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


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.


Positive Integer Abelian Group Group Element Pairwise Disjoint Arithmetic Progression 
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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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