\ge 2$ and $dim_{{\mathcal H}}(E)>\frac{d+1}{2}$, then ${\mathcal L}(\Delta_x(E))>0$ for every $x \in E$ except for a set of small Hausdorff dimension. Improvements on the size of the exceptional set were obtained by the second listed author and Liu in [@ILiu17].

Recently, in [@Liu18], Liu showed that if $dim_{{\mathcal H}}(E)>\frac{d}{2}+\frac{1}{3}$, then ${\mathcal L}(\Delta_x(E)) > 0$ for every $x \in E$ except those in a set of small Hausdorff dimension. Using Liu’s method, we are also able to bound the size of pinned distance sets.

\[main\] If $E\subset\mathbb{R}^2$ is a compact set with Hausdorff dimension larger than $\frac54$, then there is a point $x\in E$ such that its pinned distance set $\Delta_x(E)$ has positive Lebesgue measure.

Other norms
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The Falconer problem has also been studied for other norms. Suppose that $K$ is a symmetric convex body in ${\mathbb{R}}^d$ and $\| \cdot \|_K$ is the norm with unit ball $K$. We let $\Delta_{K}(E)$ be the set of distances $ \| x-y \|_{K}$ with $x,y \in E$ and we let $\Delta_{K,x}(E)$ be the set of distances $\| x - y \|_K$ with $y \in E$. If $K$ is the cube $[-1,1]^d$, then $\| \cdot \|_K$ is the $l^\infty$ norm, and it is not 