Additive Combinatorics

This book concerns the problem of finding the sum set or difference set of a given set of integers A (or, more generally, some subset of an additive group). Starting with additive sets A and B, the sum set A + B is the set of all elements of the form a + b, where a comes from A and b comes from B. Similarly, the difference set A – B is the set of all elements of the form a – b, where a comes from A and b comes from B. The authors study questions such as

• For a given additive set A, what can we say about the size of sum and difference sets A + A and A – A?
• When are they “large” or “small?”
• Conversely, if these sum or difference sets are large or small, what does that tell us about the original set A?

In general, the sum set A + B will have some structure; in particular, for a given set A, the sum sets A + A, 3A = A + A + A, 4A, …, kA, will have some structure, especially as k increases.

This is an incredibly dense book. Although the topic being covered may seem small enough, the authors provide an amazingly rich summary of the study of these problems. (Like many advanced mathematics texts, this book came about from lecture notes.) They include 388 references, 637 exercises, and they make use of a wide array of mathematical tools: probability, geometry, Fourier analysis, graph theory, ergodic theory, abstract algebra, even a little topology.

Coming in at just around 500 pages, one might think that the authors are verbose; quite the opposite: the writing style is terse. The proofs do not give every detail, so the reader does have to pay attention… and will need to have some expertise in the subject. This means that the audience for this book is rather limited. But if you’re interested in sum and difference sets, this is a great reference to have.

Donald L. Vestal is Assistant Professor of Mathematics at South Dakota State University. His interests include number theory, combinatorics, spending time with his family, and working on his hot sauce collection. He can be reached at Donald.Vestal(AT)sdstate.edu.