Chao Lin, Department of Pure Math, University of Waterloo
“Freiman’s Theorem - Part II”
For a set of numbers A, let the sum-set A + A denote {a1 + a2 : a1, a2 ∈ A}. Freiman’s theorem proves the remarkable notion that if a finite subset of integers A has a relatively small sum-set A + A, then A essentially resembles an arithmetic progression. More precisely, if |A + A| ≤ C|A| then there exists constants d, S depending only on C such that A is contained in a generalized arithmetic progression of dimension d and size ≤ S|A|. This is the first of two talks presenting the proof of Freiman’s theorem, which uses ideas ranging from graph theory, discrete Fourier analysis, and Minkowski’s geometry of numbers. For a set of numbers A, let the sum-set A + A denote {a1 + a2 : a1,a2 ∈ A}. Freiman’s theorem proves the remarkable notion that if a finite subset of integers A has a relatively small sum-set A + A, then A essentially resembles an arithmetic progression. More precisely, if |A+A| ≤ C|A| then there exists constants d,S depending only on C such that A is contained in a generalized arithmetic progression of dimension d and size ≤ S|A|. This is the first of two talks presenting the proof of Freiman’s theorem, which uses ideas ranging from graph theory, discrete Fourier analysis, and Minkowski’s geometry of numbers.