Restricted families of projections onto planes: The general case of nonvanishing geodesic curvature

Abstract

It is shown that if $\gamma :[a,b]\to S^2$ is $C^3$ with $\det (\gamma ,{\gamma }^{\prime },{\gamma }^{\prime \prime })\ne 0$, and if $A\subseteq {\mathbb{R}}^3$ is a Borel set, then $\dim {\pi }_{\theta }(A)\ge \min \{2,\dim A,\dim A/2+3/4\}$ for a.e. $\theta \in [a,b]$, where ${\pi }_{\theta }$ denotes projection onto the orthogonal complement of $\gamma (\theta )$ and “dim” refers to Hausdorff dimension. This partially resolves a conjecture of Fässler and Orponen in the range $1<\dim A\le 3/2$, which was previously known only for non-great circles. For $3/2<\dim A<5/2$, this improves the known lower bound for this problem.