A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in ${\mathbb{R}}^3$

We resolve a conjecture of Fässler and Orponen on the dimension of exceptional projections to one-dimensional subspaces indexed by a space curve in ${\mathbb{R}}^3$. We do this by obtaining sharp $L^p$ bounds for a variant of the Wolff circular maximal function over fractal sets for a class of $C^2$ curves related to Sogge’s cinematic curvature condition. A key new tool is the use of lens cutting techniques from discrete geometry.