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Israel Journal of Mathematics

, Volume 176, Issue 1, pp 269–283 | Cite as

On the number of popular differences

  • Sergei V. Konyagin
  • Vsevolod F. Lev
Article

Abstract

We prove that there exists an absolute constant c > 0 such that for any finite set A ⊆ ℤ with |A| ≥ 2 and any positive integer m < c|A|/ ln |A| there are at most m integers b > 0 satisfying |(A + b) \ A| ≤ m; equivalently, there are at most m positive integers possessing |A| −m (or more) representations as a difference of two elements of A.

This is best possible in the sense that for each positive integer m there exists a finite set A ⊆ ℤ with |A| > m log2(m/2) such that |(A+b)\A| ≤ m holds for b = 1, ..., m + 1.

Keywords

Positive Integer Abelian Group Group Element Pairwise Disjoint Arithmetic Progression 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [BS96]
    R. Balasubramanian and K. Soundararajan, On a conjecture of R. L. Graham, Acta Arithmetica 75 (1996), 1–38.zbMATHMathSciNetGoogle Scholar
  2. [EH64]
    P. Erdős and H. Heilbronn, On the addition of residue classes mod p, Acta Arithmetica 9 (1964), 149–159.MathSciNetGoogle Scholar
  3. [G32]
    R. M. Gabriel, The rearrangement of positive Fourier coefficients, Proceedings of the London Mathematical Society (2) 33 (1932), 32–51.CrossRefGoogle Scholar
  4. [G70]
    R. L. Graham, Problem 5749, The American Mathematical Monthly 77 (1970), 775.CrossRefGoogle Scholar
  5. [HL28]
    G. H. Hardy and J. E. Littlewood, Notes on the theory of series (VIII): an inequality, Journal of the London Mathematical Society 3 (1928), 105–110.CrossRefGoogle Scholar
  6. [HLP88]
    G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, second edition, Cambridge University Press, Cambridge, 1988.zbMATHGoogle Scholar
  7. [L96]
    V. F. Lev, Structure theorem for multiple addition and the Frobenius problem, Journal of Number Theory 58 (1996), 79–88.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [L98]
    V. F. Lev, Linear equations over Fp and moments of exponential sums, Duke Mathematical Journal 107 (2001), 239–263.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [L06]
    V. F. Lev, Large sum-free sets in ℤ/pℤ, Israel Journal of Mathematics 154 (2006), 221–233.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [O68]
    J. E. Olson, An addition theorem modulo p, Journal of Combinatorial Theory 5 (1968), 45–52.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [P74]
    J. M. Pollard, A generalization of the theorem of Cauchy and Davenport, Journal of the London Mathematical Society (2) 8 (1974), 460–462.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2010

Authors and Affiliations

  1. 1.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  2. 2.Department of MathematicsThe University of Haifa at OranimTivonIsrael

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