ABSTRACT

Falconer distance conjecture is one of the most famous open problems in harmonic analysis and geometric measure theory. It states that if a Euclidean subset in $\mathbb{R}^d$ has Hausdorff dimension greater than d/2, then its distance set must have positive Lebesgue measure. I will introduce the history of this conjecture, some classical results, and sketch a recent proof of this conjecture under an extra assumption that the given set has equal Hausdorff and packing dimensions.