A sharp exceptional set estimate for visibility

Abstract

A Borel set $B \subset \mathbb{R}^{n}$ is visible from $x \in \mathbb{R}^{n}$, if the radial projection of $B$ with base point $x$ has positive $\mathcal{H}^{n - 1}$ measure. I prove that if $\dim B > n - 1$, then $B$ is visible from every point $x \in \mathbb{R}^{n} \setminus E$, where $E$ is an exceptional set with dimension $\dim E \leq 2(n - 1) - \dim B$. This is the sharp bound for all $n \geq 2$. Many parts of the proof were already contained in a recent previous paper by P. Mattila and the author, where a weaker bound for $\dim E$ was derived as a corollary from a certain slicing theorem. Here, no improvement to the slicing result is obtained; in brief, the main observation of the present paper is that the proof method gives the optimal result, when applied directly to the visibility problem.