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

TL;DR

This paper establishes new lower bounds on the Hausdorff dimension of projections of sets in three-dimensional space onto planes orthogonal to curves with nonvanishing geodesic curvature, partially resolving a conjecture and improving previous bounds.

Contribution

It proves dimension estimates for projections onto planes orthogonal to curves with nonvanishing geodesic curvature, extending results to a broader class of curves and dimensions.

Findings

Dimension bounds for projections onto orthogonal planes

Partial resolution of F"assler and Orponen's conjecture

Improved lower bounds for sets with dimension between 1 and 2.5

Abstract

It is shown that if $\gamma: [a,b] \to S^2$ is $C^3$ with $\det(\gamma, \gamma', \gamma'') \neq 0$, and if $A \subseteq \mathbb{R}^3$ is a Borel set, then $\dim \pi_{\theta} (A) \geq \min\left\{ 2,\dim A, \frac{ \dim A}{2} + \frac{3}{4} \right\}$ 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 F\"assler and Orponen in the range $1< \dim A \leq 3/2$, 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.