Hyper-Atoms Applied to the Critical Pair Theory

Abstract

The isoperimetric method is often useful for proving results regarding sumsets. Here, we introduce the notion of a hyper-atom into the method, which overcomes a previous weakness when dealing with atoms that are cosets. To show the utility of this new object, we give a new isoperimetric proof of the cornerstone of classical critical pair theory: The Kemperman Structure Theorem, proved in its so-called “dual” formulation.

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References

  1. [1]

    R. Aharoni and E. Berger: Menger’s Theorem for Infinite Graphs, Invent. Math.176 (2009), 1–62.

    MathSciNet  Article  Google Scholar 

  2. [2]

    E. Balandraud: Un nouveau point de vue isopérimetrique appliqué au théorème de Kneser, Ann. Inst. Fourier58 (2008), 915–943.

    MathSciNet  Article  Google Scholar 

  3. [3]

    D. Grynkiewicz: Quasi-periodic decompositions and the Kemperman’s structure theorem, European J. Combin.26 (2005), 559–575.

    MathSciNet  Article  Google Scholar 

  4. [4]

    D. Grynkiewicz: A step beyond Kemperman’s structure theorem, Mathematika55 (2009), 67–114.

    MathSciNet  Article  Google Scholar 

  5. [5]

    D. Grynkiewicz: Structural Additive Theory, Developments in Mathematics 30, Springer (2013).

  6. [6]

    Y. O. Hamidoune: Quelques problèmes de connexité dans les graphes orientés, J. Comb. Theory, Ser. B30 (1981), 1–10.

    Article  Google Scholar 

  7. [7]

    Y. O. Hamidoune: On the connectivity of Cayley digraphs, Europ. J. Combinatorics, 5 (1984), 309–312.

    MathSciNet  Article  Google Scholar 

  8. [8]

    Y. O. Hamidoune: Subsets with small sums in abelian groups I: The Vosper property, European J. Combin.18 (1997), 541–556.

    MathSciNet  Article  Google Scholar 

  9. [9]

    Y. O. Hamidoune: An isoperimetric method in additive theory, J. Algebra179 (1996), 622–630.

    MathSciNet  Article  Google Scholar 

  10. [10]

    Y. O. Hamidoune; Some results in Additive number Theory I: The critical pair Theory, Acta Arith.96 (2000), 97–119.

    MathSciNet  Article  Google Scholar 

  11. [11]

    Y. O. Hamidoune: Some additive applications of the isopermetric approach, Annales de l’ Institut Fourier58 (2008), 2007–2036.

    Article  Google Scholar 

  12. [12]

    Y. O. Hamidoune and A. Plagne: A new critical pair theorem applied to sum-free sets, Comment. Math. Helv.79 (2004), 183–207.

    MathSciNet  Article  Google Scholar 

  13. [13]

    J. H. B. Kemperman: On small sumsets in Abelian groups, Acta Math.103 (1960), 66–88.

    MathSciNet  Article  Google Scholar 

  14. [14]

    Y. O. Hamidoune, O. Serra and G. Zémor: One some subgroup chains related to Kneser’s Theorem, J. Théor. Nr. Bordx.20 (2008), 125–130.

    Article  Google Scholar 

  15. [15]

    R. A. Lee: Proving Kneser’s theorem for finite groups by another e-transform, Proc. Amer. Math. Soc.44 (1974), 255–258.

    MathSciNet  MATH  Google Scholar 

  16. [16]

    V. F. Lev: Critical pairs in abelian groups and Kemperman’s structure theorem, Int. J. Number Theory2 (2006), 379–396.

    MathSciNet  Article  Google Scholar 

  17. [17]

    M. B. Nathanson: Additive Number Theory. Inverse problems and the geometry of sumsets, Grad. Texts in Math. 165, Springer, 1996.

  18. [18]

    T. Tao and V. H. Vu: Additive Combinatorics, Cambridge Studies in Advanced Mathematics 105 (2006), Cambridge University Press.

  19. [19]

    G. Vosper: The critical pairs of subsets of a group of prime order, J. London Math. Soc.31 (1956), 200–205.

    MathSciNet  Article  Google Scholar 

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Acknowledgement

The author is grateful to an anonymous referee for many valuable comments on the first two drafts.

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Correspondence to Yahya O. Hamidoune.

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Yahya O. Hamidoune passed away on March 11, 2011 before the final revisions for the already submitted manuscript could be completed. Final revisions have been made post mortem by David J. Gynkiewicz, Department of Mathematical Sciences, University of Memphis, Memphis TN, 38152, USA

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Hamidoune, Y.O. Hyper-Atoms Applied to the Critical Pair Theory. Combinatorica 39, 1281–1315 (2019). https://doi.org/10.1007/s00493-019-2429-5

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