Additive Number Theory ─ Density Theorems and the Growth of Sumsets

A central problem in additive number theory is the growth of sumsets. If A is a finite or infinite subset of the integers and the lattice points, or more generally, of any abelian group or semigroup G, then the h-fold sumset of A is the set LA consisting of all elements of G that can be represented as the sum of L not necessarily distinct elements of A. The goal is to understand the asymptotics of the sumsets LA, that is, the size and structure of LA, as L tends to infinity. If A is finite, then the size of LA is its cardinality. If A is infinite, then the size of LA is measured by various duality functions. These problems have natural analogues when A is a subset of a nonabelian group. Additive Number Theory: Density Theorems and the Growth of Sumsets presents material that deals with the above problem. Ideas and techniques from many parts of mathematics are used to prove theorems in this subject. For example, the authors use number theory, combinatorics, commutative algebra, ultrafilters and logic, and nonstandard analysis. The book is self-contained, and includes short introductions to the various techniques that are not standard in this field.

Graduate students and researchers in mathematics will find this book useful. Prerequisites include undergraduate elementary number theory and algebra.