A learning roadmap for Additive combinatorics. 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: Apart from Tao-Vu (which is a very useful resource) there aren't any obvious books from which to learn this subject at all comprehensively. The books Kevin recommends give an excellent survey of the state of the area in the late 1990s.
For more recent material, here are a few sets of course notes. Some of these are a bit more leisurely than Tao-Vu. 
Jacques Verstraete: http://www.wix.com/annatar0/math262
Andrew Granville: http://www.dms.umontreal.ca/~andrew/Courses/MAT6640.H10.html
Ben Green: http://www.dpmms.cam.ac.uk/~bjg23/add-combinatorics.html
Terry Tao: http://www.math.ucla.edu/~tao/254a.1.03w/
For an overview of the content of Tao-Vu, you could consult my review of it in the Bulletin of the AMS:
http://www.ams.org/journals/bull/2009-46-03/S0273-0979-09-01231-2/home.html
A: 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 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.
A: Nathanson's books:


*

*The Classical Bases

*Inverse Problems and the Geometry of Sumsets

*Elementary Methods in Number Theory


These are so standard that they're often overlooked. To work in this area, you should at least be aware of all of the results and methods in all three of these books. Not an expert, necessarily, but at least aware. It's also worth noting that the style of these books is of the "leave nothing out" variety, which can be tiring if you know too much but refreshingly honest if you know too little.
Three more books that are ubiquitous:


*

*Sequences, by Halberstam and Roth

*The Probabilistic Method, by Alon and Spencer

*Ramsey Theory, by Graham, Rothschild, and Spencer


I'm all for jumping right to current research, and I strongly advise you to have ongoing discussions with somebody, or some seminar. But the material in these six books constitutes a breathtaking panorama of the subject. 
A: Video lectures are the next best thing to seminars and discussions with others. Have a look at the videos of the workshop "Introduction to Ergodic Theory and Additive Combinatorics" from Fall 2008 at the MSRI.
