF01299737 [2] Erdos, P., Ginzburg, A., Ziv, A.: Theorem in the additive number theory. Bull Research Council Israel 10F, 41–43 (1961) [3] Gao, W.: Note on a zero-sum problem. J. Combin. Theory, Series A 95, 387–389 (2001) · Zbl 0992.11027 · doi:10.1006/jcta.2001.3181 [4] Kemnitz, A.: On a lattice point problem. Ars Combin. 16b, 151–160 (1983) [5] Rónyai, L.: On a conjecture of Kemnitz. Combinatorica 20, 569–573 (2000) · Zbl 0963.11013 · doi:10.1007/s004930070008 [6] Schmidt, W.M.: Equations Over Finite Fields, An Elementary Approach. Springer Verlag, Lecture Notes in Math (1976)