The APDE seminar on Monday, 3/10, will be given by Pablo Shmerkin (University of Brisith Columbia)in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Anuj Kumar (anujkumar@berkeley.edu).

Title: Around the sum-product problem for fractals

Abstract: A fundamental result of Bourgain asserts, in its simplest form, that if $A$ is a Borel subset of the real line with Hausdorff dimension $s\in (0,1)$, then $\dim (A+A.A)\ge s+c(s)$ for some small constant $c(s)>0$. Note that this implies, in particular, that there exist no Borel subrings of the reals of intermediate dimension. I will present some recent variants and improvements on this result, including joint work (across several papers) with N. de Saxcé, T. Orponen and H. Wang.