Israel Journal of Mathematics

, Volume 139, Issue 1, pp 349–359 | Cite as

The Erdős-Heilbronn problem in Abelian groups



Solving a problem of Erdős and Heilbronn, in 1994 Dias da Silva and Hamidoune proved that ifA is a set ofk residues modulo a primep,p≥2k−3, then the number of different elements of ℤ/pℤ that can be written in the forma+a′ wherea, a′ ∈A,aa′, is at least 2k−3. Here we extend this result to arbitrary Abelian groups in which the order of any nonzero element is at least 2k−3.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N. Alon,Combinatorial Nullstellensatz, Combinatorics, Probability and Computing8 (1999), 7–29.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    N. Alon, M. B. Nathanson and I. Z. Ruzsa,Adding distinct congruence classes modulo a prime, The American Mathematical Monthly102 (1995), 250–255.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    N. Alon, M. B. Nathanson and I. Z. Ruzsa,The polynomial method and restricted sums of congruence classes, Journal of Number Theory56 (1996), 404–417.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Y. F. Bilu, V. F. Lev and I. Z. Ruzsa,Rectification principles in additive number theory, Discrete and Computational Geometry19 (1998), 343–353.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    S. Dasgupta, Gy. Károlyi, O. Serra and B. Szegedy,Transversals of additive Latin squares, Israel Journal of Mathematics126 (2001), 17–28.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    H. Davenport,On the addition of residue classes, Journal of the London Mathematical Society10 (1935), 30–32.MATHCrossRefGoogle Scholar
  7. [7]
    J. A. Dias da Silva and Y. O. Hamidoune,Cyclic spaces for Grassmann derivatives and additive theory, The Bulletin of the London Mathematical Society26 (1994), 140–146.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    S. Eliahou and M. Kervaire,Sumsets in vector spaces over finite fields, Journal of Number Theory71 (1998), 12–39.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    S. Eliahou and M. Kervaire,Restricted sums of sets of cardinality 1+p in a vector space over F p, Discrete Mathematics235 (2001), 199–213.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    S. Eliahou and M. Kervaire,Restricted sumsets in finite vector spaces: the case p=3, Integers1 (2001), Research paper A2, 19 pages (electronic).Google Scholar
  11. [11]
    P. Erdős and R. L. Graham,Old and New Problems and Results in Combinatorial Number Theory, L’Enseignement Mathématique, Geneva, 1980.Google Scholar
  12. [12]
    G. A. Freiman,Foundations of a Structural Theory of Set Addition, Translations of Mathematical Monographs37, American Mathematical Society, Providence, RI, 1973.MATHGoogle Scholar
  13. [13]
    Y. O. Hamidoune, A. S. Lladó, and O. Serra,On restricted sums, Combinatorics, Probability and Computing9 (2000), 513–518.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Gy. Károlyi,A compactness argument in the additive theory and the polynomial method, Discrete Mathematics (2003), to appear.Google Scholar
  15. [15]
    M. Kneser,Abschätzungen der asymptotischen Dichte von Summenmengen, Mathematische Zeitschrift58 (1953), 459–484.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    V. F. Lev,Restricted set addition in groups. I: The classical setting, Journal of the London Mathematical Society (2)62 (2000), 27–40.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    V. F. Lev,Restricted set addition in groups. II: A generalization of the Erdős-Heilbronn conjecture, Electronic Journal of Combinatorics7 (2000), Research paper R4, 10 pages (electronic).Google Scholar
  18. [18]
    V. F. Lev, Personal communication.Google Scholar
  19. [19]
    M. B. Nathanson,Additive Number Theory. Inverse Problems and the Geometry of Sumsets, GTM165, Springer, Berlin, 1996.Google Scholar

Copyright information

© Hebrew University 2004

Authors and Affiliations

  1. 1.Department of Algebra and Number TheoryEötvös UniversityBudapestHungary

Personalised recommendations