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Representation of group elements as subsequence sums. (English) Zbl 1143.20032

Summary: Let \(G\) be a finite (additive written) Abelian group of order \(n\). Let \(w_1,\dots,w_n\) be integers coprime to \(n\) such that \(w_1+w_2+\cdots+w_n\equiv 0\pmod n\). Let \(I\) be a set of cardinality \(2n-1\) and let \(\zeta=\{x_i: i\in I\}\) be a sequence of elements of \(G\). Suppose that for every subgroup \(H\) of \(G\) and every \(a\in G\), \(\zeta\) contains at most \(2n-\tfrac n{|H|}\) terms in \(a+H\). Then, for every \(y\in G\), there is a subsequence \(\{y_1,\dots,y_n\}\) of \(\zeta\) such that \(y=w_1y_1+\cdots+w_ny_n\).
Our result implies some known generalizations of the Erdős-Ginzburg-Ziv Theorem.

MSC:

20K01 Finite abelian groups
11B75 Other combinatorial number theory
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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[1] Bialostocki, A.; Dierker, P.; Grynkiewicz, D.; Lotspeich, M., On some developments of the erdős – ginzburg – ziv theorem II, Acta. arith., 110, 173-184, (2003) · Zbl 1069.11007
[2] Bialostocki, A.; Lotspeich, M., Some developments of the erdős – ginzburg – ziv theorem I, Colloq. math. soc. János bolyai, 60, 97-117, (1991) · Zbl 1042.11510
[3] Bollobás, B.; Leader, I., The number of \(k\)-sums modulo \(k\), J. number theory, 78, 27-35, (1999) · Zbl 0929.11008
[4] Brakemeier, W., Ein beitrag zur additiven zahlentheorie, (1973), Thesis Braunschweig
[5] Brakemeier, W., Eine anzahlformel von zahlen modulo \(n\), Monatsh. math., 85, 277-282, (1978) · Zbl 0395.10006
[6] Caro, Y., Zero-sum Ramsey problems, a survey, Discrete math., 152, 93-113, (1996) · Zbl 0856.05068
[7] Eggleton, R.B.; Erdős, P., Two combinatorial problems in group theory, Acta arith., 21, 111-116, (1972) · Zbl 0248.20068
[8] Erdős, P.; Ginzburg, A.; Ziv, A., Theorem in the additive number theory, Bull. res. council Israel, 10, 41-43, (1961)
[9] C. Flores, O. Ordaz, On sequences with zero sum in abelian group, Volume in homage to Dr. Rodolfo A. Ricabarra, pp. 99-106, Vol. Homenaje, 1, Univ. Nac. del Sur, Bahia Blanca, 1995 (in Spanish). · Zbl 0874.11028
[10] Flores, C.; Ordaz, O., On the erdős – ginzburg – ziv theorem, Discrete math., 152, 321-324, (1996) · Zbl 0857.05073
[11] Gallardo, L.; Grekos, G., On brakemeier variant of the Erdös-Ginzburg-Ziv problem, Tatra mt. math. publ., 20, 91-98, (2000) · Zbl 0999.11016
[12] Gao, W., Addition theorems for finite abelian groups, J. number theory, 53, 241-246, (1995) · Zbl 0836.11007
[13] Gao, W.; Geroldinger, A., On long minimal zero sequences in abelian groups, Period. math. hungar., 38, 179-211, (1999) · Zbl 0980.11014
[14] Gao, W.; Geroldinger, A., Zero-sum problems in finite abelian groups: a survey, Exposition math., 24, 337-369, (2006) · Zbl 1122.11013
[15] Gao, W.; Hamidoune, Y.O., Zero sums in abelian groups, Combin. probab. comput., 7, 261-263, (1998) · Zbl 1076.11501
[16] Grynkiewicz, D.J., On a conjecture of hamidoune for subsequence sums, Integers, 5, 2, A7, (2005), 11pp. (electronic)
[17] Grynkiewicz, D.J., On a partition analog of the cauchy – davenport theorem, Acta math. hungar., 107, 161-174, (2005) · Zbl 1102.11016
[18] Grynkiewicz, D.J., An extension of the erdös – ginzburg and Ziv theorem to hypergraphs, European J. combin., 26, 1154-1176, (2005) · Zbl 1107.11015
[19] Grynkiewicz, D.J., Sumsets, zero-sums and extremal combinatorics, ph.D. dissertation, (2006), California Institute of Technology Pasadena, California
[20] Grynkiewicz, D.J., A weighted version of the erdös – ginzburg and Ziv theorems, Combinatorica, 4, 445-453, (2006) · Zbl 1121.11018
[21] Hamidoune, Y.O., On weighted sequence sums, Combin. probab. comput., 4, 363-367, (1995) · Zbl 0848.20049
[22] Hamidoune, Y.O., On weighted sums in abelian groups, Discrete math., 162, 127-132, (1996) · Zbl 0872.11016
[23] Hamidoune, Y.O.; Ordaz, O.; Ortuño, A., On a combinatorial theorem of erdös – ginzburg and Ziv, Combin. probab. comput., 7, 403-412, (1998) · Zbl 1057.05507
[24] Hamidoune, Y.O., Subsequence sums, Combin. probab. comput., 12, 413-425, (2003) · Zbl 1049.11024
[25] Hamidoune, Y.O.; Quiroz, D., On subsequence weighted products, Combin. probab. comput., 14, 485-489, (2005) · Zbl 1095.20013
[26] Hennecart, F., La fonction de brakemeier dans le problème d’ erdös – ginzburg – ziv, Acta arith., 117, 35-50, (2005) · Zbl 1063.11008
[27] Hennecart, F., Restricted addition and some developments of the erdös – ginzburg – ziv theorem, Bull. London math. soc., 37, 481-490, (2005) · Zbl 1070.05078
[28] Kneser, M., Abschätzung der asymptotischen dichte von summenmengen, Math. Z., 58, 459-484, (1953) · Zbl 0051.28104
[29] Mann, H.B., Addition theorems, (1965), Wiley New York · Zbl 0189.29701
[30] Mann, H.B., Two addition theorems, J. combin. theory, 3, 233-235, (1967) · Zbl 0189.29701
[31] M.B. Nathanson, Additive number theory: inverse problems and the geometry of sumsets. Graduate Texts in Mathematics, vol. 165, Springer, New York, 1996. · Zbl 0859.11003
[32] Olson, J.E., On a combinatorial problem of erdős – ginzburg – ziv, J. number theory, 8, 52-57, (1976) · Zbl 0333.05009
[33] Olson, J.E., An addition theorem for finite abelian group, J. number theory, 9, 63-70, (1977) · Zbl 0351.20032
[34] Ordaz, O.; Quiroz, D., On zero-free sets, Divulgaciones matemática, 14, 1-10, (2006) · Zbl 1217.20033
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