Abstract

Falconer's distance set conjecture states that if a compact subset E of R^d (d > 1) has Hausdorff dimension greater than d/2 then its distance set,

D(E):={|x-y|:x,y Î E},

has positive Lebesgue measure.

In this lecture, we will discuss the recent progress in this conjecture and closely related Fourier extension estimates relative to fractal measures.