Hello,
I'd love to learn more about the field of additive combinatorics. From what I've understand, there's a book by Tao and Vu out on the subject, and it looks fun, but I think I lack the prerequisites. Right now, I've had basic real analyis (Rudin), read the first volume of Stanley's "Enumerative combinatorics", and some algebra (some graduate). I have no experience of probability theory whatsoever, or functional analysis or ergodic theory. So I'm curious, from my background, what would be needed to reach the level where I can read and understand the book of Tao and Vu? Are there any certain books to reach that level which you can recommend?
Best regards,
CM



A learning roadmap for Additive combinatorics.
Some portions of their book should be accessible without too much background. Take a look at their sections on additive geometry, graph-theoretic methods, and algebraic methods, for example. For the bulk of the book, though, knowing some probability theory will make a big difference.


A recent book that I like and you might find more accessible is Alfred Geroldinger-Imre Z. Ruzsa, "Combinatorial Number Theory and Additive Group Theory", Birkhäuser, 2009.


From their Foreword:



> 
> This book collects the material delivered in the 2008 edition of the *DocCourse in Combinatorics and Geometry* which was devoted to the topic of *Additive Combinatorics*.
> 
> 
> The two ﬁrst parts, which form the bulk of the volume, contain the two main advanced courses, *Additive Group Theory and Non-unique Factorizations*, by Alfred Geroldinger, and *Sumsets and Structure*, by Imre Z. Ruzsa.
> 
> 
> The ﬁrst part focusses on the interplay between zero-sum problems, arising from the Erdős–Ginzburg–Ziv theorem, and nonuniqueness of factorizations in monoids and integral domains.
> 
> 
> The second part deals with structural set addition. It aims at describing the structure of sets in a commutative group from the knowledge of some properties of its sumset.
> 
> 
> The third part of the volume collects some of the seminars which accompanied the main courses and covers several aspects of contemporary methods and problems in Additive Combinatorics.
> 
> 
> 


I would recommend that you work through the second part, and see how you find the material. It should be accessible.


You may also want to take a look at Ben Green's [notes](http://www.dpmms.cam.ac.uk/%7Ebjg23/notes.html) on the structure theory of Set Addition.


Let me add: If you are mainly interested in classical additive combinatorics, as it applies to the natural numbers, then I strongly recommend Melvyn Nathanson, "Additive number theory. Vol II: Inverse problems and the geometry of sumsets", Springer, GTM 165, 1996.


