Length of sets under restricted families of projections onto lines

TL;DR

This paper proves that for a broad class of curved projections in three-dimensional space, sets with Hausdorff dimension greater than one project onto lines with positive length for almost every direction, answering a previously open question.

Contribution

It establishes a new result on the length of projections of sets under restricted families of line projections defined by a smooth curve.

Findings

Sets with Hausdorff dimension > 1 have positive length projections for almost every direction in the family.

The result applies to a class of projections defined by a nonvanishing curvature condition.

It resolves an open question posed by K"aenm"aki, Orponen, and Venieri.

Abstract

Let $\gamma: I \to S^2$ be a $C^2$ curve with $\det(\gamma, \gamma', \gamma'')$ nonvanishing, and for each $\theta \in I$ let $\rho_{\theta}$ be orthogonal projection onto the span of $\gamma(\theta)$. It is shown that if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff dimension strictly greater than 1, then $\rho_{\theta}(A)$ has positive length for a.e. $\theta \in I$. This answers a question raised by K\"aenm\"aki, Orponen and Venieri.