Preface

This book grew out of several courses given at Karl-Franzens-Universität Graz in 2008–2012 and has since been augmented with additional material. It should be of interest both to people new to the field variously known as Additive Number Theory, Additive Combinatorics, Additive Group Theory and Combinatorial Number Theory—as a basic introduction to the area—as well as to the more seasoned researcher, in view of the unified presentation of material previously only available in research articles combined with a fair amount of new material. As there seems no real consensus on whether additive problems belong to Combinatorics, Number Theory, Group Theory, or even Analysis, as they often find themselves lying between all these more established areas of mathematics, we will refer to the broader area of mathematics dealt with by this book simply as Additive Theory. In recent years, the first few comprehensive texts on what has become a rapidly developing subject have begun to be published in Additive Theory. With so few texts on the subject, I have made little attempt to compete with these already established works. Indeed, the focus of this text is specifically on those areas of Additive Theory that have not been treated in detail by previous books, and even when treating more basic results also found in other texts, I have endeavored to present such results either in greater generality or with new proof variations. Rather than focus on the great achievements in approximate results—such as Freiman’s Theorem, Szemeredi’s Theorem, and results achieved via Fourier Analytic/Ergodic Theory breakthroughs—results that require little hypotheses but, at this price, yield only rough results with imprecise constants that leave much room for future improvement, I have instead focused on the more exact and refined results in the area, results which have some satisfactory air of completeness to themselves, even if they may yet one day fit into a wider landscape. This reflects both a personal bias as well as the aforementioned wish to complement, and not compete with, the current material available. On top of this, I have not shied away from presenting important results even when they have notoriously complicated proofs. There is little prerequisite for this book apart from a solid background in undergraduate mathematics, particularly in the theory of finitely generated abelian groups and Linear Algebra, which makes the material suitable for a graduate or advanced undergraduate course. Knowledge of the basic algebraic concepts of groups, rings, fields, vector spaces and modules is assumed, though very little beyond the definitions and basics would be needed. The main prerequisite might best be described as a fair amount of mathematical maturity. I suspect that the intricacy of the more difficult proofs treated here may be the like of which a less experienced student has not yet seen. For this reason, I have taken great care to include as many details as reasonable, many more than is common in other texts, when presenting each and every proof. While this may expand the length of each proof in print, I hope that it will also reduce the time needed to absorb the complicated combinatorial arguments that often arise. Many of the chapters are self contained or come in a series of interconnected topics,