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On Kemnitz’ conjecture concerning lattice-points in the plane. (English) Zbl 1126.11011
Let f(n,k) 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(n,1)=2n-1 and conjectured by A. Kemnitz [Ars Comb. 16-B, 151–160 (1983; Zbl 0539.05008)] that f(n,2)=4n-3 who gave some partial results concerning this conjecture. In this note the author proves Kemnitz’ conjecture in general.

MSC:
11B50Sequences (mod m)
11B75Combinatorial number theory
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)