Book Review: Additive combinatorics

Additive Combinatorics ( Winter 2010 )

For A,B subsets of an additive group Z, we define A + B to be the sumset {a + b : a ∈ A, b ∈ B}, and kA to be the k-fold sum A + A + · · · + A of A. We also let A−B = {a−b : a ∈ A, b ∈ B} and b+A = {b}+A for a single element set {b}, a translate of A. Note that A − A is not 0 unless |A| = 1. We let k ¦ A = {ka : a ∈ A}, a dilate of A. There are many obvious properties of “+” that can be checked...

Additive Combinatorics : Winter 2007

Additive Combinatorics ( Winter 2005 )

For A,B subsets of an additive group Z, we define A + B to be the sumset {a + b : a ∈ A, b ∈ B}, and kA to be the k-fold sum A + A + · · · + A of A. We also let A−B = {a−b : a ∈ A, b ∈ B} and b+A = {b}+A for a single element set {b}, a translate of A. Note that A − A is not 0 unless |A| = 1. We let k ⋄ A = {ka : a ∈ A}, a dilate of A. There are many obvious properties of “+” that can be checked...

Ergodic Methods in Additive Combinatorics

Shortly after Szemerédi’s proof that a set of positive upper density contains arbitrarily long arithmetic progressions, Furstenberg gave a new proof of this theorem using ergodic theory. This gave rise to the field of combinatorial ergodic theory, in which problems motivated by additive combinatorics are addressed kwith ergodic theory. Combinatorial ergodic theory has since produced combinatori...

An Introduction to Additive Combinatorics

This is a slightly expanded write-up of my three lectures at the Additive Combinatorics school. In the first lecture we introduce some of the basic material in Additive Combinatorics, and in the next two lectures we prove two of the key background results, the Freiman-Ruzsa theorem and Roth’s theorem for 3-term arithmetic progressions. Lecture I: Introductory material 1. Basic Definitions. 2. I...