Archiv der Mathematik

, Volume 101, Issue 6, pp 501–512 | Cite as

Some exact values of the Harborth constant and its plus-minus weighted analogue

  • Luz E. Marchan
  • Oscar Ordaz
  • Dennys Ramos
  • Wolfgang A. Schmid


The Harborth constant of a finite abelian group is the smallest integer \({\ell}\) such that each subset of G of cardinality \({\ell}\) has a subset of cardinality equal to the exponent of the group whose elements sum to the neutral element of the group. The plus-minus weighted analogue of this constant is defined in the same way except that instead of considering the sum of all elements of the subset, one can choose to add either the element or its inverse. We determine these constants for certain groups, mainly groups that are the direct sum of a cyclic group and a group of order 2. Moreover, we contrast these results with existing results and conjectures on these problems.

Mathematics Subject Classification (1991)

11B30 11B75 20K01 


Finite abelian group Weighted subsum Zero-sum problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adhikari S.D. et al.: Contributions to zero-sum problems, Discrete Math. 306, 1–10 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    S. D. Adhikari and P. Rath, Davenport constant with weights and some related questions, Integers 6 (2006), A30, 6 pp.Google Scholar
  3. 3.
    Adhikari S.D., Grynkiewicz D.J., Sun Z.-W.: On weighted zero-sum sequences. Adv. in Appl. Math. 48, 506–527 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bhowmik G., Schlage-Puchta J.-Ch.: An improvement on Olson’s constant for \({\mathbb{Z}_p \oplus \mathbb{Z}_p}\). Acta Arith. 141, 311–319 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chintamani M. N. et al.: New upper bounds for the Davenport and for the Erdős–Ginzburg–Ziv constants. Arch. Math. (Basel) 98, 133–142 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Edel Y. et al.: Zero-sum problems in finite abelian groups and affine caps. Q. J. Math. 58, 159–186 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    P. Erdős, A. Ginzburg, and A. Ziv, A theorem in additive number theory, Bull. Res. Council Israel 10F (1961), 41–43.Google Scholar
  8. 8.
    Y. Fan, W. D. Gao and Q. Zhong, On the Erdős–Ginzburg–Ziv constant for finite abelian groups of high rank, J. Number Theory 131 (2011), 1864–1874.Google Scholar
  9. 9.
    Fan Y. et al.: Two zero-sum invariants on finite abelian groups. Europ. J. Comb. 34, 1331–1337 (2013)CrossRefGoogle Scholar
  10. 10.
    W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: a survey, Expo. Math. 24 (2006), 337–369.Google Scholar
  11. 11.
    W. D. Gao, A. Geroldinger, and W.A. Schmid, Inverse zero-sum problems, Acta Arith. 128 (2007), 245–279.Google Scholar
  12. 12.
    W. D. Gao and R. Thangadurai, A variant of Kemnitz conjecture, J. Combin. Theory Ser. A 107 (2004), 69–86.Google Scholar
  13. 13.
    A. Geroldinger Additive group theory and non-unique factorizations, in: Combinatorial number theory and additive group theory, A. Geroldinger and I. Ruzsa, eds., Advanced Courses in Mathematics - CRM Barcelona, Birkhäuser, 2009, pp. 1–86.Google Scholar
  14. 14.
    D. J. Grynkiewicz, Structural Additive Theory, Developments in Mathematics, Springer, 2013.Google Scholar
  15. 15.
    H. Harborth, Ein Extremalproblem für Gitterpunkte, J. Reine Angew. Math. 262/263 (1973), 356–360.Google Scholar
  16. 16.
    Kemnitz A.: On a lattice point problem. Ars Combin. 16B, 151–160 (1983)MathSciNetMATHGoogle Scholar
  17. 17.
    Ordaz O. et al.: On the Olson and the strong Davenport constants. J. Théor. Nombres Bordx. 23, 715–750 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Potechin A.: Maximal caps in AG(6,3). Des. Codes Cryptogr. 46, 243–259 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Luz E. Marchan
    • 1
  • Oscar Ordaz
    • 2
  • Dennys Ramos
    • 1
  • Wolfgang A. Schmid
    • 3
  1. 1.Departamento de Matemáticas, Decanato de Ciencias y TecnologíasUniversidad Centroccidental Lisandro AlvaradoBarquisimetoVenezuela
  2. 2.Escuela de Matemáticas y Laboratorio MoST, Centro ISYS, Facultad de CienciasUniversidad Central de VenezuelaCaracasVenezuela
  3. 3.Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS, UMR 7539, Université Paris 8VilletaneuseFrance

Personalised recommendations