A polynomial bound in Freiman's theorem

Abstract

In this paper the following improvement on Freiman's theorem on set addition is obtained (see Theorems 1 and 2 in Section 1).

Let $A\subset \mathbb{Z}$ be a finite set such that . Then A is contained in a proper d-dimensional progression P, where $d\le \left[\alpha -1\right]$ and .

Earlier bounds involved exponential dependence in α in the second estimate. Our argument combines I. Ruzsa's method, which we improve in several places, as well as Y. Bilu's proof of Freiman's theorem.