A Marstrand-type restricted projection theorem in ${\mathbb{R}}^3$

Marstrand’s projection theorem from $1954$ states that if $K \subset $\mathbb{R}$^{3}$isananalyticset,then,for$$\mathcal{H}$^{2}$almostevery$e \in S^{2}$,theorthogonalprojection$\pi_{e}(K)$of$K$ to the line spanned by $e$ has Hausdorff dimension $\min \{{\dim }_{\mathrm{H}}K,1\}$. This paper contains the following sharper version of Marstrand’s theorem. Let $V \subset $\mathbb{R}$^{3}$beany$2$-plane,whichisnotasubspace\mathrm{.}Then,for$$\mathcal{H}$^{1}$almostevery$e \in S^{2} \cap V$,theprojection$\pi_{e}(K)$hasHausdorffdimension$\min{\dim_{$\mathrm{H}$} K,1}$\mathrm{.}For$0 \leq t < \dim_{$\mathrm{H}$} K$, we also prove an upper bound for the Hausdorff dimension of those vectors $e\in S^2\cap V$ with $\dim_{$\mathrm{H}$} \rho_{e}(K) \leq t < \dim_{$\mathrm{H}$} K$.