Additive Combinatorics ( Winter 2005 )

Additive Combinatorics : Winter 2007

Finite field models in additive combinatories

Combinatorics of distance doubling maps ∗

We study the combinatorics of distance doubling maps on the circle R/Z with prototypes h(β) = 2β mod 1 and h(β) = −2β mod 1, representing the orientation preserving and orientation reversing case, respectively. In particular, we identify parts of the circle where iterates f◦n of a distance doubling map f provide ‘distance doubling behavior’. The results include well-known statements for h relat...

Additive Combinatorics and Theoretical Computer Science ∗ Luca Trevisan

Additive combinatorics is the branch of combinatorics where the objects of study are subsets of the integers or of other abelian groups, and one is interested in properties and patterns that can be expressed in terms of linear equations. More generally, arithmetic combinatorics deals with properties and patterns that can be expressed via additions and multiplications. In the past ten years, add...

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...