TITLE: A learning roadmap for Additive combinatorics.
QUESTION [11 upvotes]: 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

REPLY [9 votes]: 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

REPLY [4 votes]: 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.