Softcover ISBN:  9780821843512 
Product Code:  CRMP/43 
335 pp 
List Price:  $112.00 
MAA Member Price:  $100.80 
AMS Member Price:  $89.60 
Electronic ISBN:  9781470439576 
Product Code:  CRMP/43.E 
335 pp 
List Price:  $105.00 
MAA Member Price:  $94.50 
AMS Member Price:  $84.00 

Book DetailsCRM Proceedings & Lecture NotesVolume: 43; 2007MSC: Primary 11; Secondary 05; 42; 28; 37;
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 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.
ReadershipUndergraduates, graduate students, and research mathematicians interested in additive combinatorics.

Table of Contents

Chapters

An introduction to additive combinatorics

Elementary additive combinatorics

Many additive quadruples

An old new proof of Roth’s theorem

Bounds on exponential sums over small multiplicative subgroups

Montréal notes on quadratic Fourier analysis

Ergodic methods in additive combinatorics

The ergodic and combinatorial approaches to Szemerédi’s theorem

Cardinality questions about sumsets

Open problems in additive combinatorics

Some problems related to sumproduct theorems

Lattice points on circles, squares in arithmetic progressions and sumsets of squares

Problems in additive number theory. I

Double and triple sums modulo a prime

Additive properties of product sets in fields of prime order

Many sets have more sums than differences

Davenport’s constant for groups of the form $\mathbb {Z}_3\oplus \mathbb {Z}_3\oplus \mathbb {Z}_{3d}$

Some combinatorial group invariants and their generalizations with weights


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

Chapters

An introduction to additive combinatorics

Elementary additive combinatorics

Many additive quadruples

An old new proof of Roth’s theorem

Bounds on exponential sums over small multiplicative subgroups

Montréal notes on quadratic Fourier analysis

Ergodic methods in additive combinatorics

The ergodic and combinatorial approaches to Szemerédi’s theorem

Cardinality questions about sumsets

Open problems in additive combinatorics

Some problems related to sumproduct theorems

Lattice points on circles, squares in arithmetic progressions and sumsets of squares

Problems in additive number theory. I

Double and triple sums modulo a prime

Additive properties of product sets in fields of prime order

Many sets have more sums than differences

Davenport’s constant for groups of the form $\mathbb {Z}_3\oplus \mathbb {Z}_3\oplus \mathbb {Z}_{3d}$

Some combinatorial group invariants and their generalizations with weights