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1 results

  • 4 - Small Doubling in Abelian Groups

  • Matthew C. H. Tointon, University of Cambridge
  • Book:
    Introduction to Approximate Groups
    Published online:
    31 October 2019
    Print publication:
    14 November 2019, pp 54-80

  • Summary

    We present Green and Ruzsa’s proof of Freiman’s theorem in an arbitrary abelian group. More specifically, we show that a finite set A of small doubling inside an abelian group is contained in a relatively small coset progression of bounded rank. We introduce the basics of discrete Fourier analysis, and how it relates to sets of small doubling. We prove the Green–Ruzsa result that a set of small doubling in an arbitrary abelian group has a Freiman model in a relatively small finite abelian group. We then prove Bogolyubov’s lemma that a small iterated sum set of this model must contain a relatively large Bohr set of low rank. Combined with the material of the previous chapter, this shows that A contains a relatively large coset progression of low rank. We then deduce the main theorem of the chapter using Chang’s covering argument. In the exercises we guide the reader to a simpler version of the argument yielding the same result in the special case in which A is a set of integers.