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The structure of popular difference sets

Israel Journal of Mathematics volume 179, pages 253–278 (2010)Cite this article

Abstract

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.

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Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge, CB3 0WB, UK

    J. Wolf

  2. Institute for Advanced Study, School of Mathematics, Einstein Drive, Princeton, NJ, 08540, USA

    J. Wolf

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  1. J. Wolf
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Correspondence to J. Wolf.

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Wolf, J. The structure of popular difference sets. Isr. J. Math. 179, 253–278 (2010). https://doi.org/10.1007/s11856-010-0081-2

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  • DOI: https://doi.org/10.1007/s11856-010-0081-2

Keywords

  • Independent Random Variable
  • Arithmetic Progression
  • Total Variation Distance
  • Joint Distribution Function
  • Standard Normal Distribution Function