## 507- Plünnecke inequalities and sumset estimates

George Petridis, a student of Gowers, has found very nice new arguments for the Plünnecke-Ruzsa sumset inequality (If $A$ is a finite subset of an Abelian group $G$, and ${}|A+A|\le C|A|$, then ${}|kA-lA|\le C^{k+l}|A|$ for any $k,l\in{\mathbb N}$) and for the Plünnecke graph inequalities.

In lecture we went through the nice standard argument. But the new proofs are significantly simpler. For example, the graph inequalities are no longer needed for the sumset ones, and Menger’s theorem is no longer needed fro the graph inequalities. Gowers has posted the nice proof in his blog, with links to Petridis’s papers on the ArXiv.