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E-raamat: Introduction to Approximate Groups

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(University of Cambridge)
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Introduction to Approximate GroupsSuurem pilt
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Recent years have seen rapid progress in the field of approximate groups, with the emergence of a varied range of applications. Written by a leader in the field, this text for both beginning graduate students and researchers collects, for the first time in book form, the main concepts and techniques into a single, self-contained introduction.

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 The book also features a considerable amount of previously unpublished material, as well as numerous exercises and motivating examples.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

Arvustused

'The book now under reviews offers an excellent introduction the book is very nicely written, Researchers and fledgling researchers in this area will want to own this book.' Mark Hunacek, The Mathematical Gazette ' an aspiring student who wants to enter the world of approximate groups will surely find the first chapters of the book, which cover the fundamentals, invaluable. Moreover, anyone willing to climb the mountain that is the BGT theorem should be grateful for the webbing ladders laid out in Chapters IVVI. Less ambitious readers might still enjoy the small gems, scattered throughout the text, like Solymosi's sum-product theorem in Chapter IX or the SandersCrootSisask power set argument in Chapter X, both of which are a delight to read this is perhaps the first book that provides a systematic treatment of approximate groups as a mathematical subject. It is very likely to become one of standard texts in this rapidly developing field.' Michael Bjorklund, Bulletin of the American Mathematical Society

Muu info

Provides a comprehensive exploration of the main concepts and techniques from the young, exciting field of approximate groups.
Preface xi
1 Introduction
1(10)
1.1 Introduction
1(2)
1.2 Historical Discussion
3(3)
1.3 Bounds and Asymptotic Notation
6(1)
1.4 General Notation
7(2)
1.5 Miscellaneous Results
9(2)
2 Basic Concepts
11(24)
2.1 Large Doubling of Random Sets of Integers
11(4)
2.2 Sets of Very Small Doubling
15(1)
2.3 Iterated Sum Sets and the Pliinnecke-Ruzsa Inequalities
16(4)
2.4 Ruzsa's Covering Argument
20(3)
2.5 Small Tripling and Approximate Groups
23(4)
2.6 Stability of Approximate Groups under Basic Operations
27(3)
2.7 Freiman Homomorphisms
30(2)
Exercises
32(3)
3 Coset Progressions and Bohr Sets
35(19)
3.1 Introduction
35(4)
3.2 Small Sets and Freiman Images of Coset Progressions
39(2)
3.3 Lattices
41(3)
3.4 Convex Bodies
44(2)
3.5 Successive Minima and Minkowski's Second Theorem
46(3)
3.6 Finding Dense Coset Progressions in Bohr Sets
49(4)
Exercises
53(1)
4 Small Doubling in Abelian Groups
54(27)
4.1 Introduction
54(5)
4.2 Fourier Analysis
59(4)
4.3 Convolutions and Fourier Analysis of Sets of Small Doubling
63(3)
4.4 Dense Models for Abelian Sets of Small Doubling
66(3)
4.5 Bohr Sets in Dense Subsets of Finite Abelian Groups
69(2)
4.6 Reducing the Dimension of the Bohr Set
71(2)
4.7 Chang's Covering Argument
73(1)
4.A Dissociated Subsets of C?
74(4)
Exercises
78(3)
5 Nilpotent Groups, Commutators and Nilprogressions
81(28)
5.1 Progressions in the Heisenberg Group
81(4)
5.2 Nilpotent Groups
85(4)
5.3 Commutators
89(4)
5.4 The Collecting Process and Basic Commutators
93(5)
5.5 Commutator Forms
98(3)
5.6 Nilprogressions
101(6)
Exercises
107(2)
6 Nilpotent Approximate Groups
109(21)
6.1 Introduction and Overview of the Torsion-Free Case
109(3)
6.2 Details of the Torsion-Free Case
112(2)
6.3 Abelian p-Groups
114(4)
6.4 Multi-Variable Homomorphisms into Abelian Groups
118(2)
6.5 Placing Arbitrary Subgroups inside Normal Subgroups
120(4)
6.6 Conclusion of the General Case
124(4)
Exercises
128(2)
7 Arbitrary Approximate Groups
130(3)
7.1 The Breuillard-Green-Tao Theorem
130(3)
8 Residually Nilpotent Approximate Groups
133(14)
8.1 Introduction
133(1)
8.2 Central Extensions of Nilpotent Approximate Groups
134(2)
8.3 Bounded Normal Series for Nilpotent Approximate Groups
136(6)
8.4 From Normal to Central Subgroups
142(2)
8.5 Residually Nilpotent Groups
144(1)
Exercises
145(2)
9 Soluble Approximate Subgroups of GLn(C)
147(23)
9.1 Introduction
147(2)
9.2 The Sum-Product Phenomenon over C
149(4)
9.3 Complex Upper-Triangular Groups
153(3)
9.A Representation Theory
156(7)
9.B The Structure of Soluble Linear Groups
163(4)
Exercises
167(3)
10 Arbitrary Approximate Subgroups of GLn(C)
170(8)
10.1 Introduction
170(1)
10.2 Free Groups and the Uniform Tits Alternative
171(1)
10.3 Small Neighbourhoods of the Identity
172(3)
10.4 Approximate Subgroups of Complex Linear Groups
175(2)
Exercises
177(1)
11 Applications to Growth in Groups
178(20)
11.1 Introduction
178(4)
11.2 Finite-Index Subgroups
182(1)
11.3 A Refinement of Gromov's Theorem
183(2)
11.4 Persistence of Polynomial Growth
185(3)
11.5 Diameters of Finite Groups
188(1)
11.6 An Isoperimetric Inequality for Finite Groups
189(5)
11.A Expansion in Special Linear Groups
194(2)
Exercises
196(2)
References 198(4)
Index 202
Matthew C. H. Tointon is the Stokes Research Fellow at Pembroke College, Cambridge, affiliated to the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge. He has held postdoctoral positions at Homerton College, Cambridge, at the Université de Paris-Sud and at the Université de Neuchâtel, Switzerland. Tointon is the author of numerous research papers on approximate groups and he proved the strongest known results describing the structure of nilpotent and residually nilpotent approximate groups.
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