Structural additive theory. Based on courses given at Karl-Franzens-Universität Graz, Austria, 2008--2012
type: Item
instance of: scholarly article
title: Structural additive theory. Based on courses given at Karl-Franzens-Universität Graz, Austria, 2008--2012
zbMATH Open document ID: 1368.11109
DOI: 10.1007/978-3-319-00416-7
author: David J. Grynkiewicz
published in: Developments in Mathematics
publication date: 2013-05-10T00:00:00Z
review text: 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 presente…
reviewed by: Wolfgang A. Schmid
Mathematics Subject Classification ID: 11P70
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11B50
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 20K01
zbMATH DE Number: 6162097
zbMATH Keywords: sumsets
zbMATH Keywords: subsequence sums
zbMATH Keywords: zero-sum sequence
zbMATH Keywords: Davenport constant
zbMATH Keywords: Kneser's theorem
zbMATH Keywords: abelian groups
zbMATH Keywords: Erdős-Ginzburg-Ziv theorem
zbMATH Keywords: group algebra
zbMATH Keywords: Freiman homomorphisms
zbMATH Keywords: additive energy
zbMATH Keywords: Sidon sets
zbMATH Keywords: polynomial method
zbMATH Keywords: isoperimetric method
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-00416-7
OpenAlex ID: W590122907
description: scientific article; zbMATH DE number 6162097