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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
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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.

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 The book also features a considerable amount of previously unpublished material, as well as numerous exercises and motivating examples.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.

The Erdős–Moser Sum-free Set Problem
Part of
- Additive number theory; partitions
Tom Sanders
Journal:

Canadian Journal of Mathematics / Volume 73 / Issue 1 / February 2021

Published online by Cambridge University Press:

23 September 2019, pp. 63-107

Print publication:

February 2021
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We show that there is an absolute $c>0$ such that if $A$ is a finite set of integers, then there is a set $S\subset A$ of size at least $\log ^{1+c}|A|$ such that the restricted sumset $\{s+s^{\prime }:s,s^{\prime }\in S\text{ and }s\neq s^{\prime }\}$ is disjoint from $A$. (The logarithm here is to base $3$.)

A NOTE ON THE FREIMAN AND BALOG–SZEMERÉDI–GOWERS THEOREMS IN FINITE FIELDS
Part of
- Sequences and sets
BEN GREEN, TERENCE TAO
Journal:

Journal of the Australian Mathematical Society / Volume 86 / Issue 1 / February 2009

Published online by Cambridge University Press:

01 February 2009, pp. 61-74

Print publication:

February 2009
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We prove quantitative versions of the Balog–Szemerédi–Gowers and Freiman theorems in the model case of a finite field geometry 𝔽2n, improving the previously known bounds in such theorems. For instance, if is such that ∣A+A∣≤K∣A∣ (thus A has small additive doubling), we show that there exists an affine subspace H of 𝔽2n of cardinality such that . Under the assumption that A contains at least ∣A∣3/K quadruples with a1+a2+a3+a4=0, we obtain a similar result, albeit with the slightly weaker condition ∣H∣≫K−O(K)∣A∣.

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