A Polynomial Bound in Freiman’s Theorem

In this paper the following improvement on Freiman’s theorem on set addition is obtained. (Theorem 1 and Theorem 2 in Section 1.) Let A ⊂ Z be a finite set such that |A + A| < α|A|. Then A is contained in a proper d-dimensional progression P , where d ≤ [α− 1] and log |P | |A| < Cα2(log α)3. Earlier bounds involved exponential dependence in α in the second estimate. Our argument combines Ruzsa’s method, which we improve in several places, as well as Bilu’s proof of Freiman’s theorem. A fundamental result in the theory of set addition is Freiman’s theorem. Let A ⊂ Z be a finite set of integers with small sumset, thus assume |A + A| < α|A| (0.1) where A + A = {x + y| x, y ∈ A} (0.2) and | · | denotes the cardinality. The factor α should be thought of as a (possibly large) constant. Then Freiman’s theorem states that A is contained in a d-dimensional progression P , where