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.
61 Basic Notions
62 Independent sets sumfree subsets and Sidon sets
64 Proof of the BalogSzemerediGowers theorem
72 The Fourieranalytic approach
For each i e A we have the easy bound
28 Elementary sumproduct estimates
31 Additive groups
33 Convex bodies
34 The BrunnMinkowski inequality
36 Progressions and proper progressions
41 Basic theory
4114 Let G H be two subgroups of Z and
45 p constants Bhg sets and dissociated sets
46 The spectrum of an additive set
47 Progressions in sum sets
52 Sum sets in vector spaces
53 Freiman homomorphisms
55 Universal ambient groups
84 Cell decompositions and the distinct distances problem
the coordinate functions xi xm m
96 Kemnitzs conjecture
102 The small torsion case
105 An ergodic argument
0 fu 1 and Ezj Ez_f Finally
Z C be normalized so
115 The inﬁnitary ergodic approach
116 The hypergraph approach
117 Arithmetic progressions in the primes
124 Complete and subcomplete sequences
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absolute constant additive group additive set apply arbitrary argument arithmetic progression assume bipartite graph Bohr set cardinality claim follows claim then follows conclude conjecture Corollary coset covering lemma cyclic group deﬁne Deﬁnition denote disjoint doubling constant elements estimate Exercise exists ﬁeld ﬁnd ﬁnite additive group finite field ﬁrst Fourier transform Fourier-analytic Freiman homomorphism Freiman isomorphism function Gowers uniformity group homomorphism hence Hint hypergraph implies independent inﬁnite integer isomorphic of order lattice least Let F Let G linear lower bound multiplicative non-empty non-zero norm Observe obtain particular partition pigeonhole principle polynomial positive integers progression of length proper arithmetic progression proper progression Proposition random variable regularity lemma result Roth’s theorem Section Sidon set subgroup subset sufﬁciently large sum sets symmetric Szemer´edi’s theorem torsion-free translates triangle inequality universal ambient group upper bound vector space zero
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.