A superadditivity and submultiplicativity property for cardinalities of sumsets

Abstract

For finite sets of integers A 1,…,A n we study the cardinality of the n-fold sumset A 1+…+ A n compared to those of (n−1)-fold sumsets A 1+…+A i−1+A i+1+…+A n . We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition of elements is restricted to an addition graph between the sets.

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

References

  1. [1]

    N. Alon: Problems and results in extremal combinatorics I., Discrete Math.273 (2003), 31–53.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    P. Ballister and B. Bollobás: Projections, entropy and sumsets, Combinatorica, to appear.

  3. [3]

    B. Bollobás and A. Thomason: Projections of bodies and hereditary properties of hypergraphs, Bull. London Math. Soc.27 (1995), 417–424.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    F. R. K. Chung, R. L. Graham, P. Frankl and J. B. Shearer: Some intersection theorems for ordered sets and graphs, J. Combin. Theory Ser. A43 (1986), 23–37.

    MATH  Article  MathSciNet  Google Scholar 

  5. [5]

    T. M. Cover and J. A. Thomas: Elements of information theory, Wiley, New York - Chichester etc., 1991.

    MATH  Book  Google Scholar 

  6. [6]

    K. Gyarmati, F. Hennecart and I. Z. Ruzsa: Sums and differences of finite sets, Functiones et Approximatio37 (2007), 175–186.

    MATH  MathSciNet  Google Scholar 

  7. [7]

    T. S. Han: Nonnegative entropy measures of multivariate symmetric correlations, Inform. Contr.36 (1978), 133–156.

    MATH  Article  Google Scholar 

  8. [8]

    V. F. Lev: Structure theorem for multiple addition and the Frobenius problem, J. Number Theory58 (1996), 79–88.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    J. L. Malouf: On a theorem of Plünnecke concerning the sum of a basis and a set of positive density, J. Number Theory54 (1995), 12–22.

    MATH  Article  MathSciNet  Google Scholar 

  10. [10]

    A. Marcus and P. Tetali: Entropy relations on sums, preprint.

  11. [11]

    M. B. Nathanson: Additive number theory: Inverse problems and the geometry of sumsets, Springer, 1996.

  12. [12]

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

    MATH  MathSciNet  Google Scholar 

  13. [13]

    I. Z. Ruzsa: An application of graph theory to additive number theory, Scientia, Ser. A3 (1989), 97–109.

    MATH  MathSciNet  Google Scholar 

  14. [14]

    I. Z. Ruzsa: Addendum to: An application of graph theory to additive number theory; Scientia, Ser. A4 (1990/91), 93–94.

    Google Scholar 

  15. [15]

    I. Z. Ruzsa: Cardinality questions about sumsets, in: Additive Combinatorics (Providence, RI, USA), CRM Proceedings and Lecture Notes, vol. 43, American Math. Soc., 2007, pp. 195–205.

    MathSciNet  Google Scholar 

  16. [16]

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

    MATH  Book  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Katalin Gyarmati.

Additional information

Supported by Hungarian National Foundation for Scientific Research (OTKA), Grants No. T 43631, T 43623, T 49693.

Supported by Hungarian National Foundation for Scientific Research (OTKA), Grants No. PF-64061, T-049301, T-047276.

Supported by Hungarian National Foundation for Scientific Research (OTKA), Grants No. T 43623, T 42750, K 61908.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gyarmati, K., Matolcsi, M. & Ruzsa, I.Z. A superadditivity and submultiplicativity property for cardinalities of sumsets. Combinatorica 30, 163–174 (2010). https://doi.org/10.1007/s00493-010-2413-6

Download citation

Mathematics Subject Classification (2000)

  • 11B50
  • 11B75
  • 11P70