Near optimal bounds in Freiman's theorem

Abstract

We prove that if for a finite set $A$ of integers we have $|A+A|\le K\left|A\right|$, then $A$ is contained in a generalized arithmetic progression of dimension at most $K^{1+C(\log K{)}^{-1/2}}$ and of size at most $\exp (K^{1+C(\log K{)}^{-1/2}})\left|A\right|$ for some absolute constant $C$. We also discuss a number of applications of this result.