The APDE seminar on Monday, 4/25, will be given by Hong Wang (UCLA) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: Distance sets spanned by sets of dimension d/2

Abstract: Suppose that E is a subset of ${\mathbb{R}}^d$, its distance set is defined as $\mathrm{\Delta }(E):=\{|x-y|,x,y\in E\}$.  Joint with Pablo Shmerkin, we prove that if the packing dimension and Hausdorff dimension of $E$ both equal to $d/2$, then ${\dim }_H\mathrm{\Delta }(E)=1$. 

We also prove that if ${\dim }_{H}E\ge d/2$, then ${\dim }_H\mathrm{\Delta }(E)\ge d/2+c_d$ when $d=2,3$; and ${\underset{―}{dim}}_B\mathrm{\Delta }(E)\ge d/2+c_d$ when $d>3$  for some explicit constants $c_d$.