Authors:
Alfred Geroldinger Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstrasse 36, 8010 Graz, Austria

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David J. Grynkiewicz Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstrasse 36, 8010 Graz, Austria

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Wolfgang A. Schmid CMLS, École polytechnique, 91128 Palaiseau cedex, France

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Abstract

For a finite abelian group G and a positive integer d, let sd(G) denote the smallest integer ∊ℕ0 such that every sequence S over G of length |S|≧ has a nonempty zero-sum subsequence T of length |T|≡0 mod d. We determine sd(G) for all d≧1 when G has rank at most two and, under mild conditions on d, also obtain precise values in the case of p-groups. In the same spirit, we obtain new upper bounds for the Erdős–Ginzburg–Ziv constant provided that, for the p-subgroups Gp of G, the Davenport constant D(Gp) is bounded above by 2exp  (Gp)−1. This generalizes former results for groups of rank two.

  • [1] Adhikari, S. D., Grynkiewicz, D. J. and Sun, Z.-W., On weighted zero-sum sequences, manuscript.

  • [2] Bhowmik, G., Schlage-Puchta, J.-C. 2007 Davenport’s constant for groups of the form ℤ3⊕ℤ3⊕ℤ3d Granville, A., Nathanson, M. B., Solymosi, J. (eds.) Additive Combinatorics CRM Proceedings and Lecture Notes 43 307326 American Mathematical Society.

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  • [3] Chi, R., Ding, S., Gao, W., Geroldinger, A., Schmid, W. A. 2005 On zero-sum subsequences of restricted size. IV Acta Math. Hungar. 107 337344 .

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  • [4] Delorme, C., Ordaz, O., Quiroz, D. 2001 Some remarks on Davenport constant Discrete Math. 237 119128 .

  • [5] Edel, Y., Elsholtz, C., Geroldinger, A., Kubertin, S., Rackham, L. 2007 Zero-sum problems in finite abelian groups and affine caps Quarterly. J. Math., Oxford II. Ser. 58 159186 .

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  • [6] Freeze, M., Schmid, W. A. 2010 Remarks on a generalization of the Davenport constant Discrete Math. 310 33733389 .

  • [7] Gao, W. 2001 On zero sum subsequences of restricted size III Ars Combin. 61 6572.

  • [8] Gao, W., Geroldinger, A. 2006 Zero-sum problems in finite abelian groups: a survey Expo. Math. 24 337369.

  • [9] Gao, W., Geroldinger, A. 2007 On the number of subsequences with given sum of sequences over finite abelian p-groups Rocky Mt. J. Math. 37 15411550 .

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  • [10] Gao, W., Geroldinger, A., Schmid, W. A. 2007 Inverse zero-sum problems Acta Arith. 128 245279 .

  • [11] Gao, W., Hamidoune, Y. ould, Wang, G. 2010 Distinct lengths modular zero-sum subsequences: a proof of Graham’s conjecture J. Number Theory 130 14251431 .

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  • [12] Gao, W., Peng, J. 2009 On the number of zero-sum subsequences of restricted size Integers 9 537554 . Paper A41.

  • [13] Gao, W., Thangadurai, R. 2006 On zero-sum sequences of prescribed length Aequationes Math. 72 201212 .

  • [14] Geroldinger, A. 1993 On a conjecture of Kleitman and Lemke J. Number Theory 44 6065 .

  • [15] Geroldinger, A. 2009 Additive group theory and non-unique factorizations Geroldinger, A., Ruzsa, I. (eds.) Combinatorial Number Theory and Additive Group Theory Advanced Courses in Mathematics CRM Barcelona 186 . Birkhäuser.

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  • [16] Geroldinger, A., Grynkiewicz, D. J. 2009 On the structure of minimal zero-sum sequences with maximal cross number J. Combinatorics and Number Theory 1 926.

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  • [17] Geroldinger, A., Grynkiewicz, D. J. and Schmid, W. A., The catenary degree of Krull monoids I, J. Théor. Nombres Bordeaux, to appear.

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  • [18] Geroldinger, A., Halter-Koch, F. 2006 Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory Pure and Applied Mathematics 278 . Chapman & Hall/CRC.

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  • [19] Geroldinger, A., Liebmann, M. and Philipp, A., On the Davenport constant and on the structure of extremal sequences, Period. Math. Hungar., to appear.

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  • [20] Geroldinger, A., Schneider, R. 1992 On Davenport’s constant J. Comb. Theory, Ser. A 61 147152 .

  • [21] Girard, B., On the existence of distinct lengths zero-sum subsequences, Rocky Mt. J. Math., to appear.

  • [22] Grynkiewicz, D. J., Note on a conjecture of Graham, manuscript.

  • [23] Grynkiewicz, D. J., On a conjecture of Hamidoune for subsequence sums, Integers, 5 (2005), Paper A07, 11p.

  • [24] Kannan, S. S. and Pattanayak, S. K., Projective normality of finite group quotients and EGZ theorem, manuscript.

  • [25] Kannan, S. S., Pattanayak, S. K., Sardar, P. 2009 Projective normality of finite group quotients Proc. Am. Math. Soc. 137 863867 .

  • [26] Kubertin, S. 2005 Zero-sums of length kq in Acta Arith. 116 145152 .

  • [27] Reiher, C. 2007 On Kemnitz’ conjecture concerning lattice points in the plane Ramanujan J. 13 333337 .

  • [28] Schmid, W. A., The inverse problem associated to the Davenport constant for C2C2C2n, and applications to the arithmetical characterization of class groups, manuscript.

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  • [29] Schmid, W. A., On zero-sum subsequences in finite abelian groups, Integers, 1 (2001), Paper A01, 8p.

  • [30] Schmid, W. A., Zhuang, J. J. 2010 On short zero-sum subsequences over p-groups Ars Comb. 95 343352.

  • [31] Yuan, P., Guan, H. and Zeng, X., Normal sequences over finite abelian groups, J. Comb. Theory, Ser. A, to appear.

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Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation
1950
Volumes
per Year
3
Issues
per Year
6
Founder Magyar Tudományos Akadémia  
Founder's
Address
H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

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