Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 Introduction
- 2 Basic Concepts
- 3 Coset Progressions and Bohr Sets
- 4 Small Doubling in Abelian Groups
- 5 Nilpotent Groups, Commutators and Nilprogressions
- 6 Nilpotent Approximate Groups
- 7 Arbitrary Approximate Groups
- 8 Residually Nilpotent Approximate Groups
- 9 Soluble Approximate Subgroups of GLn(C)
- 10 Arbitrary Approximate Subgroups of GLn(C)
- 11 Applications to Growth in Groups
- References
- Index
4 - Small Doubling in Abelian Groups
Published online by Cambridge University Press: 31 October 2019
- Frontmatter
- Dedication
- Contents
- Preface
- 1 Introduction
- 2 Basic Concepts
- 3 Coset Progressions and Bohr Sets
- 4 Small Doubling in Abelian Groups
- 5 Nilpotent Groups, Commutators and Nilprogressions
- 6 Nilpotent Approximate Groups
- 7 Arbitrary Approximate Groups
- 8 Residually Nilpotent Approximate Groups
- 9 Soluble Approximate Subgroups of GLn(C)
- 10 Arbitrary Approximate Subgroups of GLn(C)
- 11 Applications to Growth in Groups
- References
- Index
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.
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- Introduction to Approximate Groups , pp. 54 - 80Publisher: Cambridge University PressPrint publication year: 2019