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 $\dim {\pi }_V(K)\ge \sigma (s)$ for almost all $V\in \mathcal{G}$. A similar result is obtained for some specific families of one-dimensional subspaces.

33 pages, 3 figures. v3: incorporated reviewer comments, to appear in Adv. Math

References in corpus (3)

• On restricted families of projections in R^3
• Application of a Fourier restriction theorem to certain families of projections in R^3
• Constancy results for special families of projections

Cited by in corpus (2)

• On restricted families of projections in R^3
• A Fubini-type theorem for Hausdorff dimension