On the critical pair theory in abelian groups: Beyond Chowla’s Theorem


We obtain critical pair theorems for subsets S and T of an abelian group such that |S + T| ≤ |S| + |T|. We generalize some results of Chowla, Vosper, Kemperman and a more recent result due to Rødseth and one of the authors.

This is a preview of subscription content, access via your institution.


  1. [1]

    N. Alon, M. B. Nathanson and I. Z. Ruzsa: The polynomial method and restricted sums of congruence classes, J. Number Theory 56 (1996), 404–417.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    A. L. Cauchy: Recherches sur les nombres, J. Ecole Polytechnique 9 (1813), 99–116.

    Google Scholar 

  3. [3]

    S. Chowla: A theorem on the addition of residue classes: applications to the number Γ(k) in Waring’s problem, Proc. Indian Acad. Sci. 2 (1935), 242–243.

    MATH  Google Scholar 

  4. [4]

    H. Davenport: On the addition of residue classes, J. London Math. Soc. 10 (1935), 30–32.

    MATH  Article  Google Scholar 

  5. [5]

    J.-M. Deshouillers and G. A. Freiman: A step beyond Kneser’s Theorem for abelian finite groups, Proc. London Math. Soc. (3) 86(1) (2003), 1–28.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    B. Green and I. Z. Ruzsa: Sets with small sumset and rectification, Bull. London Math. Soc. 38 (2006), 43–52.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

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

    MATH  MathSciNet  Google Scholar 

  8. [8]

    Y. O. Hamidoune: An isoperimetric method in Additive Theory, J. Algebra 179 (1996), 622–630.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    Y. O. Hamidoune: Some results in additive number theory I: The critical pair theory, Acta Arithmetica 96 (2000), 97–119.

    MATH  MathSciNet  Google Scholar 

  10. [10]

    Y. O. Hamidoune and Ø. J. Røseth: An inverse theorem modulo p, Acta Arithmetica 92 (2000), 251–262.

    MATH  MathSciNet  Google Scholar 

  11. [11]

    Y. O. Hamidoune, A. S. Lladó and O. Serra: On subsets with a small product in torsion-free groups, Combinatorica 18(4) (1998), 529–540.

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

    Y. O. Hamidoune: On small subset product in a group. Structure Theory of setaddition; Astérisque no. 258(xiv–xv) (1999), 281–308.

  13. [13]

    Y. O. Hamidoune and A. Plagne: A new critical pair theorem applied to sum-free sets in abelian groups, Commentarii Mathematici Helvetici 79(1) (2004), 183–207.

    MATH  Article  MathSciNet  Google Scholar 

  14. [14]

    Y. O. Hamidoune, O. Serra and G. Zémor: On the critical pair theory in ℤ/pℤ, Acta Arithmetica 121 (2006), 99–115.

    MATH  MathSciNet  Article  Google Scholar 

  15. [15]

    G. A. Freiman: Foundations of a structural theory of set addition, Transl. Math. Monographs 37, Amer. Math. Soc., Providence, RI, 1973.

    Google Scholar 

  16. [16]

    Gy. Károlyi: An inverse theorem for the restricted set addition in abelian groups, J. Algebra 290 (2005), 557–593.

    MATH  Article  MathSciNet  Google Scholar 

  17. [17]

    Gy. Károlyi: Cauchy-Davenport theorem in group extensions, Enseign. Math. (2) 51(3–4) (2005), 239–254.

    MATH  MathSciNet  Google Scholar 

  18. [18]

    J. H. B. Kemperman: On small sumsets in an abelian group, Acta Math. 103 (1960), 63–88.

    MATH  Article  MathSciNet  Google Scholar 

  19. [19]

    M. Kneser: Summenmengen in lokalkompakten abelschen Gruppen, Math. Zeit. 66 (1956), 88–110.

    MATH  Article  MathSciNet  Google Scholar 

  20. [20]

    H. B. Mann: An addition theorem of abelian groups for sets of elements, Proc. Amer. Math. Soc. 4 (1953), 423.

    Article  MathSciNet  Google Scholar 

  21. [21]

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

  22. [22]

    O. Serra and G. Zémor: On a generalization of a theorem by Vosper, Integers Electr. J. Comb. Num. Th. 0(200), A10 (electronic). http://www.integersejcnt.org/vol0.html

  23. [23]

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

  24. [24]

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

    Article  MathSciNet  MATH  Google Scholar 

  25. [25]

    G. Zémor: A generalization to noncommutative groups of a theorem of Mann, Discrete Math. 126(1–3) (1994), 365–372.

    MATH  Article  MathSciNet  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Oriol Serra.

Additional information

Supported by the Spanish Research Council under project MTM2005-08990-C02-01 and by the Catalan Research Council under project 2005SGR00256.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hamidoune, Y.O., Serra, O. & Zémor, G. On the critical pair theory in abelian groups: Beyond Chowla’s Theorem. Combinatorica 28, 441 (2008). https://doi.org/10.1007/s00493-008-2262-8

Download citation

Mathematics Subject Classification (2000)

  • 11P70