Hausdorff dimension estimates for restricted families of projections in $\mathbb{R}^3$

Abstract

This paper is concerned with restricted families of projections in $\mathbb{R}^{3}$. Let $K \subset \mathbb{R}^{3}$ be a Borel set with Hausdorff dimension $\dim K = s > 1$. If $\mathcal{G}$ is a smooth and sufficiently well-curved one-dimensional family of two-dimensional subspaces, the main result states that there exists $\sigma(s) > 1$ such that $\dim \pi_{V}(K) \geq \sigma(s)$ for almost all $V \in \mathcal{G}$. A similar result is obtained for some specific families of one-dimensional subspaces.