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 $\dim {\pi }_{\theta }(A)\ge \min \left\{2,\dim A,\frac{\dim A}{2}+\frac{3}{4}\right\}$ 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.

arXiv comment30 pages. One figure from arXiv:1911.00615 is reused. V2: two references added + minor corrections. V3: Minor corrections, accepted version

References in corpus (6)

• On restricted families of projections in R^3
• Application of a Fourier restriction theorem to certain families of projections in R^3
• A Marstrand-type restricted projection theorem in ${\mathbb{R}}^3$
• Hausdorff dimension of pinned distance sets and the $L^2$-method
• Improved bounds for restricted projection families via weighted Fourier restriction
• On the Dimension of Kakeya Sets in the First Heisenberg Group