Hyperplane Incidences and Distance Sets in Higher Dimensions

Abstract

We generalize Ren and Wang's incidence bound between points and lines in ${\mathbb{R}}^2$ \cite{RenWan23} to higher dimensions. We show how to use this incidence bound to improve the best known bound for Falconer's distance set problem in ${\mathbb{R}}^3$ and in ${\mathbb{R}}^4$. We show that if $d=3$ or $d=4$, and $E\subset {\mathbb{R}}^d$ is a Borel set of dimension ${\dim }_H(E)>d/2$, then \begin{equation*} \sup_{x\in E} \dim_H(Δ_x(E)) \geq 2/3, \end{equation*} where ${\Delta }_x(E)$ is the pinned distance set of $E$ with respect to $x$. We also show how the incidence bound can be used to generalize the planar Furstenberg set bound, to sets in ${\mathbb{R}}^d$ that contain a $t$-dimensional set of hyperplanes, each of which contains an $s$-dimensional set of points, for any $d\ge 2$, $s\in (d-2,d-1]$ and $t\in (0,d]$.