Structural additive theory

Includes index

Includes bibliography (p. 415-421) and references

1. Abelian Groups and Character Sums 2. Introduction to Sumsets 3. Simple Results for Torsion-Free Abelian Groups 4. Basic Results for Sumsets with an Infinite Summand 5. The Pigeonhole and Multiplicity Bounds 6. Periodic Sets and Kneser's Theorem 7. Compression, Complements and the 3k–4 Theorem 8. Additive Energy 9. Kemperman's Critical Pair Theory 10. Zero-Sums, Setpartitions and Subsequence Sums 11. Long Zero-Sum Free Sequences over Cyclic Groups 12. Pollard's Theorem for General Abelian Groups 13. The DeVos–Goddyn–Mohar Theorem 14. The Partition Theorem I 15. The Partition Theorem II 16. The Ψ-Weighted Gao Theorem 17. Group Algebras 18. Character and Linear Algebraic Methods 19. Character Sum and Fourier Analytic Methods 20. Freiman Homomorphisms Revisited 21. The Isoperimetric Method 22. The Polynomial Method

Nestled between number theory, combinatorics, algebra, and analysis lies a rapidly developing subject in mathematics variously known as additive combinatorics, additive number theory, additive group theory, and combinatorial number theory. Its main objects of study are not abelian groups themselves, but rather the additive structure of subsets and subsequences of an abelian group, i.e. sumsets and subsequence sums.