Leonardo Franchi

Product-free subsets of (0,1) and a strengthened Brunn–Minkowski inequality.

The third problem in Ben Green’s collection of 100 open problems asks whether an open subset of (0,1) containing no x,y,z satisfying xy=z must have measure at most 1/3. In joint work with W. Timothy Gowers and Fredy Yip, we proved this result.
In this talk, I will discuss some of the motivations behind the problem, as well as some of the new tools developed to prove a strengthened version of the one dimensional Brunn–Minkowski inequality that plays a key role in the argument.