Exceptional sets for length under restricted families of projections onto lines in $\mathbb{R}^3$

Abstract

It is shown that if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff dimension $\dim A>1$, and if $\rho_{\theta}$ is orthogonal projection to the line spanned by $\left( \cos \theta, \sin \theta, 1 \right)$, then $\rho_{\theta}(A)$ has positive length for all $\theta$ outside a set of Hausdorff dimension $\frac{3-\dim A}{2}$.