**CRM Proceedings & Lecture Notes**

Volume: 43;
2007;
335 pp;
Softcover

MSC: Primary 11;
Secondary 05; 42; 28; 37

Print ISBN: 978-0-8218-4351-2

Product Code: CRMP/43

List Price: $112.00

AMS Member Price: $89.60

MAA Member Price: $100.80

**Electronic ISBN: 978-1-4704-3957-6
Product Code: CRMP/43.E**

List Price: $105.00

AMS Member Price: $84.00

MAA Member Price: $94.50

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# Additive Combinatorics

Share this page *Edited by *
*Andrew Granville; Melvyn B. Nathanson; József Solymosi*

A co-publication of the AMS and Centre de Recherches Mathématiques

One of the most active areas in mathematics today is the rapidly emerging new topic of “additive combinatorics”. Building on Gowers' use of the Freiman–Ruzsa theorem in harmonic analysis (in particular, his proof of Szemerédi's theorem), Green and Tao famously proved that there are arbitrarily long arithmetic progressions of primes, and Bourgain and his co-authors have given non-trivial estimates for hitherto untouchably short exponential sums. There are further important consequences in group theory and in complexity theory and compelling questions in ergodic theory, discrete geometry and many other disciplines. The basis of the subject is not too difficult: it can be best described as the theory of adding together sets of numbers; in particular, understanding the structure of the two original sets if their sum is small. This book brings together key researchers from all of these different areas, sharing their insights in articles meant to inspire mathematicians coming from all sorts of different backgrounds.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Readership

Undergraduates, graduate students, and research mathematicians interested in additive combinatorics.

# Table of Contents

## Additive Combinatorics

- Cover Cover11
- Title page i2
- Contents iii4
- Preface v6
- An introduction to additive combinatorics 110
- Elementary additive combinatorics 2938
- Many additive quadruples 3948
- An old new proof of Roth’s theorem 5160
- Bounds on exponential sums over small multiplicative subgroups 5564
- Montréal notes on quadratic Fourier analysis 6978
- Ergodic methods in additive combinatorics 103112
- The ergodic and combinatorial approaches to Szemerédi’s theorem 145154
- Cardinality questions about sumsets 195204
- Open problems in additive combinatorics 207216
- Some problems related to sum-product theorems 235244
- Lattice points on circles, squares in arithmetic progressions and sumsets of squares 241250
- Problems in additive number theory. I 263272
- Double and triple sums modulo a prime 271280
- Additive properties of product sets in fields of prime order 279288
- Many sets have more sums than differences 287296
- Davenport’s constant for groups of the form ℤ₃⊕ℤ₃⊕ℤ_{3𝕕} 307316
- Some combinatorial group invariants and their generalizations with weights 327336
- Back Cover Back Cover1345