This result was striking because in previous work on the problem, there was no evidence that the Ahlfors-David case would be any easier than the general case. This approach was further developed by Keleti and Shmerkin [@KS18]. They proved very strong estimates for sets that are even roughly like Ahlfors-David regular sets. They also proved results about the Hausdorff dimension of $\Delta (E)$. For instance, if $E$ is a compact set with Hausdorff dimension strictly greater than $1$, then they proved that the Hausdorff dimension of $\Delta(E)$ is at least $.685...$. Bourgain ([@B03]) had proven that if $E$ has Hausdorff dimension at least 1, then $\Delta(E)$ has Hausdorff dimension at least $1/2 + \delta$ for some $\delta > 0$. The value of $\delta$ could be made explicit but it would be very small, and so the $.685...$ is quite striking. We will use one of the key ideas of [@O17] and [@KS18] in the proof of Theorem \[main1\].

There is a variant of the Falconer Distance Problem involving pinned distance sets. For any point $x$, the pinned distance set $\Delta_x(E)$ is defined by

$$\Delta_x(E)=\{|x-y|: y \in E\} .$$

Peres and Schlag ([@PS00]) proved that if $E \subset {\Bbb R}^d$, $d 