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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

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Just apply e^x to whatever you're studying and then you'll be doing multiplicative combinatorics, the theory of which is already well-known. –  Michael Burge Nov 21 '10 at 2:35
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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.

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Thank you your answer! What do you think would be needed to be able to read the whole book? Probability is given, as I see it, but what more? fourier analysis seems to be there too, but from what I can see it's at a high level? I'm very curious about this since I'd like to do a thesis in the subject. –  czikszentmihalyi Nov 21 '10 at 3:52
    
I would think it mostly needs certain level of mathematical comfort, since the style is very terse. Fourier analysis is used, for example, but in principle you should be able to follow most of what they do with it, if you work on your own the required equalities and inequalities that they use. With a reference on the side to check a few technical facts (a decent book on functional analysis is much more than enough, I would think), you ought to be fine. –  Andres Caicedo Nov 21 '10 at 4:06
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Also, large portions of the book are independent of one another, so you should be able to find chapters that you can follow more easily than others. All have many interesting questions and provide references, so you will have plenty of material to work with during your thesis, even if you do not read certain sections because they may need more technical prerequisites. I would recommend to read a different source for Szemeredi's theorem, though. I think that Tao and Green have many other papers on the subject where you may benefit more from the presentation. –  Andres Caicedo Nov 21 '10 at 4:08
    
Again, if the style ends up being too hard, take a look at some of the other references I mentioned. The goal of this book is to show how many results from different areas can be seen in the light of common themes and techniques, so really there is a lot of interconnection, but there is also quite a bit of independent material. –  Andres Caicedo Nov 21 '10 at 4:10
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Nathanson's books:

  1. The Classical Bases
  2. Inverse Problems and the Geometry of Sumsets
  3. 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:

  1. Sequences, by Halberstam and Roth
  2. The Probabilistic Method, by Alon and Spencer
  3. 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.

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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.

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The video lectures here: cs.princeton.edu/theory/index.php/Main/… are also relevant. –  hypercube Dec 3 '10 at 18:52
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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

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