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