Dense Difference Sets and Their Combinatorial Structure

  • Vitaly BergelsonEmail author
  • Paul Erdős
  • Neil Hindman
  • Tomasz Łuczak


We show that if a set B of positive integers has positive upper density, then its difference set D(B) has extremely rich combinatorial structure, both additively and multiplicatively. If on the other hand only the density of D(B) rather than B is assumed to be positive, one is not guaranteed any multiplicative structure at all and is guaranteed only a modest amount of additive structure.


  1. 1.
    V. Bergelson, A density statement generalizing Schur’s Theorem, J. Comb. Theory (Series A) 43 (1986), 336–343.MathSciNetGoogle Scholar
  2. 2.
    V. Bergelson, Applications of ergodic theory to combinatorics, Dissertation (in Hebrew), Hebrew University of Jerusalem: 1984.Google Scholar
  3. 3.
    V. Bergelson, Sets of recurrence of \({\mathbb{Z}}^{m}\) actions and properties of sets of differences in \({\mathbb{Z}}^{m}\), J. London Math. Soc. (2) 31 (1985), 295–304.Google Scholar
  4. 4.
    V. Bergelson and N. Hindman, On IP ∗  sets and central sets, Combinatorica 14 (1994), 269–277.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    W. Deuber, N. Hindman, I. Leader, and H. Lefmann, Infinite partition regular matrices, Combinatorica, 15 (1995), 333–355.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Anal. Math. 31 (1977), 204–256.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton Univ. Press, Princeton, 1981.zbMATHGoogle Scholar
  8. 8.
    N. Hindman, On density, translates, and pairwise sums of integers, J. Comb. Theory (Series A) 33 (1982), 147–157.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Vitaly Bergelson
    • 1
    Email author
  • Paul Erdős
    • 2
  • Neil Hindman
    • 3
  • Tomasz Łuczak
    • 4
  1. 1.Department of MathematicsOhio State UniversityColumbusUSA
  2. 2.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary
  3. 3.Department of MathematicsHoward UniversityWashington, DCUSA
  4. 4.Department of Discrete Mathematics, Faculty of Mathematics and CSAdam Mickiewicz UniversityPoznanPoland

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