Tainara Gobetti Borges, University of Pennsylvania
Event Date
2026-03-16
Event Time
02:30 pm ~ 03:30 pm
Event Location
Virtual Talk
Given a compact set $E$ in $\mathbb{R}^d$ and a point $y \in E$, define the pinned distance set at $y$ as $\Delta^y(E)=\{|x-y|\colon x\in E\}.$ Peres and Schlag proved that if the Hausdorff dimension of $E$ is larger than $(d+2)/2$, then there exists $y \in E$ such that $\Delta^y(E)$ has nonempty interior. However, such a threshold is only useful for dimensions at least 3 and does not give information when $d=2$. In this talk, we will discuss how local smoothing estimates for the wave equation can be used to get a nontrivial threshold for such a problem in the plane and to improve the known threshold in $d=3$. We proved that 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. This talk is based on joint work with B. Foster, Y. Ou, and E. Palsson.