The APDE seminar on Monday, 10/23, will be given by Kevin Ren (Princeton) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto (fpasqualotto@berkeley.edu) or Mengxuan Yang (mxyang@math.berkeley.edu).

Title: Pinned Distances in R^d

Abstract: Given a set E in R^d with Hausdorff dimension > d/2, Falconer conjectured that the set of distances between any two points in E has positive Lebesgue measure. This conjecture remains open in all dimensions, despite significant progress in the last 30 years. Building upon this progress, we show that if d >= 3 and dim_H (E) > d/2 + 1/4 – 1/(8d+4), then the distance set of E has positive Lebesgue measure. The proof uses a new radial projection theorem in R^d applied to a variant of a decoupling framework of Guth-Iosevich-Ou-Wang. Joint work with Xiumin Du, Yumeng Ou, and Ruixiang Zhang.