Mathematics Colloquium

The Falconer distance set conjecture states that if $E$ is a compact set in $\mathbb{R}^d$ with Hausdorff dimension larger than $d/2$, then its distance set, consisting of all distinct distances generated by points in $E$, should have strictly positive Lebesgue measure. This conjecture is the natural continuous analogue of the Erd\"os distinct distance conjecture, and remains open in all dimensions higher than or equal to two. In this talk we will discuss some recent partial progress towards it, which are consequences of a combination of new ideas from Fourier analysis, geometric measure theory and combinators. The talk will be based on joint works with Xiumin Du, Larry Guth, Alex Iosevich, Hong Wang, Bobby Wilson, and Ruixiang Zhang.