| Prologue | p. xi |
| The probabilistic method | p. 1 |
| The first moment method | p. 2 |
| The second moment method | p. 6 |
| The exponential moment method | p. 9 |
| Correlation inequalities | p. 19 |
| The Lovász local lemma | p. 23 |
| Janson's inequality | p. 27 |
| Concentration of polynomials | p. 33 |
| Thin bases of higher order | p. 37 |
| Thin Waring bases | p. 42 |
| Appendix: the distribution of the primes | p. 45 |
| Sum set estimates | p. 51 |
| Sum sets | p. 54 |
| Doubling constants | p. 57 |
| Ruzsa distance and additive energy | p. 59 |
| Covering lemmas | p. 69 |
| The Balog-Szemerédi-Gowers theorem | p. 78 |
| Symmetry sets and imbalanced partial sum sets | p. 83 |
| Non-commutative analogs | p. 92 |
| Elementary sum-product estimates | p. 99 |
| Additive geometry | p. 112 |
| Additive groups | p. 113 |
| Progressions | p. 119 |
| Convex bodies | p. 122 |
| The Brunn-Minkowski inequality | p. 127 |
| Intersecting a convex set with a lattice | p. 130 |
| Progressions and proper progressions | p. 143 |
| Fourier-analytic methods | p. 149 |
| Basic theory | p. 150 |
| Lp theory | p. 156 |
| Linear bias | p. 160 |
| Bohr sets | p. 165 |
| (p) constants, Bh[g] sets, and dissociated sets | p. 172 |
| The spectrum of an additive set | p. 181 |
| Progressions in sum sets | p. 189 |
| Inverse sum set theorems | p. 198 |
| Minimal size of sum sets and the e-transform | p. 198 |
| Sum sets in vector spaces | p. 211 |
| Freiman homomorphisms | p. 220 |
| Torsion and torsion-free inverse theorems | p. 227 |
| Universal ambient groups | p. 233 |
| Freiman's theorem in an arbitrary group | p. 239 |
| Graph-theoretic methods | p. 246 |
| Basic notions | p. 247 |
| Independent sets, sum-free subsets, and Sidon sets | p. 248 |
| Ramsey theory | p. 254 |
| Proof of the Balog-Szeméredi-Gowers theorem | p. 261 |
| Plünnecke's theorem | p. 267 |
| The Littlewood-Offord problem | p. 276 |
| The combinatorial approach | p. 277 |
| The Fourier-analytic approach | p. 281 |
| The Esséen concentration inequality | p. 290 |
| Inverse Littlewood-Offord results | p. 292 |
| Random Bernoulli matrices | p. 297 |
| The quadratic Littlewood-Offord problem | p. 304 |
| Incidence geometry | p. 308 |
| The crossing number of a graph | p. 308 |
| The Szemerédi-Trotter theorem | p. 311 |
| The sum-product problem in R | p. 315 |
| Cell decompositions and the distinct distances problem | p. 319 |
| The sum-product problem in other fields | p. 325 |
| Algebraic methods | p. 329 |
| The combinatorial Nullstellensatz | p. 330 |
| Restricted sum sets | p. 333 |
| Snevily's conjecture | p. 342 |
| Finite fields | p. 345 |
| Davenport's problem | p. 350 |
| Kemnitz's conjecture | p. 354 |
| Stepanov's method | p. 356 |
| Cyclotomic fields, and the uncertainty principle | p. 362 |
| Szemerédi's theorem for k = 3 | p. 369 |
| General strategy | p. 372 |
| The small torsion case | p. 378 |
| The integer case | p. 386 |
| Quantitative bounds | p. 389 |
| An ergodic argument | p. 398 |
| The Szemerédi regularity lemma | p. 406 |
| Szemerédi's argument | p. 411 |
| Szemerédi's theorem for k > 3 | p. 414 |
| Gowers uniformity norms | p. 417 |
| Hard obstructions to uniformity | p. 424 |
| Proof of Theorem 11.6 | p. 432 |
| Soft obstructions to uniformity | p. 440 |
| The infinitary ergodic approach | p. 448 |
| The hypergraph approach | p. 454 |
| Arithmetic progressions in the primes | p. 463 |
| Long arithmetic progressions in sum sets | p. 470 |
| Introduction | p. 470 |
| Proof of Theorem 12.4 | p. 473 |
| Generalizations and variants | p. 477 |
| Complete and subcomplete sequences | p. 480 |
| Proof of Theorem 12.17 | p. 482 |
| Further applications | p. 484 |
| Bibliography | p. 488 |
| Index | p. 505 |
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