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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 {\it P. Erdős, A. Ginzburg} and {\it A. Ziv} [Bull. Res. Council Israel 10F, 41--43 (1961; Zbl 0063.00009)] that $f(n,1)=2n-1$ and conjectured by {\it 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.

11B50Sequences (mod $m$)
11B75Combinatorial number theory
Full Text: DOI
[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) · Zbl 0539.05008
[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) · Zbl 0329.12001