Falconer's distance set conjecture

Let $d\geq 2$, let $E\subset \mathbb{R}^d$ be a Borel set, and write

$$\Delta(E)=\{\lvert x-y\rvert:x,y\in E\}.$$

Falconer's distance set conjecture. If $\dim_{\mathcal{H}}E>\frac{d}{2}$, then

$$\mathcal{L}^1(\Delta(E))>0.$$

This conjecture asks for the sharp Hausdorff-dimension threshold guaranteeing that the distance set has positive Lebesgue measure. Falconer proved the weaker threshold $\dim_{\mathcal{H}}E>\frac{d+1}{2}$; the conjecture remains open in all dimensions $d\geq 2$, and the exponent $\frac{d}{2}$ is known to be sharp.