New proofs of Plünnecke-type estimates for product sets in groups

Abstract

We present a new method to bound the cardinality of product sets in groups and give three applications. A new and unexpectedly short proof of the Plünnecke-Ruzsa sumset inequalities for commutative groups. A new proof of a theorem of Tao on triple products, which generalises these inequalities when no assumption on commutativity is made. A further generalisation of the Plünnecke-Ruzsa inequalities in general groups.

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

References

  1. [1]

    W. Gowers: A new way of proving sumset estimates, Blogpost, available online at http://gowers.wordpress.com/2011/02/10/a-new-way-of-proving-sumset-estimates/.

  2. [2]

    A. Granville: An introduction to additive combinatorics, In: Additive Combinatorics, CRM Proceedings & Lecture Notes (New York, 2007), A. Granville, M. Nathanson, and J. Solymosi, Eds., American Mathamatical Society, 1–27.

  3. [3]

    H. Helfgott: Growth and generation in SL 2(ℤ/pℤ), Ann. of Math. 167 (2008), 601–623.

    MathSciNet  MATH  Article  Google Scholar 

  4. [4]

    J. Malouf: On a theorem of Plünnecke concerning the sum of a basis and a set of positive density, J. Number Theory 54 (1995), 12–22.

    MathSciNet  MATH  Article  Google Scholar 

  5. [5]

    G. Petridis: Upper bounds on the cardinality of higher sumsets, Preprint, available online at arXiv:1101.5001v2, submitted to Acta Arith.

  6. [6]

    H. Plünnecke: Eigenschaften und Abschätzungen von Wirkungsfunktionen, Berichte der Gasellschaft für Mathematik und Datenverarbeitung, Bonn, 1969.

    Google Scholar 

  7. [7]

    H. Plünnecke: Eine zahlentheoretische anwendung der graphtheorie, J. Reine Angew. Math. 243 (1970), 171–183.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    I. Ruzsa: On the cardinality of A+A and A-A, In: Combinatorics (Keszthely 1976) Coll. Math. Soc. J. Bolyai, vol 18 (1978), A. Hajnal and V. Sós, eds., North-Holland-Bolyai Társulat, 933–938.

  9. [9]

    I. Ruzsa: An application of graph theory to additive number theory, Scientia, Ser. A 3 (1989), 97–109.

    MathSciNet  MATH  Google Scholar 

  10. [10]

    I. Ruzsa: Addendum to: An application of graph theory to additive number theory, Scientia, Ser. A 4 (1990/1991), 93–94.

    Google Scholar 

  11. [11]

    I. Ruzsa: An analogue of Freiman’s theorem in groups, Astérisque 258 (1999), 323–326.

    MathSciNet  Google Scholar 

  12. [12]

    I. Ruzsa: Cardinality questions about sumsets, In: Additive Combinatorics, CRM Proceedings & Lecture Notes (New York, 2007), A. Granville, M. Nathanson, and J. Solymosi, Eds., American Mathamatical Society, 195–205.

  13. [13]

    I. Ruzsa: Sumsets and structure, In: Combinatorial Number Theory and Additive Group Theory, Springer, New York, 2009.

    Google Scholar 

  14. [14]

    I. Ruzsa: Towards a noncommutative Plünnecke-type inequality, In: An Irregular Mind Szemeredi is 70, Bolyai Society Mathematical Studies, Vol. 21 (New York, 2010), I. Bárány and J. Solymosi, Eds., Springer, 591–605.

  15. [15]

    T. Tao: Product set estimates for non-commutative groups, Combinatorica, 28 (2008), 547–594.

    MathSciNet  MATH  Article  Google Scholar 

  16. [16]

    T. Tao, and V. Vu: Additive Combinatorics, Cambridge University Press, Cambridge, 2006.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Giorgis Petridis.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Petridis, G. New proofs of Plünnecke-type estimates for product sets in groups. Combinatorica 32, 721–733 (2012). https://doi.org/10.1007/s00493-012-2818-5

Download citation

Mathematics Subject Classification (2000)

  • 11P70