Tainara Borges will give a talk on 02/11/2025.

Given a compact set $E$ in the plane and a point $x$ in $E$ define the pinned distance set at $x$ as \(\Delta^{x}(E)=\{|x-y|\colon y\in E\}.\)

In this talk, we will discuss how local smoothing estimates for spherical averages can be used to get the following result: If $E$ is a compact subset of the plane with Hausdorff dimension at least $7/4$, then there exists $x$ in $E$ such that the pinned distance set at $x$ contains an interval. If time allows, we will mention how similar results are true for distance sets associated with general trees with vertices in E. This talk is based on ongoing joint work with B. Foster, Y. Ou, and E. Palsson.