Two inverse results

Abstract

Let A be a finite subset of a group G 0 with |A −1 A|≤2|A−2. We show that there are an element αA and a non-null proper subgroup H of G such that one of the following holds:

  • x −1 HyA −1 A, for all x,yA not both in

  • x Hy −1AA −1, for all x,yA not both in αH

where G is the subgroup generated by A −1 A.

Assuming that A −1 AG and that \(\left| {A^{ - 1} A} \right| < \tfrac{{5|A|}} {3} \), we show that there are a normal subgroup K of G and a subgroup H with KHA −1 A and 2|K|≥|H| such that

$A^{ - 1} AK = KA^{ - 1} A = A^{ - 1} Aand6|K| \geqslant |A^{ - 1} A| = 3|H|$

.

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

References

  1. [1]

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

    Google Scholar 

  2. [2]

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

    Google Scholar 

  3. [3]

    G. T. Diderrich: On Kneser’s addition theorem in groups, Proc. Amer. Math. Soc. 38 (1973), 443–451.

    MathSciNet  MATH  Google Scholar 

  4. [4]

    G. Freiman: Groups and the inverse problems of additive number theory, (in Russian), Number-theoretic studies in the Markov spectrum and in the structural theory of set addition (Russian), 175–183. Kalinin. Gos. Univ., Moscow, 1973.

    Google Scholar 

  5. [5]

    Y. O. Hamidoune: Sur les atomes d’un graphe orienté, C.R. Acad. Sc. Paris A 284 (1977), 1253–1256.

    MathSciNet  MATH  Google Scholar 

  6. [6]

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

    MathSciNet  MATH  Google Scholar 

  7. [7]

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

    MathSciNet  MATH  Article  Google Scholar 

  8. [8]

    Y. O. Hamidoune: On small subset product in a group, Structure Theory of setaddition, Astérisque 258 (1999), 281–308.

    MathSciNet  Google Scholar 

  9. [9]

    Y. O. Hamidoune: Some additive applications of the isopermetric approach, Annales de l’ Institut Fourier 58 (2008), 2007–2036.

    MathSciNet  MATH  Article  Google Scholar 

  10. [10]

    Y. O. Hamidoune: Topology of Cayley graphs applied to inverse additive problems, arXiv:1011.1797v1 [math.CO]

  11. [11]

    L. Husbands: Approximate groups in additive combinatorics: A review of methods and literature, Master’s dissertation, University of Bristol, September, 2009.

    Google Scholar 

  12. [12]

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

    MathSciNet  Article  Google Scholar 

  13. [13]

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

    MathSciNet  MATH  Article  Google Scholar 

  14. [14]

    J. E. Olson: On the symmetric difference of two sets in a group, European J. Combin. 18 (1984), 110–120.

    MathSciNet  MATH  Google Scholar 

  15. [15]

    J. E. Olson: On the sum of two sets in a group, J. Number Theory 18 (1984), 110–120.

    MathSciNet  MATH  Article  Google Scholar 

  16. [16]

    T. Tao: Open question: noncommutative Freiman theorem, http://terrytao.wordpress.com/2007/03/02/open-question-noncommutative-freiman-theorem

  17. [17]

    T. Tao: An elementary non-commutative Freiman theorem, http://terrytao.wordpress.com/2009/11/10/an-elementary-non-commutative-freiman-theorem.

  18. [18]

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

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yahya Ould Hamidoune.

Additional information

Yahya O. Hamidoune passed away on March 11, 2011. His friends and collaborators have helped in the final editorial work to honour his memory and have his latest results published.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hamidoune, Y.O. Two inverse results. Combinatorica 33, 217–230 (2013). https://doi.org/10.1007/s00493-013-2731-6

Download citation

Mathematics Subject Classification (2000)

  • 11B60
  • 11B34
  • 20D60