Skip to main content
Log in

Some inverse problems in zero-sum theory

  • Published:
Proceedings - Mathematical Sciences Aims and scope Submit manuscript

Abstract

Finding the exact values and bounds of a weighted generalization of different classical zero-sum constants for different weight sets and for different groups prompted a lot of research and the study of the corresponding inverse problems is also very engrossing. In this paper, we take into account a few well-known results on weighted zero-sum constants and characterize the corresponding inverse problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Adhikari S D, David C and Urroz J J, Generalizations of some zero-sum theorems, Integers 8 (2008) A52

    MathSciNet  MATH  Google Scholar 

  5. Adhikari S D and Rath P, Zero-sum problems in combinatorial number theory, Ramanujan Math. Soc. Lect. Notes Ser., vol. 2, Ramanujan Math. Soc. Mysore (2006) pp. 1–14

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

    MathSciNet  MATH  Google Scholar 

  7. Adhikari S D and Hegde S, Zero-sum constants including weights, Proc. Indian Acad. Sci. (Math. Sci.) 131 (2021) 37

    Article  MATH  Google Scholar 

  8. Adhikari S D, Hegde S, Molla M I and Sarkar S, Inverse problems related to some weighted zero-sum constants for cyclic groups, Integers 22 (2022) A7

    MathSciNet  MATH  Google Scholar 

  9. Adhikari S D, Molla M I and Paul S, Extremal sequences for some weighted zero-sum constants for cyclic groups, Combinatorial and Additive Number Theory IV (Springer Proc. in Math. & Stat.) 347 (2021) 1–10

    MathSciNet  Google Scholar 

  10. Alford W R, Granville A and Pomerance C, There are infinitely many Carmichael numbers, Ann. Math. 139(2)(3) (1994) 703–722

  11. Bialostocki A and Dierker P, On the Erdős–Ginzburg–Ziv theorem and the Ramsey numbers for stars and matchings, Discrete Math. 110(1–3) (1992) 1–8

    Article  MathSciNet  MATH  Google Scholar 

  12. Caro Y, Zero-sum problems, a survey, Discrete Math. 152 (1996) 93–113

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Erdős P, Ginzburg A and Ziv A, Theorem in the additive number theory, Bull. Research Council Israel 10F (1961) 41–43

    MathSciNet  MATH  Google Scholar 

  15. Gao W D and Geroldinger A, Zero-sum problems in finite abelian groups: a survey, Expo. Math. 24 (2006) 337–369

    Article  MathSciNet  MATH  Google Scholar 

  16. Mann H B, Addition Theorems (1965) (New York: Wiley)

    MATH  Google Scholar 

  17. Olson J E, On a combinatorial problem of finite Abelian groups I, J. Number Theory 1 (1969) 8–10

    Article  MathSciNet  MATH  Google Scholar 

  18. Rogers K, A Combinatorial problem in Abelian groups, Proc. Cambridge Phil. Soc. 59 (1963) 559–562

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Xia X, Two generalized constants related to zero-sum problems for two special weights, Integer 7 (2007) A52

    MATH  Google Scholar 

  21. Xia X and Li Z, Some Davenport constants with weights and Adhikari & Rath’s conjecture, Ars Combin. 88 (2008) 83–95

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to sincerely thank his advisor, Prof. S D Adhikari, for proposing the problems and for significantly insightful suggestions. He thanks Shruti Hegde for helpful discussions. He would also like to acknowledge CSIR, Government of India, for a research fellowship. The author extends his thanks to the referee for going through the manuscript minutely and providing several suggestions to improve the presentation of the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Md Ibrahim Molla.

Additional information

Communicated by Sanoli Gun.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Molla, M.I. Some inverse problems in zero-sum theory. Proc Math Sci 132, 63 (2022). https://doi.org/10.1007/s12044-022-00712-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12044-022-00712-4

Keywords

2010 Mathematics Subject Classification

Navigation