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The Ψ-Weighted Gao Theorem

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Structural Additive Theory

Part of the book series: Developments in Mathematics ((DEVM,volume 30))

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Abstract

The goal of this chapter is to present a fairly involved application of the DeVos-Goddyn-Mohar Theorem. Recall that the Davenport constant D(G) of an abelian group G is the minimal integer such that any sequence with |S|≥D(G) must contain a nontrivial zero-sum subsequence, while the Gao constant E(G) is the minimal integer such any sequence with |S|≥E(G) must contain a zero-sum subsequence of length |G|. We have already seen, as a simple consequence of the DeVos-Goddyn-Mohar or Partition Theorem, that

$$\mathsf{E}(G)=|G|+\mathsf{D}(G)-1 $$

for all finite abelian groups G. In this chapter, we will prove a much stronger weighted generalization of this result that also parallels some of the other exercises from previous chapters.

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References

  1. S.D. Adhikari, R. Balasubramanian, F. Pappalardi, P. Rath, Some zero-sum constants with weights. Proc. Indian Acad. Sci. Math. Sci. 118(2), 183–188 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. S.D. Adhikari, Y.G. Chen, Davenport constant with weights and some related questions II. J. Comb. Theory, Ser. A 115(1), 178–184 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. S.D. Adhikari, Y.G. Chen, J.B. Friedlander, S.V. Konyagin, F. Pappalardi, Contributions to zero-sum problems. Discrete Math. 306, 1–10 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. S.D. Adhikari, D.J. Grynkiewicz, Z.W. Sun, On weighted zero-sum sequences. Adv. Appl. Math. 48, 506–527 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. S.D. Adhikari, C. David, J. Jimenez, Generalization of some zero-sum theorems. Preprint

    Google Scholar 

  6. S.D. Adhikari, P. Rath, Davenport constant with weights and some related questions. Integers 6, A30 (2006) (electronic)

    MathSciNet  Google Scholar 

  7. J. Bass, Improving the Erdős-Ginzburg-Ziv Theorem for some non-abelian groups. J. Number Theory 126, 217–236 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Bollobás, I. Leader, The number of k-sums modulo k. J. Number Theory 78, 27–35 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Y. Caro, Zero-sum problems—a survey. Discrete Math. 152(1–3), 93–113 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Y. Edel, C. Elsholtz, A. Geroldinger, S. Kubertin, L. Rackham, Zero-sum problems in finite abelian groups and affine caps. Q. J. Math. 58, 159–186 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Y. Fan, W. Gao, Q. Zhong, On the Erdős-Ginzburg-Ziv constant of finite Abelian groups of high rank. J. Number Theory 131, 1864–1874 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. W.D. Gao, Addition theorems for finite abelian groups. J. Number Theory 53(2), 241–246 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. W. Gao, A combinatorial problem on finite abelian groups. J. Number Theory 58, 100–103 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. W.D. Gao, Y. Li, The Erdős-Ginzburg-Ziv Theorem for finite solvable groups. J. Pure Appl. Algebra 214, 898–909 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. W.D. Gao, Z. Lu, The Erdős-Ginzburg-Ziv Theorem for dihedral groups. J. Pure Appl. Algebra 214, 311–319 (2008)

    Article  MathSciNet  Google Scholar 

  16. W.D. Gao, J. Zhuang, Erdős-Ginzburg-Ziv Theorem for dihedral groups of large prime index. Eur. J. Comb. 26, 1053–1059 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Geroldinger, D.J. Grynkiewicz, On the structure of minimal zero-sum sequences with maximal cross number. J. Comb. Number Theory 1(2), 9–26 (2009)

    MathSciNet  Google Scholar 

  18. A. Geroldinger, F. Halter-Koch, Non-unique Factorizations: Algebraic, Combinatorial and Analytic Theory. Pure and Applied Mathematics: A Series of Monographs and Textbooks, vol. 278 (Chapman & Hall, an imprint of Taylor & Francis Group, Boca Raton, 2006)

    Book  Google Scholar 

  19. D.J. Grynkiewicz, E. Marchan, O. Ordaz, A weighted generalization of two theorems of Gao. Ramanujan J. 28(3), 323–340 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Y.O. Hamidoune, O. Ordaz, A. Ortuño, On a combinatorial theorem of Erdős, Ginzburg and Ziv. Comb. Probab. Comput. 7, 403–412 (1998)

    Article  MATH  Google Scholar 

  21. F. Luca, A generalization of a classical zero-sum problem. Discrete Math. 307, 1672–1678 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. H.B. Mann, Two addition theorems. J. Comb. Theory 3, 233–235 (1967)

    Article  MATH  Google Scholar 

  23. J.E. Olson, An addition theorem for finite abelian groups. J. Number Theory 9(1), 63–70 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  24. W.A. Schmid, J.J. Zhuang, On short zero-sum subsequences over p-groups. Ars Comb. 95, 343–352 (2010)

    MathSciNet  MATH  Google Scholar 

  25. R. Thangadurai, A variant of Davenport’s constant. Proc. Indian Acad. Sci. Math. Sci. 117(2), 147–158 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. X. Xia, Two generalized constants related to zero-sum problems for two special sets. Integers 7 (2007) (electronic)

    Google Scholar 

  27. P. Yuan, X. Zeng, Davenport constant with weights. Eur. J. Comb. 31(3), 677–680 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. P. Yuan, X. Zeng, Weighted Davenport’s constant and weighted EGZ Theorem. Discrete Math. 311, 1940–1947 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Graz, Austria

    David J. Grynkiewicz

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  1. David J. Grynkiewicz
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Grynkiewicz, D.J. (2013). The Ψ-Weighted Gao Theorem. In: Structural Additive Theory. Developments in Mathematics, vol 30. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00416-7_16

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