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2 - Basic Concepts
Matthew C. H. Tointon, University of Cambridge
Book:

Introduction to Approximate Groups

Published online:

31 October 2019

Print publication:

14 November 2019, pp 11-34
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Summary

We motivate the definitions of sets of small doubling and approximate groups, and introduce their basic properties. We show that random sets of integers (suitably defined) have large expected doubling. We prove Freiman’s theorem that a subset of a group of doubling less than 2/3 is close to a finite subgroup. We prove the Plünnecke–Ruzsa inequalities, Ruzsa’s triangle inequality and Ruzsa’s covering lemma. We motivate in detail the definition of an approximate group, and reduce the study of sets of small doubling to the study of finite approximate groups. We show that the notions of small tripling and approximate group are stable under intersections and group homomorphisms. We introduce Freiman homomorphisms and present their basic properties.

1 - Introduction
Matthew C. H. Tointon, University of Cambridge
Book:

Introduction to Approximate Groups

Published online:

31 October 2019

Print publication:

14 November 2019, pp 1-10
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Summary

We define sets of small doubling and approximate groups, and give brief motivation for aspects of the definitions. We give a brief overview of the history of approximate groups.

Matthew C. H. Tointon
Published online:

31 October 2019

Print publication:

14 November 2019
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Approximate groups have shot to prominence in recent years, driven both by rapid progress in the field itself and by a varied and expanding range of applications. This text collects, for the first time in book form, the main concepts and techniques into a single, self-contained introduction. The author presents a number of recent developments in the field, including an exposition of his recent result classifying nilpotent approximate groups. The book also features a considerable amount of previously unpublished material, as well as numerous exercises and motivating examples. It closes with a substantial chapter on applications, including an exposition of Breuillard, Green and Tao's celebrated approximate-group proof of Gromov's theorem on groups of polynomial growth. Written by an author who is at the forefront of both researching and teaching this topic, this text will be useful to advanced students and to researchers working in approximate groups and related areas.

SMALL DOUBLING IN ORDERED GROUPS
Part of
- Special aspects of infinite or finite groups
GREGORY FREIMAN, MARCEL HERZOG, PATRIZIA LONGOBARDI, MERCEDE MAJ
Journal:

Journal of the Australian Mathematical Society / Volume 96 / Issue 3 / June 2014

Published online by Cambridge University Press:

30 April 2014, pp. 316-325

Print publication:

June 2014
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We prove that if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ is a finite subset of an ordered group that generates a nonabelian ordered group, then $|S^2|\geq 3|S|-2$. This generalizes a classical result from the theory of set addition.

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