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 Szemeredi'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.
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ISBN-13
978-0-8218-4351-2 (9780821843512)
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Schweitzer Klassifikation
An introduction to additive combinatorics by A. Granville Elementary additive combinatorics by J. Solymosi Many additive quadruples by A. Balog An old new proof of Roth's theorem by E. Szemeredi Bounds on exponential sums over small multiplicative subgroups by P. Kurlberg Montreal notes on quadratic Fourier analysis by B. Green Ergodic methods in additive combinatorics by B. Kra The ergodic and combinatorial approaches to Szemeredi's theorem by T. Tao Cardinality questions about sumsets by I. Z. Ruzsa Open problems in additive combinatorics by E. S. Croot III and V. F. Lev Some problems related to sum-product theorems by M.-C. Chang Lattice points on circles, squares in arithmetic progressions and sumsets of squares by J. Cilleruelo and A. Granville Problems in additive number theory. I by M. B. Nathanson Double and triple sums modulo a prime by K. Gyarmati, S. Konyagin, and I. Z. Ruzsa Additive properties of product sets in fields of prime order by A. A. Glibichuk and S. V. Konyagin Many sets have more sums than differences by G. Martin and K. O'Bryant Devenport's constant for groups of the form $\mathbb{Z}_3\oplus\mathbb{Z}_3\oplus\mathbb{Z}_{3d}$ by G. Browmik and J.-C. Schlage-Puchta Some combinatorial group invariants and their generalizations with weights by S. D. Adhikari, R. Balasubramanian, and P. Rath.
