Additive Combinatorics ( Winter 2010 )
نویسنده
چکیده
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 like commutativity, associativity and the distributive law A + (B ∪ C) = (A + B) ∪ (A + C). Prove that k ¦ A ⊆ kA and classify when they are equal. Prove that |b + A| = |A|. Show that |A| ≤ |A+B| ≤ |A||B|. Describe the situations when we get equality. Improve this last upper bound for |A+A| and for |A−A|.
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