Additive Combinatorics

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Cambridge University Press, Sep 14, 2006 - Mathematics
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Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate-level 2006 text will allow students and researchers easy entry into this fascinating field. Here, the authors bring together in a self-contained and systematic manner the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as Szemerédi's theorem on arithmetic progressions, the Kakeya conjecture and Erdos distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new results.
 

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Contents

14 Correlation inequalities
19
Thus it remains to verify the S 1
25
16 Jansons inequality
27
17 Concentration of polynomials
33
the distribution of the primes
45
2
51
21 Sum sets
54
22 Doubling constants
57
6
246
61 Basic Notions
247
62 Independent sets sumfree subsets and Sidon sets
248
64 Proof of the BalogSzemerediGowers theorem
261
7
276
72 The Fourieranalytic approach
281
For each i e A we have the easy bound
306
8
308

28 Elementary sumproduct estimates
99
3
112
31 Additive groups
113
32 Progressions
119
33 Convex bodies
122
34 The BrunnMinkowski inequality
127
36 Progressions and proper progressions
143
4
149
41 Basic theory
150
4114 Let G H be two subgroups of Z and
156
45 p constants Bhg sets and dissociated sets
172
46 The spectrum of an additive set
181
47 Progressions in sum sets
189
5
198
52 Sum sets in vector spaces
211
53 Freiman homomorphisms
220
55 Universal ambient groups
233
3
311
84 Cell decompositions and the distinct distances problem
319
9
329
the coordinate functions xi xm m
333
96 Kemnitzs conjecture
354
10
369
102 The small torsion case
378
105 An ergodic argument
398
0 fu 1 and Ezj Ez_f Finally
405
Z C be normalized so
406
11
414
115 The infinitary ergodic approach
448
116 The hypergraph approach
454
117 Arithmetic progressions in the primes
463
12
470
124 Complete and subcomplete sequences
480

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Popular passages

Page 5 - Intersection Theorems for Systems of Finite Sets,
Page 9 - J. Kahn, J. Komlos, E. Szemeredi, On the probability that a random ±1 matrix is singular, J.
Page 1 - A LOWER BOUND FOR THE VOLUME OF STRICTLY CONVEX BODIES WITH MANY BOUNDARY LATTICE POINTS BY GEORGE E.
Page 12 - Additive number theory. Inverse problems and the geometry of sumsets.
Page 5 - Graham, Old and new problems and results in combinatorial number theory, Monographies de L'EnseignementMathematique 28, Universite de Geneve, L'Enseignement Mathematique, Geneva, 1980.
Page 3 - Chung. The number of different distances determined by n points in the plane. J. Combin. Theory Ser. A, 36:342354, 1984.
Page 5 - J. Esary, F. Proschan, and D. Walkup, Association of random variables with applications, Ann.
Page 1 - References 1. M. Ajtai, V. Chvatal, M. Newborn, and E. Szemeredi: Crossing-free subgraphs. Annals of Discrete Mathematics 12 (1982), 9-12 2.
Page 12 - Olson, A combinatorial problem on finite Abelian groups. I, J. Number Theory 1 (1969), 8-10.

References to this book

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About the author (2006)

Terence Tao is a Professor in the Department of Mathematics at the University of California, Los Angeles. He was awarded the Fields Medal in 2006 for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory.

Van H. Vu is a Professor in the Department of Mathematics at Rutgers University, New Jersey.

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