Falconer distance conjecture: let \(E \subset \mathbb{R}^d\) be compact with Hausdorff dimension \(\dim E > d/2\). Then its distance set \(\Delta(E) = \{|x-y| : x, y \in E\}\) has positive Lebesgue measure. (In the plane: dimension \(> 1\) forces a fat distance set; the record stands at \(5/4\).)
Falconer posed it in 1985 as the distance analogue of the Kakeya problem: how large must a set be before its distance set is large? Wolff's ½-bound (1999) stood for two decades; the 2020s brought the first improvements in the plane (record now 5/4, via decoupling and refined Strichartz estimates), while higher dimensions lag further behind. Every advance has required importing fresh harmonic analysis, and the conjectured threshold still looks distant.
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