Length of sets under restricted families of projections onto lines

Abstract

Let $\gamma: I \to S^2$ be a $C^2$ curve with $\det(\gamma, \gamma', \gamma'')$ nonvanishing, and for each $\theta \in I$ let $\rho_{\theta}$ be orthogonal projection onto the span of $\gamma(\theta)$. It is shown that if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff dimension strictly greater than 1, then $\rho_{\theta}(A)$ has positive length for a.e. $\theta \in I$. This answers a question raised by K\"aenm\"aki, Orponen and Venieri.