Falconer distance set problem using Fourier analysis

Given a set E of Hausdorff dimension s>d/2 in ℝd , Falconer conjectured that its distance set Δ(E)={|x−y|:x,y∈E} should have positive Lebesgue measure. When d is even, we show that dimHE>d/2+1/4 implies |Δ(E)|>0. This improves upon the work of Wolff, Erdogan, Du-Zhang, etc. Our tools include Orponen's radial projection theorem and refined decoupling estimates.

This is joint work with Guth, Iosevich, and Ou and with Du, Iosevich, Ou, and Zhang.