On the dimension and smoothness of radial projections

TL;DR

This paper investigates the dimension and smoothness properties of radial projections of sets and measures in Euclidean spaces, establishing conditions under which these projections have positive dimension and are absolutely continuous.

Contribution

It provides new results characterizing when radial projections have positive dimension and are smooth, including sharp bounds on exceptional sets for measures with finite energy.

Findings

Radial projections of sets with positive dimension are large unless on a line or zero-dimensional.

For measures with finite s-energy, radial projections are absolutely continuous outside a small exceptional set.

The dimension bounds on the exceptional set are proven to be sharp.

Abstract

This paper contains two results on the dimension and smoothness of radial projections of sets and measures in Euclidean spaces. To introduce the first one, assume that $E,K \subset \mathbb{R}^{2}$ are non-empty Borel sets with $\dim_{\mathrm{H}} K > 0$. Does the radial projection of $K$ to some point in $E$ have positive dimension? Not necessarily: $E$ can be zero-dimensional, or $E$ and $K$ can lie on a common line. I prove that these are the only obstructions: if $\dim_{\mathrm{H}} E > 0$, and $E$ does not lie on a line, then there exists a point in $x \in E$ such that the radial projection $\pi_{x}(K)$ has Hausdorff dimension at least $(\dim_{\mathrm{H}} K)/2$. Applying the result with $E = K$ gives the following corollary: if $K \subset \mathbb{R}^{2}$ is Borel set, which does not lie on a line, then the set of directions spanned by $K$ has Hausdorff dimension at least $(\dim_{\mathrm{H}} K)/2$. For the second result, let $d \geq 2$ and $d - 1 < s < d$. Let $\mu$ be a compactly supported Radon measure in $\mathbb{R}^{d}$ with finite $s$-energy. I prove that the radial projections of $\mu$ are absolutely continuous with respect to $\mathcal{H}^{d - 1}$ for every centre in $\mathbb{R}^{d} \setminus \operatorname{spt} \mu$, outside an exceptional set of dimension at most $2(d - 1) - s$. In fact, for $x$ outside an exceptional set as above, the proof shows that $\pi_{x\sharp}\mu \in L^{p}(S^{d - 1})$ for some $p > 1$. The dimension bound on the exceptional set is sharp.