## Additive CombinatoricsAdditive 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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### Common terms and phrases

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