Length of sets under restricted families of projections onto lines

Abstract

Let γ : I → S 2 \gamma : I \to S^2 be a C 2 C^2 curve with det ( γ , γ ′ , γ ) \det (\gamma , \gamma ’, \gamma ) nonvanishing, and for each θ ∈ I \theta \in I let ρ θ \rho _{\theta } be orthogonal projection onto the span of γ ( θ ) \gamma (\theta ) . It is shown that if A ⊆ R 3 A \subseteq \mathbb {R}^3 is a Borel set of Hausdorff dimension strictly greater than 1, then ρ θ ( A ) \rho _{\theta }(A) has positive length for a.e.  θ ∈ I \theta \in I . This answers a question raised by Käenmäki, Orponen and Venieri.