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1 Simple inverse theorems 1

1.1 Direct and inverse problems 1

1.2 Finite arithmetic progressions 7

1.3 An inverse problem for distinct summands 13

1.4 A special case 18

1.5 Small sumsets:The case |2A| ？3k-4 21

1.6 Application:The number of sums and products 29

1.7 Application:Sumsets and powers of2 31

1.8 Notes 33

1.9 Exercises 35

2 Sums of congruence classes 41

2.1 Addition in groups 41

2.2 The e-transform 42

2.3 The Cauchy-Davenport theorem 43

2.4 The Erd？s-Ginzburg-Ziv theorem 48

2.5 Vosper's theorem 52

2.6 Application:The range ofa diagonal form 57

2.7 Exponential sums 62

2.8 The Freiman-Vosper theorem 67

2.9 Notes 73

2.10 Exercises 74

3 Sums of distinct congruence classes 77

3.1 The Erd？s-Heilbronn conjecture 77

3.2 Vaodermonde determinants 78

3.3 Multidimensional ballot numbers 81

3.4 A review oflinear algebra 89

3.5 Alternating products 92

3.6 Erd？s-Heilbronn,concluded 95

3.7 The polynomial method 98

3.8 Erd？s-Heilbronn via polynomials 101

3.9 Notes 106

3.10 Exercises 107

4 Kneser's theorem for groups 109

4.1 Periodic subsets 109

4.2 The addition theorem 110

4.3 Application:The sum oftwo sets ofintegers 117

4.4 Application:Basesforfiniteandσ-finite groups 127

4.5 Notes 130

4.6 Exercises 131

5 Sums of vectors in Euclidean space 133

5.1 Sinail sumsets and hyperplanes 133

5.2 Linearly independent hyperplanes 135

5.3 Blocks 142

5.4 Proofofthe theorem 152

5.5 Notes 163

5.6 Exercises 163

6 Geometry of numbers 167

6.1 Lattices and determinants 167

6.2 Convex bodies and Minkowski's FirstTheorem 174

6.3 Application:Sums offour squares 177

6.4 Successive minima and Minkowski's second theorem 180

6.5 Bases for sublattices 185

6.6 Torsion-free abelian groups 190

6.7 An important example 194

6.8 Notes 196

6.9 Exercises 196

7 Pliinnecke's inequality 201

7.1 Pliinnecke graphs 201

7.2 Examples ofPlüinnecke graphs 203

7.3 Multiplicativityofmagnification ratios 205

7.4 Menger's theorem 209

7.5 Pliinnecke's inequality 212

7.6 Application:Estimates for sumsets in groups 217

7.7 Application:Essential components 221

7.8 Notes 226

7.9 Exercises 227

8 Freiman's theorem 231

8.1 Multidimensional arithmetic progressions 231

8.2 Freiman isomorphisms 233

8.3 Bogolyubov's method 238

8.4 Ruzsa's proof,concluded 244

8.5 Notes 251

8.6 Exercises 252

9 Applications of Freiman's theorem 255

9.1 Combinatorial number theory 255

9.2 Small sumsets and long progressions 255

9.3 The regularity iemma 257

9.4 Tbe Balog-Szemer？di theorem 270

9.5 A conjecture ofErd？s 277

9.6 The proper conjecture 278

9.7 Notes 279

9.8 Exercises 280

References 283

Index 292