##
**Structural additive theory. Based on courses given at Karl-Franzens-Universität Graz, Austria, 2008–2012.**
*(English)*
Zbl 1368.11109

Developments in Mathematics 30. Cham: Springer (ISBN 978-3-319-00415-0/hbk; 978-3-319-00416-7/ebook). xii, 426 p. (2013).

The book under review presents a well-written introduction to what its author calls additive theory, other commonly used terms are additive group theory, additive (or more recently arithmetic) combinatorics or structure theory of set addition; traditionally the problems are also considered as part of additive and combinatorial number theory.

The topics that are covered encompass a considerable range of subjects. A declared goal of the author was to focus on aspects not yet covered in other books on the subject. More concretely, there is an emphasis on results that in a way are exact as opposed to asymptotic or approximate. The book developed from lecture notes of its author. In particular, it contains also exercises and end of chapter notes.

The general outline of the book is as follows. It starts of with a chapter covering necessary prerequisites, mostly, on abelian groups. The remainder of the book is then split into three parts: Sumsets, Subsequence Sums, Advanced methods.

The first part ‘Sumsets’, after presenting fundamental notions, proceeds to cover numerous classical results, including Kneser’s theorem, Vosper’s theorem, the \(3k-4\) theorem, and even Kemperman’s structure theorem. Freiman homomorphisms, Additive Energy and Sidon sets are discussed, too.

The second part ‘Subsequence sums’ discusses the Erdős-Ginzburg-Ziv theorem, the Davenport constant, results on the structure of long zero-sum free in cyclic groups, Pollard’s Theorem, the Devos-Goddyn-Mohar theorem, the Partition Theorem, and finally weighted problems, in particular giving a general form of Gao’s theorem.

The third and final part ‘Advanced methods’ complements the first two parts by presenting various types of techniques and applications thereof. There is a chapter on group algebras, used to investigate the Davenport constant, a chapter on Snevily’s conjecture, Fourier analytic methods are used to obtain a partial \(3k-4\) theorem for groups of prime order, and the polynomial method is presented giving a proof of the Erdős-Heilbronn conjecture as application. Moreover, there is a chapter where Freiman homomorphisms and the notion of universal ambient group are discussed in detail and a chapter on the isoperimetric method.

The book is an excellent introduction to the subject. It can be used both for self-study and as a text accompanying a course. In addition, it is also valuable as a reference text for researchers in the field.

The topics that are covered encompass a considerable range of subjects. A declared goal of the author was to focus on aspects not yet covered in other books on the subject. More concretely, there is an emphasis on results that in a way are exact as opposed to asymptotic or approximate. The book developed from lecture notes of its author. In particular, it contains also exercises and end of chapter notes.

The general outline of the book is as follows. It starts of with a chapter covering necessary prerequisites, mostly, on abelian groups. The remainder of the book is then split into three parts: Sumsets, Subsequence Sums, Advanced methods.

The first part ‘Sumsets’, after presenting fundamental notions, proceeds to cover numerous classical results, including Kneser’s theorem, Vosper’s theorem, the \(3k-4\) theorem, and even Kemperman’s structure theorem. Freiman homomorphisms, Additive Energy and Sidon sets are discussed, too.

The second part ‘Subsequence sums’ discusses the Erdős-Ginzburg-Ziv theorem, the Davenport constant, results on the structure of long zero-sum free in cyclic groups, Pollard’s Theorem, the Devos-Goddyn-Mohar theorem, the Partition Theorem, and finally weighted problems, in particular giving a general form of Gao’s theorem.

The third and final part ‘Advanced methods’ complements the first two parts by presenting various types of techniques and applications thereof. There is a chapter on group algebras, used to investigate the Davenport constant, a chapter on Snevily’s conjecture, Fourier analytic methods are used to obtain a partial \(3k-4\) theorem for groups of prime order, and the polynomial method is presented giving a proof of the Erdős-Heilbronn conjecture as application. Moreover, there is a chapter where Freiman homomorphisms and the notion of universal ambient group are discussed in detail and a chapter on the isoperimetric method.

The book is an excellent introduction to the subject. It can be used both for self-study and as a text accompanying a course. In addition, it is also valuable as a reference text for researchers in the field.

Reviewer: Wolfgang A. Schmid (Paris)

### MSC:

11P70 | Inverse problems of additive number theory, including sumsets |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11B50 | Sequences (mod \(m\)) |

11B75 | Other combinatorial number theory |

20K01 | Finite abelian groups |