Local smoothing and pinned nonempty interior for distance sets
by
Endenicher Allee 60, Seminarraum 0.011 (Mathezentrum)
Given a compact set $E$ in $R^d$ and a point $y$ in $E$, define the pinned distance set of $E$ 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 a point $y$ in $E$ such that the pinned distance set of $E$ at $y$ has nonempty interior. However, their threshold is only useful when $d$ is at least $3$ and does not say anything when $d=2$. In this talk, I will show you 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 prove that if $E$ is a compact subset of the plane with Hausdorff dimension at least $7/4$, then there exists $y$ in $E$ such that the pinned distance set at $y$ contains an interval. This talk is based on joint work with B. Foster, Y. Ou, and E. Palsson.