preface
notation
1 simple inverse theorems
1.1 direct and inverse problems
1.2 finite arithmetic progressions
1.3 an inverse problem for distinct summands
1.4 a special case
1.5 small sumsets: the case 2a 3k - 4
1.6 application: the number of sums and products
1.7 application: sumsets and powers of 2
1.8 notes
1.9 exercises
2 sums of congruence classes
2.1 addition in groups
2.2 the e-transform
2.3 the cauchy-davenport theorem
2.4 the erdos——ginzburg-ziv theorem
2.5 vosper''s theorem
2.6 application: the range of a diagonal form
2.7 exponential sums
2.8 the freiman-vosper theorem
2.9 notes
2.10 exercises
3 sums of distinct congruence classes
3.1 the erd6s-heilbronn conjecture
3.2 vandermonde determinants
3.3 multidimensional ballot numbers
3.4 a review of linear algebra
3.5 alternating products
3.6 erdos-heilbronn, concluded
3.7 the polynomial method
3.8 erd6s-heilbronn via polynomials
3.9 notes
3.10 exercises
4 kneser''s theorem for groups
4.1 periodic subsets
4.2 the addition theorem
4.3 application: the sum of two sets of integers
4.4 application: bases for finite and a-finite groups
4.5 notes
4.6 exercises
5 sums of vectors in euclidean space
5.1 small sumsets and hyperplanes
5.2 linearly independent hyperplanes
5.3 blocks
5.4 proof of the theorem
5.5 notes
5.6 exercises
6 geometry of numbers
6.1 lattices and determinants
6.2 convex bodies and minkowski''s first theorem
6.3 application: sums of four squares
6.4 successive minima and minkowski''s second theorem
6.5 bases for sublattices
6.6 torsion-free abelian groups
6.7 an important example
6.8 notes
6.9 exercises
7. plunnecke''s inequality
7.1 plunnecke graphs
7.2 examples of plunnecke graphs
7.3 multiplicativity of magnification ratios
7.4 menger''s theorem
7.5 pliinnecke''s inequality
7.6 application: estimates for sumsets in groups
7.7 application: essential components
7.8 notes
7.9 exercises
8 freiman''s theorem
8.1 multidimensional arithmetic progressions
8.2 freiman isomorphisms
8.3 bogolyubov''s method
8.4 ruzsa''s proof, concluded
8.5 notes
8.6 exercises
9 applications of freiman''s theorem
9.1 combinatorial number''theory
9.2 small sumsets and long progressions
9.3 the regularity lemma
9.4 the balog-szemeredi theorem
9.5 a conjecture of erd6s
9.6 the proper conjecture
9.7 notes
9.8 exercises
references
index
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