Exceptional sets for length under restricted families of projections onto lines in $\mathbb{R}^3$

TL;DR

The paper proves that for Borel sets in three-dimensional space with dimension greater than one, the projections onto certain lines have positive length for almost all directions, with exceptions forming a set of small Hausdorff dimension.

Contribution

It establishes a new bound on the Hausdorff dimension of the exceptional set of directions where the projected length may be zero, under a specific family of projections.

Findings

Projections of sets with dimension > 1 have positive length in most directions.

The exceptional set of directions has Hausdorff dimension at most (3 - dim A)/2.

The result applies to projections onto lines spanned by vectors of a specific form.

Abstract

It is shown that if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff dimension $\dim A>1$, and if $\rho_{\theta}$ is orthogonal projection to the line spanned by $( \cos \theta, \sin \theta, 1 )$, then $\rho_{\theta}(A)$ has positive length for all $\theta$ outside a set of Hausdorff dimension at most $\frac{3-\dim A}{2}$.