On the dimension and smoothness of radial projections

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.