CRM Proceedings & Lecture Notes 2007; 335 pp; softcover Volume: 43 ISBN10: 0821843516 ISBN13: 9780821843512 List Price: US$105 Member Price: US$84 Order Code: CRMP/43
 One of the most active areas in mathematics today is the rapidly emerging new topic of "additive combinatorics". Building on Gowers' use of the FreimanRuzsa 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 coauthors have given nontrivial 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 copublished with the Centre de Recherches Mathématiques. Readership Undergraduates, graduate students, and research mathematicians interested in additive combinatorics. Table of Contents  A. Granville  An introduction to additive combinatorics
 J. Solymosi  Elementary additive combinatorics
 A. Balog  Many additive quadruples
 E. Szemerédi  An old new proof of Roth's theorem
 P. Kurlberg  Bounds on exponential sums over small multiplicative subgroups
 B. Green  Montréal notes on quadratic Fourier analysis
 B. Kra  Ergodic methods in additive combinatorics
 T. Tao  The ergodic and combinatorial approaches to Szemerédi's theorem
 I. Z. Ruzsa  Cardinality questions about sumsets
 E. S. Croot III and V. F. Lev  Open problems in additive combinatorics
 M.C. Chang  Some problems related to sumproduct theorems
 J. Cilleruelo and A. Granville  Lattice points on circles, squares in arithmetic progressions and sumsets of squares
 M. B. Nathanson  Problems in additive number theory. I
 K. Gyarmati, S. Konyagin, and I. Z. Ruzsa  Double and triple sums modulo a prime
 A. A. Glibichuk and S. V. Konyagin  Additive properties of product sets in fields of prime order
 G. Martin and K. O'Bryant  Many sets have more sums than differences
 G. Bhowmik and J.C. SchlagePuchta  Davenport's constant for groups of the form \(\mathbb{Z}_3\oplus\mathbb{Z}_3\oplus\mathbb{Z}_{3d}\)
 S. D. Adhikari, R. Balasubramanian, and P. Rath  Some combinatorial group invariants and their generalizations with weights
