Nonempty interior of pinned distance and tree sets

For a compact set $E\subset {\mathbb{R}}^d$, $d\ge 2$, consider the pinned distance set ${\mathrm{\Delta }}^y(E)=\{\mathrm{\mid }x-y\mathrm{\mid }:x\in E\}$. Peres and Schlag showed that if the Hausdorff dimension of $E$ is bigger than $\frac{d+2}{2}$ with $d\ge 3$, then there exists a point $y\in E$ such that ${\mathrm{\Delta }}^y(E)$ has nonempty interior. In this paper we obtain the first non-trivial threshold for this problem in the plane, improving on the Peres–Schlag threshold when $d=3$, and we extend the results to trees using a novel induction argument.