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 $\dim E\le 2(n-1)-\dim B$. This is the sharp bound for all $n\ge 2$. 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.

6 pages

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