Sets with small doubling in R^k and Z^k

In this talk we explore structural results about sets with small doubling in k dimensions. We start in the continuous world with a sharp stability result for the Brunn-Minkowski inequality conjectured by Figalli and Jerison and work our way to the discrete world, where we discuss the natural extension: we show that non-degenerate sets in Z^k with doubling close to 2^k are close to convex progressions i.e. convex sets intersected with a sub-lattice. This talk is based on joint work with Peter van Hintum and Hunter Spink.