Israel Journal of Mathematics

, Volume 179, Issue 1, pp 253–278 | Cite as

The structure of popular difference sets



We show that the set of popular differences of a large subset of ℤ N does not always contain the complete difference set of another large set. For this purpose we construct a so-called niveau set, which was first introduced by Ruzsa in [Ruz87] and later used in [Ruz91] to show that there exists a large subset of ℤ N whose sumset does not contain any long arithmetic progressions. In this paper we make substantial use of measure concentration results on the multi-dimensional torus and Esseen’s Inequality.


Independent Random Variable Arithmetic Progression Total Variation Distance Joint Distribution Function Standard Normal Distribution Function 
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© Hebrew University Magnes Press 2010

Authors and Affiliations

  1. 1.Department of Pure Mathematics and Mathematical StatisticsCambridgeUK
  2. 2.Institute for Advanced StudySchool of MathematicsPrincetonUSA

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